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a/docs/_sources/content/bvps/boundary-value-problems.md b/docs/_sources/content/bvps/boundary-value-problems.md new file mode 100644 index 0000000..9fa1ca3 --- /dev/null +++ b/docs/_sources/content/bvps/boundary-value-problems.md @@ -0,0 +1,3 @@ +# Boundary Value Problems + +This chapter focuses on methods for solving 2nd-order ODEs constrained by boundary conditions: boundary-value problems (BVPs). diff --git a/docs/_sources/content/bvps/eigenvalue.ipynb b/docs/_sources/content/bvps/eigenvalue.ipynb new file mode 100644 index 0000000..61bd397 --- /dev/null +++ b/docs/_sources/content/bvps/eigenvalue.ipynb @@ -0,0 +1,596 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Eigenvalue problems\n", + "\n", + "\"Eigenvalue\" means characteristic value. These types of problems show up in many areas involving boundary-value problems, where we may not be able to obtain an analytical solution, but we can identify certain characteristic values that tell us important information about the system: the eigenvalues.\n", + "\n", + "## Example: beam buckling\n", + "\n", + "Let's consider deflection in a simply supported (static) vertical beam: $y(x)$, with boundary conditions $y(0) = 0$ and $y(L) = 0$. To get the governing equation, start with considering the sum of moments around the upper pin:\n", + "\\begin{align}\n", + "\\sum M &= M_z + P y = 0 \\\\\n", + "M_z &= -P y\n", + "\\end{align}\n", + "\n", + "We also know that $M_z = E I y''$, so we can obtain\n", + "\\begin{align}\n", + "M_z = E I \\frac{d^2 y}{dx^2} &= -P y \\\\\n", + "y'' + \\frac{P}{EI} y &= 0\n", + "\\end{align}\n", + "This equation governs the stability of a beam, considering small deflections.\n", + "To simplify things, let's define $\\lambda^2 = \\frac{P}{EI}$, which gives us the ODE\n", + "\\begin{equation}\n", + "y'' + \\lambda^2 y = 0\n", + "\\end{equation}\n", + "We can get the general solution to this:\n", + "\\begin{equation}\n", + "y(x) = A \\cos (\\lambda x) + B \\sin (\\lambda x)\n", + "\\end{equation}\n", + "\n", + "To find the coefficients, let's apply the boundary conditions, starting with $x=0$:\n", + "\\begin{align}\n", + "y(x=0) &= 0 = A \\cos 0 + B \\sin 0 \\\\\n", + "\\rightarrow A &= 0 \\\\\n", + "y(x=L) &= 0 = B \\sin (\\lambda L)\n", + "\\end{align}\n", + "Now what? $B \\neq 0$, because otherwise we would have the trivial solution $y(x) = 0$. Instead, to satisfy the boundary condition, we need\n", + "\\begin{align}\n", + "B \\neq 0 \\rightarrow \\sin (\\lambda L) &= 0 \\\\\n", + "\\text{so} \\quad \\lambda L &= n \\pi \\quad n = 1, 2, 3, \\ldots, \\infty \\\\\n", + "\\lambda &= \\frac{n \\pi}{L} \\quad n = 1, 2, 3, \\ldots, \\infty\n", + "\\end{align}\n", + "$\\lambda$ give the the **eigenvalues** for this problem; as you can see, there are an infinite number, that correspond to **eigenfunctions**:\n", + "\\begin{equation}\n", + "y_n = B \\sin \\left( \\frac{n \\pi x}{L} \\right) \\quad n = 1, 2, 3, \\ldots, \\infty\n", + "\\end{equation}\n", + "\n", + "The eigenvalues and associated eigenfunctions physically represent different modes of deflection.\n", + "For example, consider the first three modes (corresponding to $n = 1, 2, 3$):" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear all; clc\n", + "\n", + "L = 1.0;\n", + "x = linspace(0, L);\n", + "subplot(1,3,1);\n", + "y = sin(pi * x / L);\n", + "plot(y, x); title('n = 1')\n", + "subplot(1,3,2);\n", + "y = sin(2 * pi * x / L);\n", + "plot(y, x); title('n = 2')\n", + "subplot(1,3,3);\n", + "y = sin(3* pi * x / L);\n", + "plot(y, x); title('n = 3')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we see different modes of how the beam will buckle. How do we know when this happens?\n", + "\n", + "Recall that the eigenvalue is connected to the physical properties of the beam:\n", + "\\begin{gather}\n", + "\\lambda^2 = \\frac{P}{EI} \\rightarrow \\lambda = \\sqrt{\\frac{P}{EI}} = \\frac{n \\pi}{L} \\\\\n", + "P = \\frac{EI}{L} n^2 \\pi^2\n", + "\\end{gather}\n", + "This means that when the combination of load force and beam properties match certain values, the beam will deflect—and buckle—in one of the modes corresponding to the associated eigenfunction.\n", + "\n", + "In particular, the first mode ($n=1$) is interesting, because this is the first one that will be encountered if a load starts at zero and increases. This is the **Euler critical load** of buckling, $P_{cr}$:\n", + "\\begin{gather}\n", + "\\lambda_1 = \\frac{\\pi}{L} \\rightarrow \\lambda_1^2 = \\frac{P}{EI} = \\frac{\\pi^2}{L^2} \\\\\n", + "P_{cr} = \\frac{\\pi^2 E I}{L^2}\n", + "\\end{gather}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: beam buckling with different boundary conditions\n", + "\n", + "Let's consider a slightly different case, where at $x=0$ the beam is supported such that $y'(0) = 0$. How does the beam buckle in this case?\n", + "\n", + "The governing equation and general solution are the same:\n", + "\\begin{align}\n", + "y'' + \\lambda^2 y &= 0 \\\\\n", + "y(x) &= A \\cos (\\lambda x) + B \\sin (\\lambda x)\n", + "\\end{align}\n", + "but our boundary conditions are now different:\n", + "\\begin{align}\n", + "y'(0) = 0 = -\\lambda A \\sin(0) + \\lambda B\\cos(0) \\\\\n", + "\\rightarrow B &= 0 \\\\\n", + "y &= A \\cos (\\lambda x) \\\\\n", + "y(L) &= 0 = A \\cos (\\lambda L) \\\\\n", + "A \\neq 0 \\rightarrow \\cos(\\lambda L) &= 0 \\\\\n", + "\\text{so} \\quad \\lambda L &= \\frac{(2n-1) \\pi}{2} \\quad n = 1,2,3,\\ldots, \\infty \\\\\n", + "\\lambda &= \\frac{(2n-1) \\pi}{2 L} \\quad n = 1,2,3,\\ldots, \\infty\n", + "\\end{align}\n", + "\n", + "Then, the critical buckling load, again corresponding to $n=1$, is\n", + "\\begin{equation}\n", + "P_{cr} = \\frac{\\pi^2 EI}{4 L^2}\n", + "\\end{equation}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Getting eigenvalues numerically\n", + "\n", + "We can only get the eigenvalues analytically if we can obtain an analytical solution of the ODE, but we might want to get eigenvalues for more complex problems too. In that case, we can use an approach based on *finite differences* to find the eigenvalues.\n", + "\n", + "Consider the same problem as above, for deflection of a simply supported beam:\n", + "\\begin{equation}\n", + "y'' + \\lambda^2 y = 0 \n", + "\\end{equation}\n", + "with boundary conditions $y(0) = 0$ and $y(L) = 0$. Let's represent this using finite differences, for a case where $L=3$ and $\\Delta x = 1$, so we have four points in our solution grid.\n", + "\n", + "The finite difference representation of the ODE is:\n", + "\\begin{align}\n", + "\\frac{y_{i-1} - 2y_i + y_{i+1}}{\\Delta x^2} + \\lambda^2 y_i &= 0 \\\\\n", + "y_{i-1} + \\left( \\lambda^2 \\Delta x^2 - 2 \\right) y_i + y_{i+1} &= 0\n", + "\\end{align}\n", + "However, in this case, we are not solving for the values of deflection ($y_i$), but instead the **eigenvalues** $\\lambda$.\n", + "\n", + "Then, we can write the system of equations using the above recursion formula and our two boundary conditions:\n", + "\\begin{align}\n", + "y_1 &= 0 \\\\\n", + "y_1 + y_2 \\left( \\lambda^2 \\Delta x^2 - 2 \\right) + y_3 &= 0 \\\\\n", + "y_2 + y_3 \\left( \\lambda^2 \\Delta x^2 - 2 \\right) + y_4 &= 0 \\\\\n", + "y_4 &= 0\n", + "\\end{align}\n", + "which we can simplify down to two equations by incorporating the boundary conditions into the equations for the two middle points, and also letting $k = \\lambda^2 \\Delta x^2$:\n", + "\\begin{align}\n", + "y_2 (k-2) + y_3 &= 0 \\\\\n", + "y_2 + y_3 (k-2) &= 0\n", + "\\end{align}\n", + "Let's modify this once more by multiplying both equations by $-1$:\n", + "\\begin{align}\n", + "y_2 (2-k) - y_3 &= 0 \\\\\n", + "-y_2 + y_3 (2-k) &= 0\n", + "\\end{align}\n", + "\n", + "Now we can represent this system of equations as a matrix equation $A \\mathbf{y} = \\mathbf{b} = \\mathbf{0}$:\n", + "\\begin{equation}\n", + "\\begin{bmatrix} 2-k & -1 \\\\ -1 & 2-k \\end{bmatrix}\n", + "\\begin{bmatrix} y_2 \\\\ y_3 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}\n", + "\\end{equation}\n", + "$\\mathbf{y} = \\mathbf{0}$ is a trivial solution to this, so instead $\\det(A) = 0$ satisfies this equation.\n", + "For our $2\\times 2$ matrix, that looks like:\n", + "\\begin{align}\n", + "\\det(A) = \\begin{vmatrix} 2-k & -1 \\\\ -1 & 2-k \\end{vmatrix} = (2-k)^2 - 1 &= 0 \\\\\n", + "k^2 - 4k + 3 &= 0 \\\\\n", + "(k-3)(k-1) &= 0\n", + "\\end{align}\n", + "so the roots of this equation are $k_1 = 1$ and $k_2 = 3$. Recall that $k$ is directly related to the eigenvalue: $k = \\lambda^2 \\Delta x^2$, and $\\Delta x = 1$ for this case, so we can calculate the two associated eigenvalues:\n", + "\\begin{align}\n", + "k_1 &= \\lambda_1^2 \\Delta x^2 = 1 \\rightarrow \\lambda_1 = 1 \\\\\n", + "k_2 &= \\lambda_2^2 \\Delta x^2 = 3 \\rightarrow \\lambda_2 = \\sqrt{3} = 1.732\n", + "\\end{align}\n", + "\n", + "Our work has given us approximations for the first two eigenvalues. We can compare these against the exact values, given in general by $\\lambda = n \\pi / L$ (which we determined above):\n", + "\\begin{align}\n", + "n=1: \\quad \\lambda_1 &= \\frac{\\pi}{L} = \\frac{\\pi}{3} = 1.0472 \\\\\n", + "n=2: \\quad \\lambda_2 &= \\frac{2\\pi}{L} = \\frac{2\\pi}{3} = 2.0944\n", + "\\end{align}\n", + "So, our approximations are close, but with some obvious error. This is because we used a fairly crude step size of $\\Delta x = 1$, dividing the domain into just three segments. By using a finer resolution, we can get more-accurate eigenvalues and also more of them (remember, there are actually an infinite number!). \n", + "\n", + "To do that, we will need to use Matlab, which offers the `eig()` function for calculating eigenvalues---essentially it is finding the roots to the polynomial given by $\\det(A) = 0$. We need to modify this slightly, though, to use the function:\n", + "\\begin{align}\n", + "\\det(A) &= 0 \\\\\n", + "\\det \\left( A^* - k I \\right) = 0\n", + "\\end{align}\n", + "where the new matrix is\n", + "\\begin{equation}\n", + "A^* = \\begin{bmatrix} 2 & -1 \\\\ -1 & 2 \\end{bmatrix}\n", + "\\end{equation}\n", + "Then, `eig(A*)` will provide the values of $k$, which we can use to find the $\\lambda$s:" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "lambda_1: 1.000\n", + "lambda_2: 1.732" + ] + } + ], + "source": [ + "clear all; clc\n", + "\n", + "dx = 1.0;\n", + "L = 3.0;\n", + "\n", + "Astar = [2 -1; -1 2];\n", + "k = eig(Astar);\n", + "\n", + "lambda = sqrt(k) / dx;\n", + "\n", + "fprintf('lambda_1: %6.3f\\n', lambda(1));\n", + "fprintf('lambda_2: %6.3f', lambda(2));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As expected, this matches with our manual calculation above. But, we might want to calculate these eigenvalues more accurately, so let's generalize this a bit and then try using $\\Delta x= 0.1$:" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "lambda_1: 1.047\n", + "lambda_2: 2.091\n", + "\n", + "Error in lambda_1: 0.05%\n" + ] + } + ], + "source": [ + "clear all; clc\n", + "\n", + "dx = 0.1;\n", + "L = 3.0;\n", + "x = 0 : dx : L;\n", + "n = length(x) - 2;\n", + "\n", + "Astar = zeros(n,n);\n", + "for i = 1 : n\n", + " if i == 1\n", + " Astar(1,1) = 2;\n", + " Astar(1,2) = -1;\n", + " elseif i == n\n", + " Astar(n,n-1) = -1;\n", + " Astar(n,n) = 2;\n", + " else\n", + " Astar(i,i-1) = -1;\n", + " Astar(i,i) = 2;\n", + " Astar(i,i+1) = -1;\n", + " end\n", + "end\n", + "k = eig(Astar);\n", + "\n", + "lambda = sqrt(k) / dx;\n", + "\n", + "fprintf('lambda_1: %6.3f\\n', lambda(1));\n", + "fprintf('lambda_2: %6.3f\\n\\n', lambda(2));\n", + "\n", + "err = abs(lambda(1) - (pi/L)) / (pi/L);\n", + "fprintf('Error in lambda_1: %5.2f%%\\n', 100*err);" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: mass-spring system\n", + "\n", + "Let's analyze the motion of masses connected by springs in a system:\n", + "
\n", + "
\n", + " \"mass-spring\n", + "
Figure: System with two masses connected by springs
\n", + "
\n", + "
\n", + "First, we need to write the equations of motion, based on doing a free-body diagram on each mass:\n", + "\\begin{align}\n", + "m_1 \\frac{d^2 x_1}{dt^2} &= -k x_1 + k(x_2 - x_1) \\\\\n", + "m_2 \\frac{d^2 x_2}{dt^2} &= -k (x_2 - x_1) - k x_2\n", + "\\end{align}\n", + "We can condense these equations a bit:\n", + "\\begin{align}\n", + "x_1^{\\prime\\prime} - \\frac{k}{m_1} \\left( -2 x_1 + x_2 \\right) &= 0 \\\\\n", + "x_2^{\\prime\\prime} - \\frac{k}{m_2} \\left( x_1 - 2 x_2 \\right) &= 0\n", + "\\end{align}\n", + "\n", + "To proceed, we can assume that the masses will move in a sinusoidal fashion, with a shared frequency but separate amplitude:\n", + "\\begin{align}\n", + "x_i &= A_i \\sin (\\omega t) \\\\\n", + "x_i^{\\prime\\prime} &= -A_i \\omega^2 \\sin (\\omega t)\n", + "\\end{align}\n", + "We can plug these into the ODEs:\n", + "\\begin{align}\n", + "\\sin (\\omega t) \\left[ \\left( \\frac{2k}{m_1} - \\omega^2 \\right) A_1 - \\frac{k}{m_1} A_2 \\right] &= 0 \\\\\n", + "\\sin (\\omega t) \\left[ -\\frac{k}{m_2} A_1 + \\left( \\frac{2k}{m_2} - \\omega^2 \\right) A_2 \\right] &= 0\n", + "\\end{align}\n", + "or\n", + "\\begin{align}\n", + "\\left( \\frac{2k}{m_1} - \\omega^2 \\right) A_1 - \\frac{k}{m_1} A_2 &= 0 \\\\\n", + "-\\frac{k}{m_2} A_1 + \\left( \\frac{2k}{m_2} - \\omega^2 \\right) A_2 &= 0\n", + "\\end{align}\n", + "Let's put some numbers in, and try to solve for the eigenvalues: $\\omega^2$.\n", + "Let $m_1 = m_2 = 40 $ kg and $k = 200$ N/m.\n", + "\n", + "Now, the equations become\n", + "\\begin{align}\n", + "\\left( 10 - \\omega^2 \\right) A_1 - 5 A_2 &= 0 \\\\\n", + "-5 A_1 + \\left( 10 - \\omega^2 \\right) A_2 &= 0\n", + "\\end{align}\n", + "or $A \\mathbf{y} = \\mathbf{0}$, which we can represent as\n", + "\\begin{equation}\n", + "\\begin{bmatrix} 10-\\omega^2 & -5 \\\\ -5 & 10-\\omega^2 \\end{bmatrix}\n", + "\\begin{bmatrix} A_1 \\\\ A_2 \\end{bmatrix} = \n", + "\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}\n", + "\\end{equation}\n", + "Here, $\\omega^2$ are the eigenvalues, and we can find them with $\\det(A) = 0$:\n", + "\\begin{align}\n", + "\\det(B) &= 0 \\\\\n", + "\\det (B^* - \\omega^2 I) &= 0\n", + "\\end{align}" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "omega_1 = 2.24 rad/s\n", + "omega_2 = 3.87 rad/s\n" + ] + } + ], + "source": [ + "clear all; clc\n", + "\n", + "Bstar = [10 -5; -5 10];\n", + "omega_squared = eig(Bstar);\n", + "omega = sqrt(omega_squared);\n", + "\n", + "fprintf('omega_1 = %5.2f rad/s\\n', omega(1));\n", + "fprintf('omega_2 = %5.2f rad/s\\n', omega(2));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We find there are two modes of oscillation, each associated with a different natural frequency. Unfortunately, we cannot calculate independent and unique values for the amplitudes, but if we insert the values of $\\omega$ into the above equations, we can find relations connecting the amplitudes:\n", + "\\begin{align}\n", + "\\omega_1: \\quad A_1 &= A_2 \\\\\n", + "\\omega_2: \\quad A_1 &= -A_2\n", + "\\end{align}\n", + "\n", + "So, for the first mode, we have the two masses moving in sync with the same amplitude. In the second mode, they move with opposite (but equal) amplitude. With the two different frequencies, they also have two different periods:" + ] + }, + { + "cell_type": "code", + "execution_count": 42, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "t = linspace(0, 3);\n", + "subplot(1,5,1)\n", + "plot(sin(omega(1)*t), t); hold on\n", + "plot(0,0, 's');\n", + "set (gca, 'ydir', 'reverse' )\n", + "box off; set(gca,'Visible','off')\n", + "\n", + "subplot(1,5,2)\n", + "plot(sin(omega(1)*t), t); hold on\n", + "plot(0,0, 's');\n", + "set (gca, 'ydir', 'reverse' )\n", + "text(-2.5,-0.2, 'First mode')\n", + "box off; set(gca,'Visible','off')\n", + "\n", + "subplot(1,5,4)\n", + "plot(-sin(omega(2)*t), t); hold on\n", + "plot(0,0, 's');\n", + "set (gca, 'ydir', 'reverse' )\n", + "box off; set(gca,'Visible','off')\n", + "\n", + "subplot(1,5,5)\n", + "plot(sin(omega(2)*t), t); hold on\n", + "plot(0,0, 's');\n", + "set (gca, 'ydir', 'reverse' )\n", + "box off; set(gca,'Visible','off')\n", + "text(-2.7,-0.2, 'Second mode')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can confirm that the system would actually behave in this way by setting up the system of ODEs and integrating based on initial conditions matching the amplitudes of the two modes.\n", + "\n", + "For example, let's use $x_1 (t=0) = x_2(t=0) = 1$ for the first mode, and $x_1(t=0) = 1$ and $x_2(t=0) = -1$ for the second mode. We'll use zero initial velocity for both cases. \n", + "\n", + "Then, we can solve by converting the system of two 2nd-order ODEs into a system of four 1st-order ODEs:" + ] + }, + { + "cell_type": "code", + "execution_count": 57, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Created file '/Users/niemeyek/projects/ME373-book/content/bvps/masses.m'.\n" + ] + } + ], + "source": [ + "%%file masses.m\n", + "function dxdt = masses(t, x)\n", + "% this is a function file to calculate the derivatives associated with the system\n", + "\n", + "m1 = 40;\n", + "m2 = 40;\n", + "k = 200;\n", + "\n", + "dxdt = zeros(4,1);\n", + "\n", + "dxdt(1) = x(2);\n", + "dxdt(2) = (k/m1)*(-2*x(1) + x(3));\n", + "dxdt(3) = x(4);\n", + "dxdt(4) = (k/m2)*(x(1) - 2*x(3));" + ] + }, + { + "cell_type": "code", + "execution_count": 65, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear all; clc\n", + "\n", + "% this is the integration for the system in the first mode\n", + "[t, X] = ode45('masses', [0 3], [1.0 0.0 1.0 0.0]);\n", + "subplot(1,5,1)\n", + "plot(X(:,1), t); \n", + "ylabel('displacement (m)'); xlabel('time (s)')\n", + "set (gca, 'ydir', 'reverse' )\n", + "%box off; set(gca,'Visible','off')\n", + "\n", + "subplot(1,5,2)\n", + "plot(X(:,3), t); xlabel('time (s)')\n", + "set (gca, 'ydir', 'reverse' )\n", + "text(-4,-0.2, 'First mode')\n", + "\n", + "% this is the integration for the system in the second mode\n", + "[t, X] = ode45('masses', [0 3], [1.0 0.0 -1.0 0.0]);\n", + "subplot(1,5,4)\n", + "plot(X(:,1), t);\n", + "ylabel('displacement (m)'); xlabel('time (s)')\n", + "set (gca, 'ydir', 'reverse' )\n", + "%box off; set(gca,'Visible','off')\n", + "\n", + "subplot(1,5,5)\n", + "plot(X(:,3), t); xlabel('time (s)')\n", + "set (gca, 'ydir', 'reverse' )\n", + "text(-4,-0.2, 'Second mode')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This shows that we get either of the pure modes of motion with the appropriate initial conditions.\n", + "\n", + "What about if the initial conditions *don't* match either set of amplitude patterns?" + ] + }, + { + "cell_type": "code", + "execution_count": 56, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "[t, X] = ode45('masses', [0 3], [0.25 0.0 0.75 0.0]);\n", + "subplot(1,5,1)\n", + "plot(X(:,1), t);\n", + "%plot(0,0, 's');\n", + "set (gca, 'ydir', 'reverse' )\n", + "%box off; set(gca,'Visible','off')\n", + "\n", + "subplot(1,5,2)\n", + "plot(X(:,3), t);\n", + "set (gca, 'ydir', 'reverse' )\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this case, the resulting motion will be a complicated superposition of the two modes." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.11" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/docs/_sources/content/bvps/finite-difference.ipynb b/docs/_sources/content/bvps/finite-difference.ipynb new file mode 100644 index 0000000..e8fc766 --- /dev/null +++ b/docs/_sources/content/bvps/finite-difference.ipynb @@ -0,0 +1,660 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Finite difference method\n", + "\n", + "\n", + "## Finite differences\n", + "\n", + "Another method of solving boundary-value problems (and also partial differential equations, as we'll see later) involves **finite differences**, which are numerical approximations to exact derivatives.\n", + "\n", + "Recall that the exact derivative of a function $f(x)$ at some point $x$ is defined as:\n", + "\\begin{equation}\n", + "f^{\\prime}(x) = \\frac{df}{dx}(x) = \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x) - f(x)}{\\Delta x}\n", + "\\end{equation}\n", + "\n", + "So, we can *approximate* this derivative using a finite difference (rather than an infinitesimal difference as in the exact derivative):\n", + "\\begin{equation}\n", + "f^{\\prime}(x) \\approx \\frac{f(x+\\Delta x) - f(x)}{\\Delta x}\n", + "\\end{equation}\n", + "which involves some error. This is a **forward difference** for approximating the first derivative.\n", + "We can also approximate the first derivative using a **backward difference**:\n", + "\\begin{equation}\n", + "f^{\\prime}(x) \\approx \\frac{f(x) - f(x - \\Delta x)}{\\Delta x}\n", + "\\end{equation}\n", + "\n", + "To understand the error involved in these differences, we can use Taylor's theorem to obtain Taylor series expansions:\n", + "\\begin{align}\n", + "f(x + \\Delta x) &= f(x) + \\Delta x \\, f^{\\prime}(x) + \\Delta x^2 \\frac{1}{2!} f^{\\prime\\prime}(x) + \\cdots \\\\\n", + "\\rightarrow \\frac{f(x + \\Delta x) - f(x)}{\\Delta x} &= f^{\\prime}(x) + \\mathcal{O}\\left( \\Delta x \\right) \\\\\n", + "f(x - \\Delta x) &= f(x) - \\Delta x \\, f^{\\prime}(x) + \\Delta x^2 \\frac{1}{2!} f^{\\prime\\prime}(x) + \\cdots \\\\\n", + "\\rightarrow \\frac{f(x) - f(x - \\Delta x)}{\\Delta x} &= f^{\\prime}(x) + \\mathcal{O}\\left( \\Delta x \\right) \\\\\n", + "\\end{align}\n", + "where the $\\mathcal{O}()$ notation stands for \"order of magnitude of\". So, we can see that each of these approximations is *first-order accurate*." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Second-order finite differences\n", + "\n", + "We can obtain higher-order approximations for the first derivative, and an approximations for the second derivative, by combining these Taylor series expansions:\n", + "\\begin{align}\n", + "f(x + \\Delta x) &= f(x) + \\Delta x \\, f^{\\prime}(x) + \\Delta x^2 \\frac{1}{2!} f^{\\prime\\prime}(x) + \\mathcal{O}\\left( \\Delta x^3 \\right) \\\\\n", + "f(x - \\Delta x) &= f(x) - \\Delta x \\, f^{\\prime}(x) + \\Delta x^2 \\frac{1}{2!} f^{\\prime\\prime}(x) + \\mathcal{O}\\left( \\Delta x^3 \\right)\n", + "\\end{align}\n", + "\n", + "Subtracting the Taylor series for $f(x+\\Delta x)$ by that for $f(x-\\Delta x)$ gives:\n", + "\\begin{align}\n", + "f(x + \\Delta x) - f(x - \\Delta x) &= 2 \\Delta x \\, f^{\\prime}(x) + \\mathcal{O}\\left( \\Delta x^3 \\right) \\\\\n", + "f^{\\prime}(x) &= \\frac{f(x + \\Delta x) - f(x - \\Delta x)}{2 \\Delta x} + \\mathcal{O}\\left( \\Delta x^2 \\right)\n", + "\\end{align}\n", + "which is a *second-order accurate* approximation for the first derivative.\n", + "\n", + "Adding the Taylor series for $f(x+\\Delta x)$ to that for $f(x-\\Delta x)$ gives:\n", + "\\begin{align}\n", + "f(x + \\Delta x) + f(x - \\Delta x) &= 2 f(x) + \\Delta x^2 f^{\\prime\\prime}(x) + \\mathcal{O}\\left( \\Delta x^3 \\right) \\\\\n", + "f^{\\prime\\prime}(x) &= \\frac{f(x + \\Delta x) - 2 f(x) + f(x - \\Delta x)}{\\Delta x^2} + \\mathcal{O}\\left( \\Delta x^2 \\right)\n", + "\\end{align}\n", + "which is a *second-order accurate* approximation for the second derivative." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Solving ODEs with finite differences\n", + "\n", + "We can use finite differences to solve ODEs by substituting them for exact derivatives, and then applying the equation at discrete locations in the domain. This gives us a system of simultaneous equations to solve.\n", + "\n", + "For example, let's consider the ODE\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + x y^{\\prime} - x y = 2 x \\;,\n", + "\\end{equation}\n", + "with the boundary conditions $y(0) = 1$ and $y(2) = 8$.\n", + "\n", + "First, we *discretize* the continuous domain: divide it into a number of discrete segments. For now, let's choose $\\Delta x = 0.5$, which creates four segments and thus five points: $x_1 = 0, x_2 = 0.5, x_3 = 1.0, x_4 = 1.5, x_5 = 2.0$. \n", + "\n", + "Our goal is then to find approximate values of $y(x)$ at these points: $y_1$ through $y_5$. So, we have five unknowns, and need five equations to solve for them. We can use the ODE to provide these equations, by replacing the derivatives with finite differences, and applying the equation at particular discrete locations.\n", + "\n", + "Recall that $y(x)$ is a function just like $f(x)$, and so we can apply the above finite difference equations to $y(x)$ and $y(x+\\Delta x)$. Now that we have points, or nodes, at locations separated by $\\Delta x$, we can consider a point $x_i$ where $y(x_i) = y_i$, $y(x_i + \\Delta x) = y(x_{i+1}) = y_{i+1}$, and $y(x_i - \\Delta x) = y(x_{i-1}) = y_{i-1}$.\n", + "\n", + "To do this, we'll follow a few steps:\n", + "\n", + "1.) Replace exact derivatives in the original ODE with finite differences, and apply the equation at a particular location $(x_i, y_i)$.\n", + "\n", + "For our example, this gives:\n", + "\\begin{equation}\n", + "\\frac{y_{i+1} - 2y_i + y_{i-1}}{\\Delta x^2} + x_i \\left( \\frac{y_{i+1} - y_{i-1}}{2 \\Delta x}\\right) - x_i y_i = 2 x_i\n", + "\\end{equation}\n", + "which applies at location $(x_i, y_i)$.\n", + "\n", + "2.) Next, rearrange the equation into a *recursion formula*:\n", + "\\begin{equation}\n", + "y_{i-1} \\left(1 - x_i \\frac{\\Delta x}{2}\\right) + y_i \\left( -2 -\\Delta x^2 x_i \\right) + y_{i+1} \\left(1 + x_i \\frac{\\Delta x}{2}\\right) = 2 x_i \\Delta x^2\n", + "\\end{equation}\n", + "We can use this equation to get an equation for each of the interior points in the domain.\n", + "\n", + "For the first and last points—the boundary points—we already have equations, given by the boundary conditions.\n", + "\n", + "3.) Set up system of linear equations\n", + "\n", + "Applying the recursion formula to the interior points, and the boundary conditions for the boundary points, we can get a system of simultaneous linear equations:\n", + "\\begin{align}\n", + "y_1 &= 1 \\\\\n", + "y_1 (0.875) + y_2 (-2.125) + y_3 (1.125) &= 0.25 \\\\\n", + "y_2 (0.75) + y_3 (-2.25) + y_4 (1.25) &= 0.5 \\\\\n", + "y_3 (0.625) + y_4 (-2.375) + y_5 (1.375) &= 0.75 \\\\\n", + "y_5 &= 8\n", + "\\end{align}\n", + "\n", + "This is a system of five equations and five unknowns, which we can solve! But, solving using substitution would be painful, so let's represent this system of equations using a matrix and vectors:\n", + "\\begin{equation}\n", + "\\begin{bmatrix}\n", + "1 & 0 & 0 & 0 & 0 \\\\\n", + "0.875 & -2.125 & 1.125 & 0 & 0 \\\\\n", + "0 & 0.75 & -2.25 & 1.25 & 0 \\\\\n", + "0 & 0 & 0.625 & -2.375 & 1.375 \\\\\n", + "0 & 0 & 0 & 0 & 1\n", + "\\end{bmatrix} \n", + "\\begin{bmatrix} y_1 \\\\ y_2 \\\\ y_3 \\\\ y_4 \\\\ y_5 \\end{bmatrix} = \n", + "\\begin{bmatrix} 1 \\\\ 0.25 \\\\ 0.5 \\\\ 0.75 \\\\ 8 \\end{bmatrix}\n", + "\\end{equation}\n", + "or, more compactly, $A \\mathbf{y} = \\mathbf{b}$.\n", + "\n", + "4.) Solve the linear system of equations\n", + "\n", + "The final step is just to solve. We can do this in Matlab with `y = A \\ b`. (This is equivalent to `y = inv(A)*b`, but faster.)" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "A = [1.0 0 0 0 0;\n", + " 0.875 -2.125 1.125 0 0;\n", + " 0 0.75 -2.25 1.25 0;\n", + " 0 0 0.625 -2.375 1.375;\n", + " 0 0 0 0 1];\n", + "\n", + "b = [1.0; 0.25; 0.5; 0.75; 8.0];\n", + "\n", + "x = [0 : 0.5 : 2];\n", + "y = A \\ b;\n", + "plot(x, y, 'o-');" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Matlab implementation\n", + "\n", + "Of course, all of this will be easier if we implement in Matlab in a general way. We'll use a `for` loop to populate the coefficient matrix $A$ and right-hand-side vector $\\mathbf{b}$:" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear all; clc\n", + "\n", + "dx = 0.5;\n", + "x = [0 : dx : 2];\n", + "n = length(x);\n", + "A = zeros(n,n); b = zeros(n,1);\n", + "\n", + "for i = 1 : n\n", + " if i == 1\n", + " A(1,1) = 1;\n", + " b(1) = 1;\n", + " elseif i == n\n", + " A(n,n) = 1;\n", + " b(n) = 8;\n", + " else\n", + " A(i, i-1) = 1 - x(i)*dx/2;\n", + " A(i, i) = -2 - x(i)*dx^2;\n", + " A(i, i+1) = 1 + x(i)*dx/2;\n", + " b(i) = 2*x(i)*dx^2;\n", + " end\n", + "end\n", + "y = A \\ b;\n", + "plot(x, y, 'o-')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This looks good, but we can get a more-accurate solution by reducing our step size $\\Delta x$:" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear all; clc\n", + "\n", + "dx = 0.001;\n", + "x = [0 : dx : 2];\n", + "n = length(x);\n", + "A = zeros(n,n); b = zeros(n,1);\n", + "\n", + "for i = 1 : n\n", + " if i == 1\n", + " A(1,1) = 1;\n", + " b(1) = 1;\n", + " elseif i == n\n", + " A(n,n) = 1;\n", + " b(n) = 8;\n", + " else\n", + " A(i, i-1) = 1 - x(i)*dx/2;\n", + " A(i, i) = -2 - x(i)*dx^2;\n", + " A(i, i+1) = 1 + x(i)*dx/2;\n", + " b(i) = 2*x(i)*dx^2;\n", + " end\n", + "end\n", + "y = A \\ b;\n", + "plot(x, y)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Boundary conditions\n", + "\n", + "We will encounter four main kinds of boundary conditions. Consider the ODE $y^{\\prime\\prime} + y = 0$, on the domain $0 \\leq x \\leq L$.\n", + "\n", + "- First type, or Dirichlet, boundary conditions specify fixed values of $y$ at the boundaries: $y(0) = a$ and $y(L) = b$.\n", + "- Second type, or Neumann, boundary conditions specify values of the derivative at the boundaries: $y^{\\prime}(0) = a$ and $y^{\\prime}(L) = b$.\n", + "- Third type, or Robin, boundary conditions specify a linear combination of the function value and its derivative at the boundaries: $a \\, y(0) + b \\, y^{\\prime}(0) = g(0)$ and $a \\, y(L) + b \\, y^{\\prime}(L) = g(L)$, where $g(x)$ is some function.\n", + "- Mixed boundary conditions, which combine any of these three at the different boundaries. For example, we could have $y(0) = a$ and $y^{\\prime}(L) = b$.\n", + "\n", + "Whichever type of boundary condition we are dealing with, the goal will be to construct an equation representing the boundary condition to incorporate in our system of equations.\n", + "\n", + "If we have a fixed value boundary condition, such as $y(0) = a$, then this equation is straightforward:\n", + "\\begin{equation}\n", + "y_1 = a\n", + "\\end{equation}\n", + "where $y_1$ is the first point in the grid of points, corresponding to $x_1 = 0$. (We saw this in the example above.) In Matlab, we can implement this equation with\n", + "```OCTAVE\n", + "A(1,1) = 1;\n", + "b(1) = a;\n", + "```\n", + "\n", + "If we have a fixed derivative boundary condition, such as $y^{\\prime}(0) = 0$, then we need to use a finite difference to represent the derivative. When the boundary condition is at the starting location, $x=0$, the easiest way to do this is with a **forward difference**:\n", + "\\begin{align}\n", + "y^{\\prime}(0) \\approx \\frac{y_2 - y_1}{\\Delta x} &= 0 \\\\\n", + "-y_1 + y_2 &= 0\n", + "\\end{align}\n", + "We can implement this in Matlab with\n", + "```OCTAVE\n", + "A(1,1) = -1;\n", + "A(1,2) = 1;\n", + "b(1) = 0;\n", + "```\n", + "\n", + "When we have this sort of derivative boundary condition at the right side of the domain, at $x=L$, then we can use a **backward difference** to represent the derivative:\n", + "\\begin{align}\n", + "y^{\\prime}(L) \\approx \\frac{y_n - y_{n_1}}{\\Delta x} &= 0 \\\\\n", + "-y_{n-1} + y_n &= 0\n", + "\\end{align}\n", + "where $y_n$ is the final point ($x_n = L$) and $y_{n-1}$ is the second-to-last point ($x_{n-1} = L - \\Delta x$). We can implement this in Matlab with\n", + "```OCTAVE\n", + "A(n,n-1) = -1;\n", + "A(n,n) = 1;\n", + "b(n) = 0;\n", + "```\n", + "\n", + "If we have a linear combination of a fixed value and fixed derivative, like $a \\, y(0) + b \\, y^{\\prime}(0) = c$, then we can combine the above approaches using a forward difference:\n", + "\\begin{align}\n", + "a y(0) + b y^{\\prime}(0) \\approx a y_1 + b \\frac{y_2 - y_1}{\\Delta x} &= c \\\\\n", + "(a \\Delta x - b) y_1 + b y_2 &= c \\Delta x\n", + "\\end{align}\n", + "and in Matlab:\n", + "```OCTAVE\n", + "A(1,1) = a*dx - b;\n", + "A(1,2) = b;\n", + "b(1) = c * dx;\n", + "```\n", + "\n", + "### Using central differences for derivative BCs\n", + "\n", + "When a boundary condition involves a derivative, we can use a *central difference* to approximate the first derivative; this is more accurate than a forward or backward difference.\n", + "\n", + "Consider the formula for a central difference at $x=0$, applied for the boundary condition $y^{\\prime}(0) = 0$:\n", + "\\begin{align}\n", + "y^{\\prime}(0) \\approx \\frac{y_2 - y_0}{2 \\Delta x} &= 0 \\\\\n", + "y_0 &= y_2\n", + "\\end{align}\n", + "where $y_0$ is an imaginary, or ghost, node *outside* the domain. We can't actually keep this point in our implementation, because it isn't a real point.\n", + "\n", + "We still need an equation *for* the point at the boundary, $y_1$. To get this, we'll apply the regular recursion formula, normally used at interior points:\n", + "\\begin{align}\n", + "a y_{i-1} + b y_i + c y_{i+1} = f(x_i) \\\\\n", + "a y_0 + b y_1 + c y_2 = f(x_1) \\;,\n", + "\\end{align}\n", + "where $a$, $b$, $c$, and $f(x)$ depend on the problem. Normally we wouldn't use this at the boundary node, $y_1$, because it references a point outside the domain to the left—but we have an equation for that! From above, based on the boundary condition, we have $y_0 = y_2$. If we incorporate that into the recursion formula, we can eliminate the ghost node $y_0$:\n", + "\\begin{align}\n", + "a y_2 + b y_1 + c y_2 &= f(x_1) \\\\\n", + "b y_1 + (a + c) y_2 &= f(x_1) \\,\n", + "\\end{align}\n", + "which is the equation we can actually use at the boundary point.\n", + "In Matlab, this looks like \n", + "```OCTAVE\n", + "A(1,1) = b;\n", + "A(1,2) = a + c;\n", + "b(1) = f(x(1));\n", + "```" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: nonlinear BVP\n", + "\n", + "So far we've seen how to handle a linear boundary value problem, but what if we have a **nonlinear** BVP? This is going to be trickier, because our work so far relies on using linear algebra to solve the system of (linear) equations.\n", + "\n", + "For example, consider the 2nd-order ODE\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} = 3y + x^2 + 100 y^2\n", + "\\end{equation}\n", + "with the boundary conditions $y(0) = y(1) = 0$. This is nonlinear, due to the $y^3$ term on the right-hand side.\n", + "\n", + "To solve this, let's first convert it into a discrete form, by replacing the second derivative with a finite difference and any $x$/$y$ present with $x_i$ and $y_i$. We'll also move any constants (i.e., terms that don't contain $y_i$) and the nonlinear term to the right-hand side:\n", + "\\begin{equation}\n", + "\\frac{y_{i-1} - 2y_i + y_{i+1}}{\\Delta x^2} - 3y_i = x_i^2 + 100 y_i^2\n", + "\\end{equation}\n", + "where the boundary conditions are now $y_1 = 0$ and $y_n = 0$, with $n$ as the number of grid points. We can rearrange and simplify into our recursion formula:\n", + "\\begin{equation}\n", + "y_{i-1} + y_i \\left( -2 - 3 \\Delta x^2 \\right) + y_{i+1} = x_i^2 \\Delta x^2 + 100 \\Delta x^2 y_i^2\n", + "\\end{equation}\n", + "\n", + "The question is: how do we solve this now? The nonlinear term involving $y_i^3$ on the right-hand side complicates things, but we know how to set up and solve this *without* the nonlinear term. We can use an approach known as **successive iteration**:\n", + "\n", + "1. Solve the ODE without the nonlinear term to get an initial \"guess\" to the solution for $y$.\n", + "2. Then, incorporate that guess solution in the nonlinear term on the right-hand side, treating it as a constant. We can call this $y_{\\text{old}}$. Then, solve the full system for a new $y$ solution.\n", + "3. Check whether the new $y$ matches $y_{\\text{old}}$ with some tolerance. For example, check whether $\\max\\left(\\left| y - y_{\\text{old}} \\right| \\right) < $ some tolerance, such as $10^{-6}$. If this is true, then we can consider the solution *converged*. If it is not true, then set $y_{\\text{old}} = y$, and repeat the process starting at step 2.\n", + "\n", + "Let's implement that process in Matlab:" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Number of iterations: 16\n" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "%% Initial setup\n", + "clear all; clc\n", + "\n", + "dx = 0.01;\n", + "x = 0 : dx : 1;\n", + "n = length(x);\n", + "\n", + "A = zeros(n, n);\n", + "b = zeros(n, 1);\n", + "\n", + "%% First, solve the problem without the nonlinear term:\n", + "for i = 1 : n\n", + " if i == 1 % x = 0 boundary condition\n", + " A(1,1) = 1;\n", + " b(1) = 0;\n", + " elseif i == n % x = L boundary condition\n", + " A(n,n) = 1;\n", + " b(n) = 0;\n", + " else % interior nodes, use recursion formula\n", + " A(i, i-1) = 1;\n", + " A(i, i) = -2 - 3*dx^2;\n", + " A(i, i+1) = 1;\n", + " b(i) = x(i)^2 * dx^2;\n", + " end\n", + "end\n", + "% get solution without nonlinear term\n", + "y = A \\ b;\n", + "\n", + "plot(x, y, '--'); hold on\n", + "\n", + "%% Now, set up iterative process to solve while incorporating nonlinear terms\n", + "iter = 1;\n", + "y_old = zeros(n, 1);\n", + "while max(abs(y - y_old)) > 1e-6\n", + " y_old = y;\n", + " % A matrix is not changed, but the b vector does\n", + " for i = 2 : n - 1\n", + " b(i) = x(i)^2 * dx^2 + 100*(dx^2)*(y_old(i)^2);\n", + " end\n", + " \n", + " y = A \\ b;\n", + " iter = iter + 1;\n", + "end\n", + "\n", + "fprintf('Number of iterations: %d\\n', iter);\n", + "plot(x, y)\n", + "legend('Initial solution', 'Final solution')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Another option is just to set our \"guess\" for the $y$ solution to be zero, rather than solve the problem in two steps:" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Number of iterations: 17\n" + ] + } + ], + "source": [ + "%% Initial setup\n", + "clear all; clc\n", + "\n", + "dx = 0.01;\n", + "x = 0 : dx : 1;\n", + "n = length(x);\n", + "\n", + "A = zeros(n, n);\n", + "b = zeros(n, 1);\n", + "\n", + "%% Set up the coefficient matrix, which does not change\n", + "for i = 1 : n\n", + " if i == 1 % x = 0 boundary condition\n", + " A(1,1) = 1;\n", + " b(1) = 0;\n", + " elseif i == n % x = L boundary condition\n", + " A(n,n) = 1;\n", + " b(n) = 0;\n", + " else % interior nodes, use recursion formula\n", + " A(i, i-1) = 1;\n", + " A(i, i) = -2 - 3*dx^2;\n", + " A(i, i+1) = 1;\n", + " b(i) = x(i)^2 * dx^2;\n", + " end\n", + "end\n", + "\n", + "% just use zeros as our initial guess for the solution\n", + "y = zeros(n, 1);\n", + "\n", + "%% Successive iteration\n", + "iter = 1;\n", + "y_old = 100 * rand(n, 1); % setting this to some random values, just to enter the while loop\n", + "while max(abs(y - y_old)) > 1e-6\n", + " y_old = y;\n", + " % A matrix is not changed, but the b vector does\n", + " for i = 2 : n - 1\n", + " b(i) = x(i)^2 * dx^2 + 100*(dx^2)*(y_old(i)^2);\n", + " end\n", + " \n", + " y = A \\ b;\n", + " iter = iter + 1;\n", + "end\n", + "\n", + "fprintf('Number of iterations: %d\\n', iter);" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This made our process take slightly more iterations, because the initial guess was slightly further away from the final solution. For other problems, having a bad initial guess could make the process take much longer, so coming up with a good initial guess may be important." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: heat transfer through a fin\n", + "\n", + "Let's now consider a more complicated example: heat transfer through an extended surface (a fin).\n", + "\n", + "
\n", + "
\n", + " \"Heat\n", + "
Figure: Geometry of a heat transfer fin
\n", + "
\n", + "
\n", + "\n", + "In this situation, we have the temperature of the body $T_b$, the temperature of the ambient fluid $T_{\\infty}$; the length $L$, width $w$, and thickness $t$ of the fin; the thermal conductivity of the fin material $k$; and convection heat transfer coefficient $h$.\n", + "\n", + "The boundary conditions can be defined in different ways, but generally we can say that the temperature of the fin at the wall is the same as the body temperature, and that the fin is insulated at the tip. This gives us\n", + "\\begin{align}\n", + "T(x=0) &= T_b \\\\\n", + "q(x=L) = 0 \\rightarrow \\frac{dT}{dx} (x=0) &= 0\n", + "\\end{align}\n", + "\n", + "Our goal is to solve for the temperature distribution $T(x)$. To do this, we need to set up a governing differential equation. Let's do a control volume analysis of heat transfer through the fin:\n", + "\n", + "
\n", + "
\n", + " \"Control\n", + "
Figure: Control volume for heat transfer through the fin
\n", + "
\n", + "
\n", + "\n", + "Given a particular volumetric slice of the fin, we can define the heat transfer rates of conduction through the fin and convection from the fin to the air:\n", + "\\begin{align}\n", + "q_{\\text{conv}} &= h P \\left( T - T_{\\infty} \\right) dx \\\\\n", + "q_{\\text{cond}, x} &= -k A_c \\left(\\frac{dT}{dx}\\right)_{x} \\\\\n", + "q_{\\text{cond}, x+\\Delta x} &= -k A_c \\left(\\frac{dT}{dx}\\right)_{x+\\Delta x} \\;,\n", + "\\end{align}\n", + "where $P$ is the perimeter (so that $P \\, dx$ is the heat transfer area to the fluid) and $A_c$ is the cross-sectional area.\n", + "\n", + "Performing a balance through the control volume:\n", + "\\begin{align}\n", + "q_{\\text{cond}, x+\\Delta x} &= q_{\\text{cond}, x} - q_{\\text{conv}} \\\\\n", + "-k A_c \\left(\\frac{dT}{dx}\\right)_{x+\\Delta x} &= -k A_c \\left(\\frac{dT}{dx}\\right)_{x} - h P \\left( T - T_{\\infty} \\right) dx \\\\\n", + "-k A_c \\frac{\\left.\\frac{dT}{dx}\\right|_{x+\\Delta x} - \\left.\\frac{dT}{dx}\\right|_{x}}{dx} &= -h P ( T - T_{\\infty} ) \\\\\n", + "\\lim_{\\Delta x \\rightarrow 0} : -k A_c \\left. \\frac{d^2 T}{dx^2} \\right|_x &= -h P (T - T_{\\infty}) \\\\\n", + "\\frac{d^2 T}{dx^2} &= \\frac{h P}{k A_c} (T - T_{\\infty}) \\\\\n", + "\\frac{d^2 T}{dx^2} &= m^2 (T - T_{\\infty})\n", + "\\end{align}\n", + "then we have as a governing equation\n", + "\\begin{equation}\n", + "\\frac{d^2 T}{dx^2} - m^2 (T - T_{\\infty}) = 0 \\;,\n", + "\\end{equation}\n", + "where $m^2 = (h P)/(k A_c)$.\n", + "\n", + "We can obtain an exact solution for this ODE. For convenience, let's define a new variable, $\\theta$, which is a normalized temperature:\n", + "\\begin{equation}\n", + "\\theta \\equiv T - T_{\\infty}\n", + "\\end{equation}\n", + "where $\\theta^{\\prime} = T^{\\prime}$ and $\\theta^{\\prime\\prime} = T^{\\prime\\prime}$.\n", + "This gives us a new governing equation:\n", + "\\begin{equation}\n", + "\\theta^{\\prime\\prime} - m^2 \\theta = 0 \\;.\n", + "\\end{equation}\n", + "This is a 2nd-order homogeneous ODE, which looks a lot like $y^{\\prime\\prime} + a y = 0$. The exact solution is then\n", + "\\begin{align}\n", + "\\theta(x) &= c_1 e^{-m x} + c_2 e^{m x} \\\\\n", + "T(x) &= T_{\\infty} + c_1 e^{-m x} + c_2 e^{m x}\n", + "\\end{align}\n", + "We'll use this to look at the accuracy of a numerical solution, but we will not be able to find an exact solution for more complicated versions of this problem.\n", + "\n", + "We can also solve this numerically using the finite difference method. Let's replace the derivative with a finite difference:\n", + "\\begin{align}\n", + "\\frac{d^2 T}{dx^2} - m^2 (T - T_{\\infty}) &= 0 \\\\\n", + "\\frac{T_{i-1} - 2T_i + T_{i+1}}{\\Delta x^2} - m^2 \\left( T_i - T_{\\infty} \\right) &= 0\n", + "\\end{align}\n", + "which we can rearrange into a recursion formula:\n", + "\\begin{equation}\n", + "T_{i-1} + T_i \\left( -2 - \\Delta x^2 m^2 \\right) + T_{i+1} = -m^2 \\Delta x^2 \\, T_{\\infty}\n", + "\\end{equation}\n", + "This gives us an equation for all the interior nodes; we can use the above boundary conditions to get equations for the boundary nodes. For the boundary condition at $x=L$, $T^{\\prime}(x=L) = 0$, let's use a *backward difference*:\n", + "\\begin{align}\n", + "T_1 &= T_b \\\\\n", + "\\frac{T_n - T_{n-1}}{\\Delta x} = 0 \\rightarrow - T_{n-1} + T_n &= 0\n", + "\\end{align}\n", + "\n", + "Combining all these equations, we can construct a linear system: $A \\mathbf{T} = \\mathbf{b}$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Heat transfer with radiation\n", + "\n", + "Let's now consider a more-complicated case, where we also have radiation heat transfer occuring along the length of the fin. Now, our governing ODE is\n", + "\\begin{equation}\n", + "\\frac{d^2 T}{dx^2} - \\frac{h P}{k A_c} \\left(T - T_{\\infty}\\right) - \\frac{\\sigma \\epsilon P}{h A_c} \\left(T^4 - T_{\\infty}^4 \\right) = 0\n", + "\\end{equation}\n", + "\n", + "This is a bit trickier to solve because of the nonlinear term involving $T^4$. But, we can handle it via the iterative solution method discussed above." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.7" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/docs/_sources/content/bvps/shooting-method.ipynb b/docs/_sources/content/bvps/shooting-method.ipynb new file mode 100644 index 0000000..1ba9df2 --- /dev/null +++ b/docs/_sources/content/bvps/shooting-method.ipynb @@ -0,0 +1,460 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Shooting Method\n", + "\n", + "Boundary-value problems are also ordinary differential equations—the difference is that our two constraints are at boundaries of the domain, rather than both being at the starting point.\n", + "\n", + "For example, consider the ODE\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + xy^{\\prime} - xy = 2x\n", + "\\end{equation}\n", + "with the boundary conditions $y(0)=1$ and $y(2)=8$.\n", + "\n", + "The numerical methods we have already discussed (e.g., Forward Euler, Runge-Kutta) require values of $y$ and $y^{\\prime}$ at the starting point, $x=0$. So we can't use these directly because we are missing $y^{\\prime}(0)$. \n", + "\n", + "But, what if we could *guess* a value for the missing initial condition, then integrate towards the second boundary condition using one of our familiar numerical methods, and then adjust our guess if necessary and repeat? This concept is the **shooting method**.\n", + "\n", + "The shooting method algorithm is:\n", + "\n", + "1. Guess a value of the missing initial condition; in this case, that is $y'(0)$.\n", + "2. Integrate the ODE like an initial-value problem, using our existing numerical methods, to get the given boundary condition(s); in this case, that is $y(L)$.\n", + "3. Assuming your trial solution for $y(L)$ does not match the given boundary condition, adjust your guess for $y'(0)$ and repeat.\n", + "\n", + "Now, this algorithm will not work particularly well if all your guesses are random/uninformed. Fortunately, we can use linear interpolation to inform a third guess based on two initial attempts:\n", + "\\begin{align}\n", + "\\text{guess 3} &= \\text{guess 2} + m \\left( \\text{target} - \\text{solution 2} \\right) \\\\\n", + "m &= \\frac{\\text{guess 1} - \\text{guess 2}}{\\text{solution 1} - \\text{solution 2}}\n", + "\\end{align}\n", + "where \"target\" is the target boundary condition—in this case, $y(L)$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: linear ODE\n", + "\n", + "Let's try solving the given ODE using the shooting method:\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + xy^{\\prime} - xy = 2x\n", + "\\end{equation}\n", + "with the boundary conditions $y(0)=1$ and $y(2)=8$.\n", + "\n", + "First, we need to convert this 2nd-order ODE into a system of two 1st-order ODEs, where we can define $u = y'$:\n", + "\\begin{align}\n", + "y' &= u \\\\\n", + "u' &= 2x + xy - xu\n", + "\\end{align}" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Created file '/Users/niemeyek/projects/ME373-book/content/bvps/shooting_rhs.m'.\n" + ] + } + ], + "source": [ + "%%file shooting_rhs.m\n", + "function dydx = shooting_rhs(x, y)\n", + "\n", + "dydx = zeros(2,1);\n", + "dydx(1) = y(2);\n", + "dydx(2) = 2*x - x*y(2) + x*y(1);" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Solution 1: 6.00\n", + "Solution 2: 11.96\n", + "Guess 3: 2.01\n", + "Solution 3: 8.00\n", + "Target: 8.00\n" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear all; clc\n", + "\n", + "% target boundary condition\n", + "target = 8;\n", + "\n", + "% Pick a guess for y'(0) of 1\n", + "guess1 = 1;\n", + "[X, Y] = ode45('shooting_rhs', [0 2], [1 guess1]);\n", + "solution1= Y(end,1);\n", + "fprintf('Solution 1: %5.2f\\n', solution1);\n", + "\n", + "% Pick a second guess for y'(0) of 4\n", + "guess2 = 4;\n", + "[X, Y] = ode45('shooting_rhs', [0 2], [1 guess2]);\n", + "solution2 = Y(end,1);\n", + "fprintf('Solution 2: %5.2f\\n', solution2);\n", + "\n", + "% now use linear interpolation to find a new guess\n", + "m = (guess1 - guess2)/(solution1 - solution2);\n", + "guess3 = guess2 + m*(target-solution2);\n", + "fprintf('Guess 3: %5.2f\\n', guess3);\n", + "\n", + "[X, Y] = ode45('shooting_rhs', [0 2], [1 guess3]);\n", + "solution3 = Y(end,1);\n", + "fprintf('Solution 3: %5.2f\\n', solution3);\n", + "fprintf('Target: %5.2f\\n', target);\n", + "\n", + "plot(X, Y(:,1)); axis([0 2 0 9])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As you can see, using linear interpolation, we are able to find the correct guess for the missing initial condition $y'(0)$ with in just three steps. This works so well because this is a *linear* ODE. If we had a nonlinear ODE, it would take more tries, as we'll see shortly." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: nonlinear ODE\n", + "\n", + "We can use the shooting method to solve a famous fluids problem: the [Blasius boundary layer](https://en.wikipedia.org/wiki/Blasius_boundary_layer).\n", + "\n", + "
\n", + "
\n", + " \"Laminar\n", + "
Figure: Laminar boundary layer, taken from https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg
\n", + "
\n", + "
\n", + "\n", + "To get to a solveable ODE, we start with the conservation of momentum equation (i.e., Navier–Stokes equation) in the $x$-direction:\n", + "\\begin{equation}\n", + "u \\frac{\\partial u}{\\partial x} + v \\frac{\\partial u}{\\partial y} = \\nu \\frac{\\partial^2 u}{\\partial y^2}\n", + "\\end{equation}\n", + "and the conservation of mass equation:\n", + "\\begin{equation}\n", + "\\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} = 0 \\;,\n", + "\\end{equation}\n", + "where $u$ is the velocity component in the $x$-direction, $v$ is the velocity component in the $y$-direction, and $\\nu$ is the fluid's kinematic viscosity. The boundary conditions are that $u = v = 0$ at $y=0$, and that $u = U_{\\infty}$ as $y \\rightarrow \\infty$, where $U_{\\infty}$ is the free-stream velocity.\n", + "\n", + "Blasius solved this problem by converting the PDE into an ODE, by recognizing that the boundary layer thickness is given by $\\delta(x) \\sim \\sqrt{\\frac{x \\nu}{U_{\\infty}}}$, and then nondimensionalizing the position coordinates using a similarity variable\n", + "\\begin{equation}\n", + "\\eta = y \\sqrt{\\frac{U_{\\infty}}{2 \\nu x}}\n", + "\\end{equation}\n", + "\n", + "By introducing the stream function, $\\psi (x,y)$, we can ensure the continuity equation is satisfied:\n", + "\\begin{equation}\n", + "u = \\frac{\\partial \\psi}{\\partial y} \\;, \\quad v = -\\frac{\\partial \\psi}{\\partial x}\n", + "\\end{equation}\n", + "\n", + "Let's check this, using SymPy:" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "True\n" + ] + } + ], + "source": [ + "%%python\n", + "import sympy as sym\n", + "sym.init_printing()\n", + "x, y, u, v = sym.symbols('x y u v')\n", + "\n", + "# Streamfunction\n", + "psi = sym.Function(r'psi')(x,y)\n", + "\n", + "# Define u and v based on the streamfunction\n", + "u = psi.diff(y)\n", + "v = -psi.diff(x)\n", + "\n", + "# Check the continuity equation:\n", + "print(u.diff(x) + v.diff(y) == 0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Using the boundary layer thickness and free-stream velocity, we can define the dimensionlesss stream function $f(\\eta)$:\n", + "\\begin{equation}\n", + "f(\\eta) = \\frac{\\psi}{U_{\\infty}} \\sqrt{\\frac{U_{\\infty}}{2 \\nu x}}\n", + "\\end{equation}\n", + "which relates directly to the velocity components:\n", + "\\begin{align}\n", + "u &= \\frac{\\partial \\psi}{\\partial y} = \\frac{\\partial \\psi}{\\partial f} \\frac{\\partial f}{\\partial \\eta} \\frac{\\partial \\eta}{\\partial y} \\\\\n", + " &= U_{\\infty} \\sqrt{\\frac{2 \\nu x}{U_{\\infty}}} \\cdot f^{\\prime}(\\eta) \\cdot \\sqrt{\\frac{U_{\\infty}}{2 \\nu x}} \\\\\n", + "u &= U_{\\infty} f^{\\prime} (\\eta) \\\\\n", + "v &= -\\frac{\\partial \\psi}{\\partial x} = -\\left( \\frac{\\partial \\psi}{\\partial x} + \\frac{\\partial \\psi}{\\partial \\eta} \\frac{\\partial \\eta}{\\partial x} \\right) \\\\\n", + " &= \\sqrt{\\frac{\\nu U_{\\infty}}{2x}} \\left( \\eta f^{\\prime} - f \\right)\n", + "\\end{align}\n", + "\n", + "We can insert these into the $x$-momentum equation, which leads to an ODE for the dimensionless stream function $f(\\eta)$:\n", + "\\begin{equation}\n", + "f^{\\prime\\prime\\prime} + f f^{\\prime\\prime} = 0 \\;,\n", + "\\end{equation}\n", + "with the boundary conditions $f = f^{\\prime} = 0$ at $\\eta = 0$, and $f^{\\prime} = 1$ as $\\eta \\rightarrow \\infty$.\n", + "\n", + "This is a 3rd-order ODE, which we can solve by converting it into three 1st-order ODEs:\n", + "\\begin{align}\n", + "y_1 &= f \\quad y_1^{\\prime} = y_2 \\\\\n", + "y_2 &= f^{\\prime} \\quad y_2^{\\prime} = y_3 \\\\\n", + "y_3 &= f^{\\prime\\prime} \\quad y_3^{\\prime} = -y_1 y_3\n", + "\\end{align}\n", + "and we can use the shooting method to solve by recognizing that we have two initial conditions, $y_1(0) = y_2(0) = 0$, and are missing $y_3(0)$. We also have a target boundary condition: $y_2(\\infty) = 1$.\n", + "\n", + "(Note: obviously we cannot truly integrate over $0 \\leq \\eta < \\infty$. Instead, we just need to choose a large enough number. In this case, using 10 is sufficient.)\n", + "\n", + "Let's create a function to evaluate the derivatives:" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Created file '/Users/niemeyek/projects/ME373-book/content/bvps/blasius_rhs.m'.\n" + ] + } + ], + "source": [ + "%%file blasius_rhs.m\n", + "function dydx = blasius_rhs(eta, y)\n", + "\n", + "dydx = zeros(3,1);\n", + "\n", + "dydx(1) = y(2);\n", + "dydx(2) = y(3);\n", + "dydx(3) = -y(1) * y(3);" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "First, let's try the same three-step approach we used for the simpler example, taking two guesses and then using linear interpolation to find a third guess:" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\n", + "ans =\n", + "\n", + " 3x3 table\n", + "\n", + " tries guesses solutions\n", + " _____ _______ _________\n", + "\n", + " 1 1 1.6553 \n", + " 2 0.1 0.3566 \n", + " 3 0.54587 1.1056 \n", + "\n", + "Target: 1.00\n" + ] + } + ], + "source": [ + "clear all; clc\n", + "\n", + "target = 1.0;\n", + "\n", + "guesses = zeros(3,1);\n", + "solutions = zeros(3,1);\n", + "\n", + "guesses(1) = 1;\n", + "[eta, F] = ode45('blasius_rhs', [0 10], [0 0 guesses(1)]);\n", + "solutions(1) = F(end, 2);\n", + "\n", + "guesses(2) = 0.1;\n", + "[eta, F] = ode45('blasius_rhs', [0 10], [0 0 guesses(2)]);\n", + "solutions(2) = F(end, 2);\n", + "\n", + "m = (guesses(1) - guesses(2))/(solutions(1) - solutions(2));\n", + "guesses(3) = guesses(2) + m*(target - solutions(2));\n", + "\n", + "[eta, F] = ode45('blasius_rhs', [0 10], [0 0 guesses(3)]);\n", + "solutions(3) = F(end, 2);\n", + "\n", + "tries = [1; 2; 3];\n", + "table(tries, guesses, solutions)\n", + "fprintf('Target: %5.2f\\n', target);" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "So, for this problem, using linear interpolation did *not* get us the correct solution on the third try. This is because the ODE is nonlinear. But, you can see that we are converging towards the correct solution—it will just take more tries.\n", + "\n", + "Rather than manually take an unknown (and potentially large) number of guesses, let's automate this with a `while` loop:" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\n", + "ans =\n", + "\n", + " 7x3 table\n", + "\n", + " tries guesses solutions\n", + " _____ _______ _________\n", + "\n", + " 1 1 1.6553 \n", + " 2 0.1 0.3566 \n", + " 3 0.54587 1.1056 \n", + " 4 0.48301 1.019 \n", + " 5 0.46922 0.99951 \n", + " 6 0.46957 1 \n", + " 7 0.46957 1 \n", + "\n", + "Number of iterations required: 7" + ] + } + ], + "source": [ + "clear all; clc\n", + "\n", + "target = 1.0;\n", + "\n", + "% get these arrays of stored values started.\n", + "% note: I'm only doing this to make it easier to show a table of values\n", + "% at the end; otherwise, there's no need to store these values.\n", + "tries = [1; 2; 3];\n", + "guesses = zeros(3,1);\n", + "solutions = zeros(3,1);\n", + "\n", + "guesses(1) = 1;\n", + "[eta, F] = ode45('blasius_rhs', [0 10], [0 0 guesses(1)]);\n", + "solutions(1) = F(end, 2);\n", + "\n", + "guesses(2) = 0.1;\n", + "[eta, F] = ode45('blasius_rhs', [0 10], [0 0 guesses(2)]);\n", + "solutions(2) = F(end, 2);\n", + "\n", + "num = 2;\n", + "solutions(3) = -1000.; % doing this to kick off the while loop\n", + "while abs(target - solutions(num)) > 1.e-9\n", + " num = num + 1;\n", + " m = (guesses(num-2) - guesses(num-1))/(solutions(num-2) - solutions(num-1));\n", + " guesses(num) = guesses(num-1) + m*(target - solutions(num-1));\n", + " [eta, F] = ode45('blasius_rhs', [0 1e3], [0 0 guesses(num)]);\n", + " solutions(num) = F(end, 2);\n", + " tries(num) = num;\n", + " \n", + " % we should probably set a maximum number of iterations, just to prevent\n", + " % an infinite while loop in case something goes wrong\n", + " if num >= 1e4\n", + " break\n", + " end\n", + "end\n", + "\n", + "table(tries, guesses, solutions)\n", + "fprintf('Number of iterations required: %d', num)" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "%plot -r 200\n", + "plot(F(:, 2), eta); ylim([0 5])\n", + "xlabel(\"f^{\\prime}(\\eta) = u/U_{\\infty}\")\n", + "ylabel('\\eta')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can see that this plot of $\\eta$, the $y$ position normalized by the boundary-layer thickness, vs. nondimensional velocity matches the original figure." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.7" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/docs/_sources/content/contributing.md b/docs/_sources/content/contributing.md new file mode 100644 index 0000000..7f2d72d --- /dev/null +++ b/docs/_sources/content/contributing.md @@ -0,0 +1,225 @@ +# Contributing to Jupyter Book + +Welcome to the `jupyter-book` repository! We're excited you're here and want to contribute. + +These guidelines are designed to make it as easy as possible to get involved. +If you have any questions that aren't discussed below, please let us know by opening an [issue][link_issues]! + +Before you start you'll need to set up a free [GitHub][link_github] account and sign in. +Here are some [instructions][link_signupinstructions]. + +Already know what you're looking for in this guide? Use the TOC to the right +to navigate this page! + +## Joining the conversation + +`jupyter-book` is a young project maintained by a growing group of enthusiastic developers— and we're excited to have you join! +Most of our discussions will take place on open [issues][link_issues]. + +As a reminder, we expect all contributors to `jupyter-book` to adhere to the [Jupyter Code of Conduct][link_coc] in these conversations. + +## Contributing through GitHub + +[git][link_git] is a really useful tool for version control. +[GitHub][link_github] sits on top of git and supports collaborative and distributed working. + +You'll use [Markdown][markdown] to chat in issues and pull requests on GitHub. +You can think of Markdown as a few little symbols around your text that will allow GitHub +to render the text with formatting. +For example you could write words as bold (`**bold**`), or in italics (`*italics*`), +or as a [link][rick_roll] (`[link](https://https://youtu.be/dQw4w9WgXcQ)`) to another webpage. + +GitHub has a helpful page on +[getting started with writing and formatting Markdown on GitHub][writing_formatting_github]. + + +## Understanding issues, milestones and project boards + +Every project on GitHub uses [issues][link_issues] slightly differently. + +The following outlines how the `jupyter-book` developers think about these tools. + +**Issues** are individual pieces of work that need to be completed to move the project forwards. +A general guideline: if you find yourself tempted to write a great big issue that +is difficult to describe as one unit of work, please consider splitting it into two or more issues. + +Issues are assigned [labels](#issue-labels) which explain how they relate to the overall project's +goals and immediate next steps. + + +### Issue labels + +The current list of labels are [here][link_labels] and include: + +* [![Help Wanted](https://img.shields.io/badge/-help%20wanted-159818.svg)][link_helpwanted] *These issues contain a task that a member of the team has determined we need additional help with.* + + If you feel that you can contribute to one of these issues, we especially encourage you to do so! + +* [![Good First Issue](https://img.shields.io/badge/-good%20first%20issue-blueviolet.svg)][link_helpwanted] *These issues contain a task that a member of the team thinks could be a good entry point to the project.* + + If you're new to the `jupyter-book` project, we think that this is a great place for your first contribution! + +* [![Bugs](https://img.shields.io/badge/-bugs-fc2929.svg)][link_bugs] *These issues point to problems in the project.* + + If you find new a bug, please give as much detail as possible in your issue, including steps to recreate the error. + If you experience the same bug as one already listed, please add any additional information that you have as a comment. + +* [![Enhancement](https://img.shields.io/badge/-enhancement-84b6eb.svg)][link_enhancement] *These issues are asking for enhancements to be added to the project.* + + Please try to make sure that your enhancement is distinct from any others that have already been requested or implemented. + If you find one that's similar but there are subtle differences please reference the other request in your issue. + +* [![Question](https://img.shields.io/badge/-question-DE8BE7.svg)][link_question] *These are questions that users and contributors have asked.* + + Please check the issues (especially closed ones) to see if your question has been asked and answered before. + If you find one that's similar but there are subtle differences please reference the other request in your issue. + +## Repository Structure of Jupyter Book + +This section covers the general structure of the +[Jupyter Book repository](https://github.com/jupyter/jupyter-book), and +explains which pieces are where. + +The Jupyter Book repository contains two main pieces: + +### The command-line tool and Python package + +This is used to help create and build books. +It can be found at [`./jupyter_book`](https://github.com/jupyter/jupyter-book/tree/master/jupyter_book). +* **The `page` module builds single pages**. This module is meant to be self-contained for + converting single `.ipynb`/`.md`/etc pages into HTML. Jupyter Book uses this module when + building entire books, but the module can also be used on its own (it's what `jupyter-book page` uses). + You can find the module at: [`jupyter_book/page`](https://github.com/jupyter/jupyter-book/tree/master/jupyter_book/page). +* **The `create.py` and `build.py` create and build a book**. They connect with the CLI and + are used to process multiple pages and stitch them together into a static website template. + +### The template SSG website + +This is used when generating new books. This website defines the structure of +the site that is created when you run `jupyter-book create`. It contains the Javascript, CSS, and +HTML structure of a book. It can be found at +[`jupyter_book/book_template`](https://github.com/jupyter/jupyter-book/tree/master/jupyter_book/book_template). +* The [`_includes/`](https://github.com/jupyter/jupyter-book/tree/master/jupyter_book/book_template/_includes) + folder contains core HTML and javascript files for the site. For example, + [`_includes/head.html`](https://github.com/jupyter/jupyter-book/blob/master/jupyter_book/book_template/_includes/head.html) contains the HTML for the header of each page, which is where CSS and JS files are linked. +* The [`assets/`](https://github.com/jupyter/jupyter-book/tree/master/jupyter_book/book_template/assets) + folder contains static CSS/JS files that don't depend on site configuration. +* The [`_sass/`](https://github.com/jupyter/jupyter-book/tree/master/jupyter_book/book_template/_sass) + folder contains all of the book and page CSS rules. This is stitched together in a single CSS file + at build time (SCSS is a way to split up CSS rules among multiple files). Within this folder, the + [`_sass/page/`](https://github.com/jupyter/jupyter-book/tree/master/jupyter_book/book_template/_sass/page) folder + has CSS files for a single page of content, while the other folders/files contain CSS rules for + the whole book. +* The [`content/`](https://github.com/jupyter/jupyter-book/tree/master/jupyter_book/book_template/content) + folder contains the content for the Jupyter Book documentation (e.g., the [markdown for this page](https://github.com/jupyter/jupyter-book/tree/master/jupyter_book/book_template/content/contributing.md)). + +### An example + +Here are a few examples of how this code gets used to help you get started. + +* when somebody runs `jupyter-book create mybook/`, the `create.py` module is used to generate an empty template using the template in `jupyter_book/book_template/`. +* when somebody runs `jupyter-book build mybook/`, the `build.py` module to loop through your page content files, + and uses the `page/` module to convert each one into HTML and places it in `mybook/_build`. + +Hopefully this explanation gets you situated and helps you understand how the pieces all fit together. +If you have any questions, feel free to [open an issue asking for help](https://github.com/jupyter/jupyter-book/issues/new)! + +## Making a change + +We appreciate all contributions to `jupyter-book`, but those accepted fastest will follow a workflow similar to the following: + +**1. Comment on an existing issue or open a new issue referencing your addition.** + +This allows other members of the jupyter-book development team to confirm that you aren't overlapping with work that's currently underway and that everyone is on the same page with the goal of the work you're going to carry out. + +[This blog][link_pushpullblog] is a nice explanation of why putting this work in up front is so useful to everyone involved. + +**2. [Fork][link_fork] the [jupyter-book repository][link_jupyter-book] to your profile.** + +This is now your own unique copy of jupyter-book. +Changes here won't effect anyone else's work, so it's a safe space to explore edits to the code! + +Make sure to [keep your fork up to date][link_updateupstreamwiki] with the master repository. + +**3. Make the changes you've discussed.** + +Try to keep the changes focused. +We've found that working on a [new branch][link_branches] makes it easier to keep your changes targeted. + +**4. Submit a [pull request][link_pullrequest].** + +A member of the development team will review your changes to confirm that they can be merged into the main code base. +When opening the pull request, we ask that you follow some [specific conventions](#pull-requests). +We outline these below. + +### Pull Requests + +To improve understanding pull requests "at a glance", we encourage the use of several standardized tags. +When opening a pull request, please use at least one of the following prefixes: + +* **[BRK]** for changes which break existing builds or tests +* **[DOC]** for new or updated documentation +* **[ENH]** for enhancements +* **[FIX]** for bug fixes +* **[REF]** for refactoring existing code +* **[STY]** for stylistic changes +* **[TST]** for new or updated tests, and + +You can also combine the tags above, for example if you are updating both a test and +the documentation: **[TST, DOC]**. + +Pull requests should be submitted early and often! + +If your pull request is not yet ready to be merged, please open your pull request as a draft. +More information about doing this is [available in GitHub's documentation][link_drafts]. +This tells the development team that your pull request is a "work-in-progress", +and that you plan to continue working on it. + +When your pull request is Ready for Review, you can select this option on the PR's page, +and a project maintainer will review your proposed changes. + + +## Recognizing contributors + +We welcome and recognize all contributions from documentation to testing to code development. +You can see a list of current contributors in the [contributors tab][link_contributors]. + +## Thank you! + +You're awesome. + +
+ +*— Based on contributing guidelines from the [STEMMRoleModels][link_stemmrolemodels] project.* + +[link_git]: https://git-scm.com +[link_github]: https://github.com/https://github.com/jupyter/governance/blob/master/conduct/code_of_conduct.md +[link_jupyter-book]: https://github.com/jupyter/jupyter-book +[link_signupinstructions]: https://help.github.com/articles/signing-up-for-a-new-github-account + +[writing_formatting_github]: https://help.github.com/articles/getting-started-with-writing-and-formatting-on-github +[markdown]: https://daringfireball.net/projects/markdown +[rick_roll]: https://www.youtube.com/watch?v=dQw4w9WgXcQ +[restructuredtext]: http://docutils.sourceforge.net/rst.html#user-documentation +[sphinx]: http://www.sphinx-doc.org/en/master/index.html +[readthedocs]: https://docs.readthedocs.io/en/latest/index.html + +[link_issues]: https://github.com/jupyter/jupyter-book/issues +[link_coc]: https://github.com/jupyter/governance/blob/master/conduct/code_of_conduct.md + +[link_labels]: https://github.com/jupyter/jupyter-book/labels +[link_bugs]: https://github.com/jupyter/jupyter-book/labels/bug +[link_helpwanted]: https://github.com/jupyter/jupyter-book/labels/help%20wanted +[link_enhancement]: https://github.com/jupyter/jupyter-book/labels/enhancement +[link_question]: https://github.com/jupyter/jupyter-book/labels/question + +[link_pullrequest]: https://help.github.com/articles/creating-a-pull-request/ +[link_fork]: https://help.github.com/articles/fork-a-repo/ +[link_pushpullblog]: https://www.igvita.com/2011/12/19/dont-push-your-pull-requests/ +[link_updateupstreamwiki]: https://help.github.com/articles/syncing-a-fork/ +[link_branches]: https://help.github.com/articles/creating-and-deleting-branches-within-your-repository/ + +[link_drafts]: https://help.github.com/articles/about-pull-requests/#draft-pull-requests + +[link_contributors]: https://github.com/jupyter/jupyter-book/graphs/contributors +[link_stemmrolemodels]: https://github.com/KirstieJane/STEMMRoleModels diff --git a/docs/_sources/content/first-order.md b/docs/_sources/content/first-order.md new file mode 100644 index 0000000..e64b6f3 --- /dev/null +++ b/docs/_sources/content/first-order.md @@ -0,0 +1,104 @@ +# Solutions to 1st-order ODEs + +## 1. Solution by direct integration + +When equations are of this form, we can directly integrate: + +\begin{align} +\frac{dy}{dx} &= y^{\prime} = f(x) \\ +\int dy &= \int f(x) dx \\ +y(x) &= \int f(x) dx + C +\end{align} + +For example: +\begin{align} +\frac{dy}{dx} &= x^2 \\ +y(x) &= \frac{1}{3} x^3 + C +\end{align} + +While these problems look simple, there may not be an obvious closed-form solution to all: + +\begin{align} +\frac{dy}{dx} &= e^{-x^2} \\ +y(x) &= \int e^{-x^2} dx + C +\end{align} + +(You may recognize this as leading to the error function, $\text{erf}$: +$\frac{1}{2} \sqrt{\pi} \text{erf}(x) + C$, +so the exact solution to the integral over the range $[0,1]$ is 0.7468.) + +## 2. Solution by separation of variables + +If the given derivative is a separate function of $x$ and $y$, then we can solve via separation of variables: +\begin{align} +\frac{dy}{dx} &= f(x) g(y) = \frac{h(x)}{j(y)} \\ +\int \frac{1}{g(y)} dy &= \int f(x) dx +\end{align} + +For example, consider this problem: +\begin{equation} +y^{\prime} = \frac{dy}{dx} = 1 + y^2 \\ +\end{equation} +We can separate this into a problem that looks like $f(y) dy = g(x) dx$, where $dy = \frac{1}{1+y^2}$ and $g(x) = 1$. +\begin{align} +\int \frac{dy}{1 + y^2} &= \int dx \\ +\arctan y &= x + c \\ +y(x) &= \tan(x+c) +\end{align} + +Unfortunately, not every separable ODE can be integrated: +\begin{align} +\frac{dy}{dx} &= \frac{e^x / 2 + 5}{y^2 + \cos y} \\ +(y^2 + \cos y) dy &= (e^x / 2 + 5) dx +\end{align} + +## 3. General solution to linear 1st-order ODEs + +Given a general linear 1st-order ODE of the form +\begin{equation} +\frac{dy}{dx} + p(x) y = q(x) +\end{equation} +we can solve by integration factor: +\begin{equation} +y(x) = e^{-\int p(x) dx} \left[ \int e^{\int p(x) dx} q(x) dx + C \right] +\end{equation} + +For example, in this equation +\begin{equation} +y^{\prime} + xy - 5 e^x = 0 +\end{equation} +after rearranging to the standard form +\begin{equation} +y^{\prime} + xy = 5 e^x +\end{equation} +we see that $p(x) = x$ and $q(x) = 5e^x$. + +## 4. Solution to nonlinear 1st-order ODEs + +Given a general nonlinear 1st-order ODE +\begin{equation} +\frac{dy}{dx} + p(x) y = q(x) y^a +\end{equation} +where $a \neq 1$ and $a$ is a constant. This is known as the Bernoulli equation. + +We can solve by transforming to a linear equation, by changing the dependent variable from $y$ to $z$: +\begin{align} +\text{let} \quad z &= y^{1-a} \\ +\frac{dz}{dx} &= (1-a) y^{-a} \frac{dy}{dx} +\end{align} +Multiply the original equation by $(1-a) y^{-a}$: +\begin{align} +(1-a) y^{-a} \frac{dy}{dx} + (1-a) y^{-a} p(x) y &= (1-a) y^{-a} q(x) y^a \\ +\frac{dz}{dx} + p(x) (1-a) z &= q(x) (1-a) \;, +\end{align} +which is now a *linear* first-order ODE, that looks like +\begin{equation} +\frac{dz}{dx} + p(x)^{\prime} z = q(x)^{\prime} +\end{equation} +where $p(x)^{\prime} = (1-a) p(x)$ and $q(x)^{\prime} = (1-a)q(x)$. + +We can solve this using the integrating-factor approach discussed above. Then, once we have $z(x)$, we can find $y(x)$: +\begin{align} +z &= y^{1-a} \\ +y &= z^{\frac{1}{1-a}} +\end{align} \ No newline at end of file diff --git a/docs/_sources/content/installing-jupyter.ipynb b/docs/_sources/content/installing-jupyter.ipynb new file mode 100644 index 0000000..d1ade9c --- /dev/null +++ b/docs/_sources/content/installing-jupyter.ipynb @@ -0,0 +1,115 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Installing Jupyter for Matlab\n", + "\n", + "This guide assumes you have Matlab already installed. I'll be assuming you have Matlab R2019_b installed.\n", + "\n", + "To use Jupyter notebooks with Matlab, you need to install Jupyter (which relies on Python) and the Matlab engine for Python.\n", + "\n", + "## macOS/Linux\n", + "\n", + "1. I recommend you [install Anaconda](https://docs.anaconda.com/anaconda/install/) to manage your Python environment—it makes installing and managing packages very easy. This comes with the Jupyter notebook.\n", + "\n", + "2. Create and activate an environment called `jmatlab` for Jupyter with Python 3.7:\n", + "\n", + "```bash\n", + "$ conda create -vv -n jmatlab python=3.7 jupyter\n", + "$ conda activate jmatlab\n", + "```\n", + "\n", + "Before the second command, you may need to tell your shell (e.g., bash, zsh) about conda, by doing `conda init zsh` for example. If you get an error with the `conda activate` command, your terminal should tell you to do this.\n", + "\n", + "You should activate this environment whenever you want to run Jupyter.\n", + "\n", + "3. Install the [Matlab kernel for Jupyter](https://github.com/Calysto/matlab_kernel):\n", + "\n", + "```bash\n", + "$ pip install matlab_kernel\n", + "```\n", + "\n", + "4. Install the [Python engine for Matlab](https://www.mathworks.com/help/matlab/matlab_external/install-the-matlab-engine-for-python.html):\n", + "\n", + "```bash\n", + "$ cd /Applications/MATLAB_R2019b.app/extern/engines/python\n", + "$ python setup.py install\n", + "```\n", + "\n", + "If you are using a different version of Matlab, then that path will be different.\n", + "\n", + "5. Run Jupyter notebook:\n", + "\n", + "```bash\n", + "$ jupyter notebook\n", + "```\n", + "\n", + "and create a new Matlab notebook with \"New\" then \"Matlab\" under \"Notebook:\".\n", + "\n", + "## Windows\n", + "\n", + "1. [Install Anaconda for Windows](https://docs.anaconda.com/anaconda/install/windows/)\n", + "\n", + "2. Open the Anaconda Prompt\n", + "\n", + "(to be continued)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Try it out\n", + "\n", + "Now try it out:" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "This is running Matlab!\n" + ] + } + ], + "source": [ + "disp('This is running Matlab!')" + ] + } + ], + "metadata": { + "file_extension": ".py", + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.7" + }, + "mimetype": "text/x-python", + "name": "python", + "npconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": 3 + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/docs/_sources/content/intro.md b/docs/_sources/content/intro.md new file mode 100644 index 0000000..eb8fb01 --- /dev/null +++ b/docs/_sources/content/intro.md @@ -0,0 +1,3 @@ +# Introduction + +This website is an interactive [Jupyter Book](https://jupyterbook.org/intro.html) for ME 373, Mechanical Engineering Methods, taught at Oregon State University. \ No newline at end of file diff --git a/docs/_sources/content/numerical-methods/error.md b/docs/_sources/content/numerical-methods/error.md new file mode 100644 index 0000000..5ec0948 --- /dev/null +++ b/docs/_sources/content/numerical-methods/error.md @@ -0,0 +1,32 @@ +# Error + +Applying the trapezoidal rule and Simpson's rule introduces the concept of **error** in numerical solutions. + +In our work so far, we have come across two obvious kinds of error, that we'll come back to later: + +- **local truncation error**, which represents how "wrong" each interval/step is compared with the exact solution; and +- **global truncation error**, which is the sum of the truncation errors over the entire method. + +In any numerical solution, there are five main sources of error: + +1. Error in input data: this comes from measurements, and can be *systematic* (for example, due to uncertainty in measurement devices) or *random*. + +2. Rounding errors: loss of significant digits. This comes from the fact that computers cannot represent real numbers exactly, and instead use a floating-point representation. + +3. Truncation error: due to an infinite process being broken off. For example, an infinite series or sum ending after a finite number of terms, or discretization error by using a finite step size to approximate a continuous function. + +4. Error due to simplifications in a mathematical model: *"All models are wrong, but some are useful"* (George E.P. Box) All models make some idealizations, or simplifying assumptions, which introduce some error with respect to reality. For example, we may assume gases are continuous, that a spring has zero mass, or that a process is frictionless. + +5. Human error and machine error: there are many potential sources of error in any code. These can come from typos, human programming errors, errors in input data, or (more rarely) a pure machine error. Even textbooks, tables, and formulas may have errors. + +### Absolute and relative error + +We can also differentiate between **absolute** and **relative** error in a quantity. If $y$ is an exact value and $\tilde{y}$ is an approximation to that value, then we have + +- absolute error: $\Delta y = | \tilde{y} - y |$ +- relative error: $\frac{\Delta y}{y} = \left| \frac{\tilde{y} - y}{y} \right|$ + +If $y$ is a vector, then we can define error using the maximum of the elements: +\begin{equation} +\max_i \frac{ |y_i - \tilde{y}_{i} |}{|y_i|} +\end{equation} \ No newline at end of file diff --git a/docs/_sources/content/numerical-methods/initial-value-methods.ipynb b/docs/_sources/content/numerical-methods/initial-value-methods.ipynb new file mode 100644 index 0000000..67c37e7 --- /dev/null +++ b/docs/_sources/content/numerical-methods/initial-value-methods.ipynb @@ -0,0 +1,491 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Numerical Solutions of 1st-order ODEs\n", + "\n", + "For numerically solving definite integrals ($\\int_a^b f(x) dx$) we have methods like the trapezoidal rule and Simpson's rule. When we need to solve 1st-order ODEs of the form\n", + "\\begin{equation}\n", + "y^{\\prime} = \\frac{dy}{dx} = f(x, y)\n", + "\\end{equation}\n", + "for $y(x)$, we need other methods. All of them will work by starting at the initial conditions, and then using information provided by the ODE to march forward in the solution, based on an increment (i.e., step size) $\\Delta x$.\n", + "\n", + "For example, let's say we want to solve \n", + "\\begin{equation}\n", + "\\frac{dy}{dx} = 4 x - \\frac{2 y}{x} \\;, \\quad y(1) = 1\n", + "\\end{equation}\n", + "This problem is fairly simple, and we can find the general and particular solutions to compare our numerical results against:\n", + "\\begin{align}\n", + "\\text{general: } y(x) &= x^2 + \\frac{x}{x^2} \\\\\n", + "\\text{particular: } y(x) &= x^2\n", + "\\end{align}\n", + "\n", + "## Forward Euler method\n", + "\n", + "Recall that the derivative, $y^{\\prime}$, is the same as the slope. At the starting point, $(x,y) = (1,1)$, where $y^{\\prime} = 2$, this looks like:" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "format compact\n", + "%plot inline\n", + "\n", + "x = linspace(1, 3);\n", + "y = x.^2;\n", + "plot(x, y); hold on\n", + "plot([1, 2], [1, 3], '--')\n", + "legend(['Solution'], ['Slope at start'])\n", + "hold off" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Remember that the slope, or derivative, is\n", + "\\begin{equation}\n", + "\\text{slope} = \\frac{\\text{rise}}{\\text{run}} = \\frac{\\Delta y}{\\Delta x}\n", + "\\end{equation}\n", + "\n", + "Let's consider the initial condition—the starting point—as $(x_i, y_i)$, and the next point in our numerical solution is $(x_{i+1}, y_{i+1})$, where $i$ represents an index starting at 1 and ending at the number of steps $N$. Our step size is then $\\Delta x = x_{i+1} - x_i$.\n", + "\n", + "Based on our (very simple) approximation to the first derivative based on slope, we can relate the derivative to our two points:\n", + "\\begin{equation}\n", + "\\left(\\frac{dy}{dx}\\right)_{i} = \\frac{y_{i+1} - y_i}{x_{i+1} - x_i} = \\frac{y_{i+1} - y_i}{\\Delta x}\n", + "\\end{equation}\n", + "Then, solve this for our unknown:\n", + "\\begin{equation}\n", + "y_{i+1} = y_i + \\left(\\frac{dy}{dx}\\right)_i \\Delta x\n", + "\\end{equation}\n", + "This is the **Forward Euler method**.\n", + "\n", + "Based on a given step size $\\Delta x$, we'll use this formula (called a *recursion* formula) to march forward and obtain the full solution over given $x$ domain. That will look something like this:" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear x y\n", + "x_exact = linspace(1, 3);\n", + "y_exact = x_exact.^2;\n", + "plot(x_exact, y_exact); hold on\n", + "\n", + "% our derivative function, dy/dx\n", + "f = @(x,y) 4*x - (2*y)/x;\n", + "\n", + "dx = 0.5;\n", + "x = 1 : dx : 3;\n", + "y(1) = 1;\n", + "for i = 1 : length(x)-1\n", + " y(i+1) = y(i) + f(x(i), y(i))*dx;\n", + "end\n", + "plot(x, y, 'o--', 'MarkerFaceColor', 'r')\n", + "\n", + "legend(['Exact solution'], ['Numerical solution'], 'Location','northwest')\n", + "hold off" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Another way to obtain the recursion formula for the Forward Euler method is to use a Taylor series expansion.\n", + "Recall that for well-behaved functions, the Taylor series expansion says\n", + "\\begin{equation}\n", + "y(x + \\Delta x) = y(x) + \\Delta x y^{\\prime}(x) + \\frac{1}{2} \\Delta x^2 y^{\\prime\\prime}(x) + \\frac{1}{3!} \\Delta x^3 y^{\\prime\\prime\\prime}(x) \\dots \\;.\n", + "\\end{equation}\n", + "This is exact for an infinite series. We can apply this formula to our (unknown) solution $y_i$ and cut off the terms of order $\\Delta x^2$ and higher; the derivative $y^{\\prime}$ is given by our original ODE.\n", + "This gives us the same recursion formula as above:\n", + "\\begin{equation}\n", + "\\therefore y_{i+1} \\approx y_i + \\left( \\frac{dy}{dx}\\right)_i \\Delta x\n", + "\\end{equation}\n", + "where we can now see that we are introducing some error on the order of $\\Delta x^2$ at each step. This is the *local truncation error*. The *global error* is the accumulation of error over all the steps, and is on the order of $\\Delta x$. Thus, the Forward Euler method is a **first-order** method, because its global error is on the order of the step size to the first power: error $\\sim \\mathcal{O}(\\Delta x)$.\n", + "\n", + "Forward Euler is also an **explicit** method, because its recursion formula is explicity defined for $y_{i+1}$. (You'll see when that may not be the case soon.)\n", + "\n", + "In general, for an $n$th-order method:\n", + "\\begin{align}\n", + "\\text{local error } &\\sim \\mathcal{O}(\\Delta x^{n+1}) \\\\\n", + "\\text{global error } &\\sim \\mathcal{O}(\\Delta x^{n})\n", + "\\end{align}\n", + "(This only applies for $\\Delta x < 1$; in cases where you have a $\\Delta x > 1$, you should nondimensionalize the problem based on the domain size such that $0 \\leq x \\leq 1$.)\n", + "\n", + "Applying the Forward Euler method then requires:\n", + "\n", + "1. Have a given first-order ODE: $\\frac{dy}{dx} = y^{\\prime} = f(x,y)$. Complex and/or nonlinear problems are fine!\n", + "2. Specify the step size $\\Delta x$ (or $\\Delta t$).\n", + "3. Specify the domain over which to integrate: $x_1 \\leq x \\leq x_n$\n", + "4. Specify the initial condition: $y(x=x_1) = y_1$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's do another example:\n", + "\\begin{equation}\n", + "y^{\\prime} = 8 e^{-x}(1+x) - 2y\n", + "\\end{equation}\n", + "with the initial condition $y(0) = 1$, and the domain $0 \\leq x \\leq 7$. This is a linear 1st-order ODE that we can find the analytical solution for comparison:\n", + "\\begin{equation}\n", + "y(x) = e^{-2x} (8 x e^x + 1)\n", + "\\end{equation}\n", + "\n", + "To solve, we'll create an anonymous function for the derivative and then incorporate that into our Forward Euler code. We'll start with $\\Delta x = 0.2$." + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "\n", + "f = @(x,y) 8*exp(-x)*(1 + x) - 2*y;\n", + "\n", + "dx = 0.2;\n", + "x = 0 : dx : 7;\n", + "n = length(x);\n", + "y(1) = 1;\n", + "\n", + "% Forward Euler loop\n", + "for i = 1 : n - 1\n", + " y(i+1) = y(i) + dx*f(x(i), y(i));\n", + "end\n", + "\n", + "x_exact = linspace(0, 7);\n", + "y_exact = exp(-2.*x_exact).*(8*x_exact.*exp(x_exact) + 1);\n", + "plot(x_exact, y_exact); hold on\n", + "plot(x, y, 'o--')\n", + "legend('Exact solution', 'Forward Euler solution')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Notice the visible error in that plot, which is between 0.2–0.25, or in other words $\\mathcal{O}(\\Delta x)$.\n", + "\n", + "How can we reduce the error? Just like with the trapezoidal rule, we have two main options:\n", + "\n", + " - Reduce the step size $\\Delta x$\n", + " - Choose a higher-order (i.e., more accurate) method\n", + " \n", + "The downside to reducing $\\Delta x$ is the increased number of steps we then have to take, which may make the solution too computationally expensive. A more-accurate method would have less error per step, which might allow us to use the same $\\Delta x$ but get a better solution. Let's next consider some better methods.\n", + "\n", + "## Heun's method\n", + "\n", + "Heun's method is a **predictor-corrector** method; these work by *predicting* a solution at some intermediate location and then using that information to get a better overall answer at the next location (*correcting*). Heun's uses the Forward Euler method to predict the solution at $x_{i+1}$, then uses the average of the slopes at $y_i$ and the predicted $y_{i+1}$ to get a better overall answer for $y_{i+1}$.\n", + "\n", + "\\begin{align}\n", + "\\text{predictor: } y_{i+1}^p &= y_i + \\Delta x f(x_i, y_i) \\\\\n", + "\\text{corrector: } y_{i+1} &= y_i + \\frac{\\Delta x}{2} \\left( f(x_i, y_i) + f(x_{i+1}, y_{i+1}^p) \\right)\n", + "\\end{align}\n", + "\n", + "Heun's method is second-order accurate, meaning the global error is $\\mathcal{O}(\\Delta x^2)$ and explicit.\n", + "\n", + "Let's see this method in action:" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Maximum error: 0.055" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "\n", + "f = @(x,y) 8*exp(-x)*(1 + x) - 2*y;\n", + "\n", + "dx = 0.2;\n", + "x = 0 : dx : 7;\n", + "n = length(x);\n", + "y(1) = 1;\n", + "\n", + "% Heun's method loop\n", + "for i = 1 : n - 1\n", + " y_p = y(i) + dx*f(x(i), y(i));\n", + " y(i+1) = y(i) + (dx/2)*(f(x(i), y(i)) + f(x(i+1), y_p));\n", + "end\n", + "\n", + "x_exact = linspace(0, 7);\n", + "y_exact = exp(-2.*x_exact).*(8*x_exact.*exp(x_exact) + 1);\n", + "plot(x_exact, y_exact); hold on\n", + "plot(x, y, 'o--')\n", + "legend('Exact solution', \"Heun's method solution\")\n", + "fprintf('Maximum error: %5.3f', abs(max(y_exact) - max(y)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Notice how the error is visibly smaller than for the Forward Euler method–the maximum error is around 0.05, which is very close to $\\Delta x^2 = 0.04$.\n", + "\n", + "## Midpoint method\n", + "\n", + "The midpoint method, also known as the modified Euler method, is another predictor-corrector method, that instead predicts the solution at the midpoint ($x + \\Delta x/2$):\n", + "\\begin{align}\n", + "y_{i + \\frac{1}{2}} &= y_i + \\frac{\\Delta x}{2} f(x_i, y_i) \\\\\n", + "y_{i+1} &= y_i + \\Delta x f \\left( x_{i+\\frac{1}{2}} , y_{i + \\frac{1}{2}} \\right)\n", + "\\end{align}\n", + "\n", + "Like Heun's method, the midpoint method is explicit and second-order accurate:" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Maximum error: 0.050" + ] + }, + { + "data": { + "image/png": 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Tp5qe6dWrF/Vn+/jx4//5z3++9tprjY2Nzz77LNXyQAgZMGDAuXPntm/f/uTJkwULFrzxxhv29vYfffSRXC5/8uQJl8v94osvvL29S0tLqfO0NAj461//Onfu3NjY2A0bNph+dEt19urVa+zYsdHR0QEBAY2NjYMGDUpISLCzszOd5C9/+UvTV40bN65pYd7e3s2+L0pUVNT06dOdnZ3t7e3d3d2Tk5OlUmkr727UqFETJ04cNmxY//79Q0JCLN6av78/dVizlRQWFrb1/09shu0nek5sumb/1VKJM2+hr3u4r8jiu6Y7SLQ6w4ErpQP53Kit22ipE+jCtH+x7VVfX6/T6UwXjc6ePRseHl5SUnL//n0nJyfq+gflyZMnZWVlbm5u3ToN1dDQUFVV5eLi0vaXNFuYxfuyUF5e/uTJE/N7Wlt/dw8ePHjmmWfMP422V/JUbN9+AiOknlCjVKTs+SxyxpTIQYQQQjSE+Fr+GUvNrvCkMgEhArnhi7TLmgg/r8Scnq8WoGPs7e2b/a3t7u5u8YydnV3TJ7tc796925VGpIXCWnpfFItrYC2dxKSNyzH0zEfENAikbpeprm78bNOI5TsE4wOpZ4xlxRZhY3EHiYTP/WD+K/erZW8vjiwYsXhfqA81uQfAIhMmTLDobQNoHVsDSa/XnzhxQqPRuLm5zZo1q5W/X+iVqa5O+t8N744LcP8tjYjZ7Yr9pGOMZcWEkBqlwjFATt3AaCwr5gjF1J32s76fJvCOCkjM3Rf6sr+3E21vAwCg+7EykBoaGubNm6fT6fz8/DIyMvbt2/fNN980HTgzwaJjNzL6XBHIT1g8L5BHad7y00uz7V09CSE8qYzKJEJIP+kYKpn0quyhIQsCgr1efd45Nl1z3tspJtir598CAEDPYGUgnT9//ubNm2fOnBGJRHfv3p0wYYJSqZTLn3L3Xw/LVFcvOnZDEz2mRhnVtPv2wa5V1BiIemgsKy6JmdNPOoaataPGT3WqLLflOwgh/t5OklCfgMRcbZUBPXjWLSEhge4SgMUY3rPwVKwMpP79+7/55psikYgQwuPx7OzsHBwc6C7qD7Q6Q8Cnucq3hpPf0kWvyjZdIqJuV/SM/X3YRN1BQoUQ9Ux9eYn78vgHu1bxpDKBPErC52qix8SmaxYdK0QmWSu2/zYB6CQWt33rdLpdu3YplUpvb++EhIS+fftaHPDGG2/4+f261HEP/0894NPrMcFe5ld9zBe41KuyBfIo80UwW2IsK67JVNQoFZ6xJzhCMdURvv9qqSZ6TLfUDQBWxzTszsnJOXToEL3FtI6VIyTKkydP+vTpIxKJCgsL8/PzfX19LQ7w8/Oj5U/O2HSNv7eTRQ9Cx25XpLobyG8rkknkUQt9RYQQr01ZyCQAaAsWjbxZGUh6vZ4Q4uLisnbtWkLI3/72tz179jQNJFpkqqsz1dXKt4Z14TmpjW1KYuYQQviu4sVZimmiYW8vPkL1OHTfnfkAAD2JlWvZbdq0KTQ01PTQ29v77t27NNZjLuDT3O64xsMRiqlrTvd3rXT0l49YFi2QR61+MqWrdmADAKAdKwNp1KhRRUVFX3zxRUVFxcWLF5OTk2Wyp1+P6QEBn15/L8irm25i5QjFhJB+0jE1mQpCSEywl8SZ+7/X67tkBzYAANqxcspuxowZP/74Y3x8/NatW+3s7KZOnbpq1Sq6i6Im66q6drLOAtXdYHq40Fe06Fih0SlksXINJu4AgO1YOUIihKxZsyYvL++7777Ly8vbtm0bj8ejuyISm66hJuuolYHMt6XpKgJ5VKUijhoqEUIkfO6+UJ+Xvo3NfTao9RcCADAfWwOJENK7d283N7c+ffrQXQghhFC7GVFreD/YtYpaJrXLf0rTHdjcSq+PdahNeCYE2ykBANuxcsqOUahtI5JK3E8Ee1UqMioVcW28x6gDmt4/q1dlu0fG7xP5BCTmKiOGYw1WAGAvBFJnVSrivvVbW+DmJZAPo5b/qVEquvWKTtOTSwiJCfJadKywW69gAQB0KxZP2TEBtW3ElrvuMcFeVKubZ+wJWtrewn1F/t5OsemaHv65AABdBYHUKdRgyN/b2d/bydQCJ5BH1SgVPV/MQl9Rprp6/5XSnv/RAACdh0DqFIE8avv/WxMT5HU/YaVpJu3BrlXURhI9jGq6O3DlvlZn6PmfDgDQSQikTrnM9RllKHQrvV6T+etuRtRK3nTdFSThc5VvDQtIzKXlpwMAdAaaGjrl8G37V6O+9VFu5riKTbvqWWx91PPCR4qwSwUAsA4CqVP2XyndF+pTcrKkm2486hhqBYdMdTV2PQcAFsGUXcfFpmveC/IylhUby4qZk0Y1SkVj9Kv7Qn0WHbuBi0kAwCIIpI7bf7V0oa+IuhOW7lp+5xgg50lljdGvnhAo/+/DDZWKOCy9CgCsgEDqoP1XSv29nSV8rvne5AxBrS0k4XNPeoUlPBOCLSoAgBUQSB104Mr9hb6iGqWCJ5WZVjtlAupeXc/YEzVKxQmXjEx19cWfHbBFBQAwHwKpI6iVTP29nepUWf2kzNpKnLpXl9rQr0apWOjrvujYDbru1QUAaDsEUkccuFK60NedEGK6/Yg5qC0qCCEcodg9Mv7Vva/5ezvnbFzGtDoBACwgkDpi/5XScF9RjVLh6M+43/LmW1TwpDKeVLb05sfG8pKCEYvpLg0AoDUIpHaj0ogQUpOpeCbgdbrLsURtUVGnyjL11/HzUgpG/G3RsRt0lwYA0BrcGNtu59XVr3o7EULqVFme0hNPPZ4W5p3o/aRjpivirr8aFJuuiQn2orEqAIBWYITUbpnqKn9v5/sJK92X07xEUBtRdyZFVidlqquxsSwAMBYCqX1Mtx+5R8azqE1AII9yK70u76fBxB0AMBYCqX1M83XswhGKHf3l0zVHJM48bJgEAMyEQGofar6O7io6ghrPbbNLiz2twRp3AMBACKR2MM3X0V1IB7kt38HPS4kexok9jZ3OAYBxEEjtwNL5OhOOUCyQR7106j2tzoDuBgBgGgRSO7B3vs7EMUAu4fO22aWhuwEAmAaB1FZsn68zoSbuwgbWLzpWSHctAAC/QyC1Fdvn60yoibtlNz/BxB0AMAoCqa2sYL7OhOq4m/XoO0zcAQBzIJDayjrm60zclu+YfuuIR30FbksCAIbAWnZtYn2/tamJu2OZB7fsPj1d8+sCd4zaix0AbA0CqU0OXLlvfcuSViri3CPjBSXuq6sM+0J9jGXFmgg/r8QcuusCABuFKbs2yVRXDb9z+n7CSroL6TLUTuc8qWyhr4jqbuAIxdjpHABohEB6ul+342Pk7kcdRu10TgiR8LkxwV5UdwN2OgcAGiGQno5q+K5TZfGkMrpr6TKmnc4JIf7eTtSiqw92rWLREuYAYGUQSC2qUSpKYuZUKuIczux6de9rDNytvDPMdzonhOwL9TmVkmYsK0ZfAwDQBU0NLaKu+fOksp05GavqrjwT8Lo1XfOndjqvVMTVqbIIIQ6ErKjO+NZvbQTdhQGAzUIgNc90zX//ldJZtRd4Q2TUw0pFnDWNIczfi+eE5f84Vuijrva3igUpAIB1MGXXPNM1//Pqar9fCvtJxxBrv+Yv4XMX+rrHpmNnCgCgB0NHSI8fP05OTv7pp59cXFxee+01d3d3iwNyc3Pz8vJMDwMDAwcOHNiFBVDX/N0j4zPVVQsMhdSlfqu/5h/uKzpw5T7VVUh3LQBgc5g4QmpoaJg/f358fHxdXd2///3vKVOmaLVai2OOHDly9OjRS78pKyvr2hpM1/y1OoNHfTkhRK/KtoVr/vtCfbB9HwDQgokjpIyMjIKCgtTU1Oeee06v10+aNOmrr75as2aN+TFFRUVLly6dPXt2N9VAXfO/tnvTR5UFHFdxpSJOr8p2j4zvph/HHBI+19/bedGxwn2hPnTXAgC2hYmBVFlZOXLkyOeee44QwuPxRCJRVVWV+QFGo/HWrVtOTk5paWnOzs4jR47kcDjdUUmCU8hg1ypHX1+rHxiZGMuK/3k+/A336Ey1CN0NANCTmBhIoaGhoaGh1NdXr17Nz8+fO3eu+QFqtbqhoWHlypUeHh4lJSVisfjQoUMuLi4W58nJ+b1FOzIysgOVZKqrljrWclzFHXgtS1GLrk4/cCQ2Xez/1jC6ywGAzkpISKC+yMnJ6dhvwh7DxECiNDY2Hj16dPPmzZMnT54xY4b5t+rr62fOnLlixQqxWHz79u3XX389ISHhvffesziDn59fZz59rc5ACHG7d50nje7wSdiIJ5WFDVQkqbIy1V4YJAGwHcNDyBwTmxoIIWVlZQsWLNi+ffu6deu2bdvWq1cv8+8OHjx4y5YtYrGYEDJw4MBp06bl5+d3eQ3UjnzG8mKO0IZGSOS3QdJHlZ9j+z4A6ElMDCS9Xj9//nxCyKlTp+bOnWuRRoQQhUKxdetW08Pa2truuIZ0Xl39sVuBla0Y1EY8qWzQi4OG3TljfRtBAQBjMTGQjh8/Xl5evn79+traWo1Go9FoKisrCSHR0dGHDx8mhDg6Ou7duzcpKenRo0cZGRmpqalBQUFdXkamuqpOlUXdEmuD3JbvWPEw6Yu0K3QXAgC2gonXkK5du1ZXVzdr1izTMwsWLIiOjk5JSamrqwsLC5s0adKNGzdiYmLWr1/ft2/fsLCw8PDwLi9DqzPoS7Jtp7/OAkcoHhqyYHrakUXHBqEFHAB6ABMDaefOnc0+X1BQYPr6nXfeefvtt3U6nUAg6N27d5fXQK1WYDxhcxeQzDn6y+VKxTpVFlrAAaAHMHHKro3s7e2FQmF3pBEh5Ly6epSh0Gbn6yhUd8Oym59ggTsA6AFMHCExQaa6apVPrb2rJ92F0M+jvnxx1uprdoESPpf8cYFwAIAuxOIRUrfS6gxDQxbYwlpBratUxLkvj/eor5hTGSCQRzn6yzURfnQXBQDWCYHUjEx1tb+3M91V0I/aFMoxQD40ZMGym5/sv1LKEYqpTaHoLg0ArBACqRlanZ6anrJxpk2hBPKosQ4/n0pJI9a+KRQA0AiB1Izz6upX0VT226ZQ1NdDQxZO1xxZdKzQ6jeFAgC6IJCaQS0aRHcV9DNtCkUIcQyQ+3s7D762xxY2hQIAWqDLrhlanQFTduS3TaEqFXF1qixCiL2r5/TMI7vHbNtMd2EAYJUQSJawgbcFi/GQ/npWpnoK7pMFgC6HKTtLt3UGiTOGR80TyKNmPbqw4cvv6C4EAKwQAsmStsowEPN1LeAIxSOWRk/XHMEq4ADQ5RBIltDR0DqeVCbnaagWcACALoRAsoSOhtZRnQ79pGMWHSukuxYAsCoIpD9AR0MbxQR5ZaqrqF3eAQC6BALpD6hbYn+a8ye6C2E6CZ8bE+QVexqrgANAl0Eg/UGmump48Rnb3La8vcJ9RZnqqkx1Nd2FAICVQCD9gVZnEN7LtfFtkNouJsgLWyUBQFdBIP2OWuRbr8rmSWV018IO1PU2DJIAoEsgkH5HLfJtLLfpbcvbKybYa9GxG3RXAQDWAIH0O2xb3gH+3k7+3s6YuAOAzkMg/YGxvBjblrdXTJDX/qulaAEHgE5CIP0uU12FEVIHSPjc8JEitIADQCchkH6n1Rnc7l1HR0MHLPQVYYQEAJ2E7Sd+Ra3RYDyBjoZ2q1Eq7DMV20TD4tYcpvrusIMfAHQARki/e6hU4JbYDqhUxAnkUYNeHORWer1gxGJHf7kmwo/uogCAfRBIvzqvrvZ/HpvOtVulIo4nlVH/+Xs7J+7/P45QzJPKKhVxdJcGACyDQPoVOho6pkapoCboOEKxQB617OYn+6+UCuRRNUoF3aUBAMsgkH6FjoaOEcijTIMhnlQ26MVB+UkHH+xa5RiAyU8AaB8EEiG/LRqENRo6gCeV6VXZelU29dBt+Y7pmiM3f7yJvgYAaC902RFCiFand85LQUdDB1D79VUq4upUWdQzEmfuZa5Pprra3xvX5ACgHRBIhBByXl091vFnDI86zHw85Ogv5+xatfPfaf7vzKOxJABgHUzZ/aq+vITjikDqAhyh2NFfPiR3D1YBB4B2QSAR8luLHToauopjgJxqAae7EABgEwQSIYRodQaP+nJM2XUht+U7qBZwugsBANZAIP26aNALJ+7RXYhV4QjFPuMCkXEOcQAAIABJREFUcHssALQdAgm6i0AeNbz4zIYvv6O7EABgBwQSOa+ufhUNyt2AIxSPWBo95NpeLAQOAG2BQCKZ6ip/b2e6q7BOPKlslKHwVEoa3YUAAAuw9T6kx48fJycn//TTTy4uLq+99pq7u3uHT6XVGSR8bhfWBiYcodg9Mv6ljcsyxwfiPlkAaB0rR0gNDQ3z58+Pj4+vq6v797//PWXKFK1W2+GzYXjUragF7lK++IzuQgCA6VgZSBkZGQUFBV9++eX//u//JiUlOTg4fPXVVx0+G4ZH3Y1a4A4t4ADQOlYGUmVl5ciRI5977jlCCI/HE4lEVVVVHTtVbLpG4oxA6l5UCzjvyHq6CwEARmPlNaTQ0NDQ0FDq66tXr+bn58+dO7fpYTk5OaavIyMjmz2VtsqAFrseIJBHCd+elrg/KSI8hO5aAGxLQkIC9UVOTk5LvwkZgpWBRGlsbDx69OjmzZsnT548Y8aMpgf4+fk99dPPVFfFBHl1T4HwO6oFnJO0B90NAD2M4SFkjq2BVFZWFhUVVVhYuG7dutDQ0F69erXr5TVKRU2mgieVTb+lcTibXfnH9aqh+1zb/cEQXxH1NT5zADDHykDS6/Xz5893c3M7deqUq6trB85QqYhzj4x/IBr2zc3cT+RjjGXFmgg/r8Scp78SOqpSESeY9+HJAuco+TBCCD5zALDAyqaG48ePl5eXr1+/vra2VqPRaDSaysrKtr+8UhHHk8p4UlmmumpvQYSxrJgjFPOkMiy81n2oz9xnfGBMsJfXpixCCD5zALDAykC6du1aXV3drFmzJv1m9+7dbX95jVJhmiwyLfItkEfVKBXdUi6Yfeb+3k4SZ15suobgMweAP2LllN3OnTs783KBPIqastOrsvtJx1BPPti1yjEAW5h3F9NnTgjZF+oTkJi70Fdkj88cAMywcoTUSTypTK/K1quyjeXF9q6ehBC9KttYVoxr7N3H9JkTQiR8bvhIUeL+JHzmAGCOlSOkTuIIxV6JOZWKuOF3Mur5vEpFnF6VTf3xDt3E9JnXqbIIIdN0hsLcDMPSD+muCwAYxBYDiSKQR11MuzI0JASzRj3GNB4SEFIw4m9TTms04+mtCAAYxBan7CiZ6mpsW06jcF+RxJmHBe4AwMR2A0mr03vUV1DXkIAWyreGxZ7W0F0FADCF7QbSbZ0BIyTa+Xs7LzpWSHcVAMAIthtIxvJijivSiGYxQV5anSFTXU13IQBAP9sNpDpVNk8qo7sKWyfhc48+OYD1GgCA2HIgaXUGuksAQggRyKP4eSlnswvoLgQAaGajgaTVGfx+KTQt0wA0onamwCAJAGw0kDLVHdxhFroDTyob61ibuD+J7kIAgE42GkiEkFGGQlxDYgiOUCyQR03K2YzuBgBbZqOBdB6/+BiG2hAEE3cAtsxGA0mLm5CYB90NADbORgNJWJqLjgamobobuEfW0V0IANDDRhdXNZaV2A/BokFMJCy9fnxV+ATZEOoh9qcAsB22OELKVFd7Yr6OkSoVcQJ51ChDYcGIxQJ5lKO/XBPhR3dRANBDbDGQtDr9zxOX409vpqlUxPGkMoE8ytTdwBGK0ekAYDtsMZBu6wwSZy7dVYClGqWC+iuB6m6gdqYQyKNqlAq6SwOAnmCLgaStMgzkI5AYRyCPMg2MRiyNfv7gEkLIg12rsIMigI2wxUDKVFeF+4rorgIs8aQyvSpbr8omhDgGyCV83qWNy4xlxZhcBbARNtplBwzEEYq9EnMqFXF1qixCiL2rJz9TIVgeT3ddANBDbDGQsM43k5mPh/pJx5xKSXsdU3YAtsHmpuwy1dX+3s50VwFtQi26eiolje5CAKAn2FwgaXV6CToaWIIjFLst3/HSqVgMagFsgc0FEnq+2YUjFDsGyO/vWkl3IQDQ7WwukNDzzTqO/vL6suLCCxl0FwIA3cv2AklnkPB5dFcB7cARij1j/2/Kd/gzAsDK2V4gVekxZcc6Ej43fKRo0bFCugsBgG5kc4EkvHfd/uP5dFcB7bbQV6TVGbClLIAVs61A0uoMHvUV9q7YeIJ9JHxuTLDXomM36C4EALqLbQVSproKG0+wl7+3k7+3MybuAKyVbQUSIcSjoYLjikBiq5ggL9yTBGCtbCuQbusMHhghsZmEz13o6+61KYvuQgCg69lWIGmrcA2J9cJ9Rf7ezrHpGroLAYAuZluLq2aqq9ZjhMR+H7sVpOz57Jom0LQKFLaoALACthVIHvW4gGQNKhVxP09Yu7rOSykfRggxlhVrIvy8EnPorgsAOsW2puyE967zpDK6q4BOqVTE8aSyiPAQQgg1cccRinlSGbXbLACwF6NHSDdu3Dh37tyKFSuafis3NzcvL8/0MDAwcODAga2fDd1Z1qFGqfCMPUEI2RfqE5CYu9BXJOFzBfKokpg5mLgDYDXmjpAqKiref//99PT0Zr975MiRo0ePXvpNWVnZU0+Im5Csg0AeRQ2GJHzu1UHZ1ELgD3atcsQ+fgAsx9ARUkhISFFRUUNDw6BBg5o9oKioaOnSpbNnz27XaT0aKjiur3RFgUAbanZOr8rmSWWO/nI35Zz7CSuNZcXUsAkA2IuhgbRx48bHjx+npKRcvXq16XeNRuOtW7ecnJzS0tKcnZ1HjhzJ4XCees7z6uoFhkJcQ2I7jlDslZhTqYirU2URQnhSWU2mwn15PN11AUBnMTSQpFIpIeTatWvNBpJarW5oaFi5cqWHh0dJSYlYLD506JCLi4vFYTk5v7ddRUZGdmvB0MPMLxdxhOKaTAWm7ACalZCQQH2Rk5PD8N+EDA2k1tXX18+cOXPFihVisfj27duvv/56QkLCe++9Z3GYn5+f+aePm5CslaO/XK/KrlTEoakBoCmGh5A55jY1tGLw4MFbtmwRi8WEkIEDB06bNi0/P/+prxLeu95POqb7q4OexhGK3ZbvqFEq9KpsumsBgI5jZSApFIqtW7eaHtbW1rblGhIWDbJiHKFYII+6n7CS7kIAoOPYFEjR0dGHDx8mhDg6Ou7duzcpKenRo0cZGRmpqalBQUGtvxYbu1k9nlTGEYpxeywAezE9kHr16mX62tR0N2nSpDfffDMmJmb48OErV64MCwsLDw9v/Txand7vl0JM2VkxTNwBsB2jmxqWLFmyZMkS08OCggLT1++8887bb7+t0+kEAkHv3r2feqrbOgO3W2oEBjFN3GFdOwA2YvoIqRX29vZCobAtaUQI0VYZ3JbvQGew1XMMkPOkMlxMAmAjFgdSu2h1BgmfR3cV0BME8ii9KhsTdwCsYzOBVKWXOGPSziZwhGL3yHgMkgBYx2YCSWcwbeYGVo8nlTkGyJFJAOzC6KaGLoQ0sjUcV3GlIq6+vMS0eiHWcQBgOJsYIWWqqyXOuIBkW6iVhIxlxY7+coE8ytFfronwo7soAGiNTQSSVqfHCMmmULvKCuRRjgHyB7tWEewqC8AGNhFIt3UGdDTYlBqlgpqgo/4vlUMCeVSNUkFzZQDQMpsIJG2VYSBGSLbEtKssIcS0fAN2lQVgOJsIJLA1PKnMdCuSafkGY1kx+hoAmMwmuuwy1VUxQV50VwE9x2JXWULIXXuXghGLI+gtCwBaZROBBLbJfDzkOWH5P44V8q6UhvuKaCwJAFphE1N2w+6c4R5ZR3cVQCcJn7sv1Cf2tEarM9BdCwA0z/oDSaszYOMJIIRI+NyYIK+AxFy6CwGA5ll/IGWqq+guAZgi3FcUPlK06Fgh3YUAQDOsP5AIIaMMhab1Y8DGxQR7aXUG7CAMwEDWH0i3dQaP+nKOUEx3IcAU+0J9Fh27gYtJAExj/YGkrcLvHfgDXEwCYCbrDyThvevoaAALuJgEwEDWH0jG8mJ7V0+6qwDGWegr0uoM+6+U0l0IAPzK+m+MrS8r5gzDMg1gScLn/tvnZsrHq6+NCzQtBo+1hQBoZP2B5NFQwXF9he4qgIkqFXGz18e/nNyoWTaGEGIsK9ZE+Hkl5tBdF4CNsvIpO63OgJ5vaBa1ZxJPKrs6KLskZg7BnkkAdLPyQMJdsdAS055Jjv5ygj2TABjABqbscBMSNIfaM8k9Mp4jFLst30ENkvSqbOyZBEAXKx8h1ZeXcFyRRtAMiz2TPGNP1CgV2DMJgEZWPkIS3svFBSRoVtM9kzhCsbGs2FhWjCE1AC2sPJC0VYZRPLqLAAazGA/lJx0siZmDRjsAWlj5lJ0blmmA9nD0l+c+G3Q/YSXdhQDYIisPJCxkB+0i4XNnLP77f3+8ieZvgJ5n5YGEm5CgvThCsd8/ducnHUQmAfQwaw6kzP9WoecbOoBquiu8kEH14AFAz7DmQNJWGT6cc47uKoCVBr04qGjye/cTVhrLiumuBcBWWHMgAXTGkim+P09cTt0wCwA9wJoDSaszSJy5dFcBLIamO4CeZM33IRl5/IF8BBJ0nITP9Vj892ufbdJH+JkvKYTVHAC6gzUHUn0/gYSP22KhUzhCsdu969TXVA5hlwqAbmLNU3ZGngBTdtBJ1C4V5k132KUCoJuwO5Bu3Lixc+fOlr5r7CeQYMoOOofapYIjFP93wRempjvsUgHQHVgcSBUVFe+//356ejrdhYA1o3apIISE+4oE8qiSmDnGsuIHu1ZhlwqALsfWa0ghISFFRUUNDQ2DBg2y+FaNUlGTqeBJZdvsSisV1wkuQUMnULNzelU2TypzDJBrqwxUI7hn7Am6SwOwNmwNpI0bNz5+/DglJeXq1asW36J2XeNJZT88KRTIfXAJGjrDYpcKD0Jq6C4JwFqxNZCkUikh5Nq1axaBRF2Cptavy7l8OaHiHCHk2Ue9HRVxGCdBh5n/4xHIoxYdK5Ska2KCvWgsCaCNEhISqC9ycnIiIyPpLaZ1bA2kltQoFaa5lLee+5n69I1lM0ti5iCQoKvEBHktOlZIkEnABgwPIXMsbmpolukStDlcgoauJeFz94X6ZKqrY9M1dNcCYD2sLZB4UplelW2+SLNelW0sK8bwCLoWMgmgy1nblJ35JWhv7eVKxS96VbZ7ZDzddYEVojJp0bHCV9XV/t5OdJcDwHqsD6RevXo1fZIaD6nLEibLWTN5CmxkyiQS7IVMAugkdgfSkiVLlixZQncVYNOoTHpn/ebn3QvMtyfGLDFAe7E7kACYQMLnxtmlRT35++YJYdRqVbj7DaADrK2pAaDnUXe/bV4VFpCYq9UZCBZgBegQBBJAZ1ELsEr4XE30GFMmYQFWgPZCIAF0lvndbzFBXlQm4e43gPZCIAF0lvndb+G+opggr7U7DtepstDXANAuaGoA6CyLBVinEzLBLlsrGibACooA7YFAAugaFtlTlF1QeGSdD/q/AdoMU3YA3WKCbIhh3keFFzLQawfQRggkgO5CZVJ+0kFkEkBbIJAAutEE2ZBxn5w8lZKGTAJ4KgQSQPfiCMWvb9+PTAJ4KjQ1AHQ7KpM+2frx66o5hBAseQfQLAQSQE/gCMVvr/mf79+e5jMuwBRCWPIOwBym7AB6SE2mYtCLgw7ftr+fsJJ6BkveAZhDIAH0kBqlwm35jtnrPzh8274kZg71JJa8AzBBIAH0EGrJOwmfO3v9BzypTBPhZywrxpJ3ACYIJIAeYlryTsLn/jxhuWOAvCRmjrGsGH0NABQ0NQD0EPMl7xwIIYTctXchhDhiyTsAQggCCaCHmWfPOHnU2eyC/I/nD0X/NwCm7ADoRS3lsP9Kqan1DsBmIZAAaEbdonT4tj3V5kB3OQC0QSAB0I8jFM9e/4HuzzOoNge6ywGgB64hATCChM8l8qjCPw0zvuXXTzoGywuBDUIgATCFhM+VzJiiOSWOejJ5steUcF8RwfJCYEswZQfAIJWKOJ5UtnlV2IEr92PTNQTLC4EtQSABMEiNUiGQR0n43H2hPvuvllKZhOWFwEYgkAAYhFpeiBAi4XOVEcMJIV6bsrC8ENgIBBIAg5iWFyKESPjcmGCv/+dx/+aPNxOeCaG7NIBuh6YGAAYxX16Iema6Jvvn5Tv2f1dKCIkJ9qK1OoDuhUACYJymfd5KqeHAlVKvTVnKiOESPpeWqgC6G6bsAFiAmr4LHykKSMylOh0ArA9GSACsERPstdBXFJCYSwiJfJikV2Xj/lmwJggkADahuu8OXCmtVMQ5+stNIYT7Z8EKYMoOgGUkfG7kw6Qy0bDC75WmG2Zx/yxYAQQSAPvUKBV+/9g97pOTmepqTYQf1SaO+2eB7TBlB8A+1P2z7pHxI5ZGn700w3vjkqEhC/SqbNw/C6yGERIA+5jun5XwuUum+I775OTZ7ALTrUsALIUREgD7NL1/dqxjbe6kGIekT33KigXyKI5QTG+FAB3A0EBqaGhITk4uLCzk8/nTp08Xiy3/15Wbm5uXl2d6GBgYOHDgwJ6tEYBmFn3enoQYZ0xJ2fP58Jg5jgFydIED6zA0kCIjIy9dujRx4sTLly9//vnnSUlJzz33nPkBR44cyc/P9/L6dSWVwYMHI5AAOELx7PUbv0ibwT2yzl+VTc3s4V4lYAsmBpJKpcrIyEhMTAwMDHz8+PG0adP27NmzadMm82OKioqWLl06e/ZsuooEYKwlU3y1o1MfKOIccK8SsAoTmxry8vJ4PJ6/vz8hpE+fPoGBgeazc4QQo9F469YtJyentLS07Oxso9FIT6EATCXhcyV8bj/pmP/+eFMT4Uc9iXuVgOGYOELS6XRCodDO7tewdHd31+l05geo1eqGhoaVK1d6eHiUlJSIxeJDhw65uLhYnCcn5/e/BCMjI7u7bABGqVEqPGNP1Nu7OpzdpYnwo64qCeRRJTFzMHFnUxISEqgvcnJyGP6bkImB9OTJE/OHdnZ2DQ0N5s/U19fPnDlzxYoVYrH49u3br7/+ekJCwnvvvWdxHj8/P4Z/+gDdh7pXSRIZT+RRjv7ylD2fD4/wI4TgXiVbw6Jfg0ycsuvbt29dXZ3pYW1tLZf7h/X2Bw8evGXLFqr1buDAgdOmTcvPz+/pKgGYzXyvP45QPGJpdMGIv2mrDFjNARiLiSMksVhcXl5eUVFBzcIVFhZatH0rFIrbt2+vWbOGelhbW8vhcGgoFIDBLO5VciBkeHH2z8t3FFUZAjdlhY8UYbs/YBomBtKYMWP69ev3/vvvr1279vr162fPnl23bh0hJDo6+uWXXw4LC3N0dNy7d6+3t3dwcHBOTk5qaurKlSvprhqAiZpeLvIhRDnOsOhY4f5NWftCXx5+53RNpgKt4cAETAwkJyenLVu2bNiwITAw0M7OburUqWFhYYSQlJSUurq6sLCwSZMm3bhxIyYmZv369X379g0LCwsPD6e7agDWkPC5yreGaXWGRccK//ndJvflOzzHB1LfQms40IiJgUQImThx4oQJEyoqKhwcHEwXkAoKCkwHvPPOO2+//bZOpxMIBL1796apTAAWk/C5J1wybkplbxU4+9dpqBk8U2s4xknQ85jY1EDp1auXq6urRTuDOXt7e6FQiDQC6LAapWLE0uh9oT6EkIBPr1Obo2MbC6ALcwMJALrbr63hfG5MsBcVS16bsnI2LkNrONACgQRgu8xbw6lYSnvF4FFfMfKmzLTvH0CPYeg1JADoAU23sXBQZQsi45WiYWcvJXA/XDfWodY9Mp4nldUoFWjGg+6GQAKwdU2jRfLbCq1nLxVQsUQIoZKJOgDNeNAdMGUHAM37bTva1Nxng7RVhvsJK03NDlinFboDAgkAWiPhc4ffOe0Ze+Lc9IRTKWmaCD8qh9CMB10OU3YA8BQCeVTdmV1LIuPJlP1nswseZiqwTit0BwQSADwFNTtHbT47QTaEyIbUKIdUKuIKL2Q8X1b8TMDr1LUlND5AJyGQAOApmjbj6VXZ7pHxnq6ehd8rz364YaxDrWOAvEapQOMDdAYCCQDapNnhztCQBY7+8rOXCl469Z4HIXWqLHtXT45QTLAKEbQfmhoAoFOoZjyP+grDvA+1OkPOxmXUEkQEjQ/QThghAUAXEMij6vJSfCLjtToDuVLqtSmLEHLh8XY0PkDbIZAAoAuYGh8kUllMsFdMsFfhhQzjkeLxfd7xP1a40Ffk7+1EHYneB2gJAgkAukBLqxBppLL9V0pj0zWLqvTUNrWVijj0PkCzEEgA0GWaHeuE+4rCfUVaneHAldK4Nas9uT5FJe7/j2RTmYTeBzBBUwMA9ARqNfGQ2gvDl0UTQq7t/kAT4Xc/YaVelY3eB6BghAQAPYda9CEmMp4EpxrLik+lpPF3fyAsvc5xFdcoFaYOCFxnsk0IJADoOeaLPnCE4hlLlupVQ+8nrPx54vLLKWmjFHE8qayfdAyuM9kmBBIA9JyWFn3wksqGhiwwlhUXfq90UMQRQioVcY7+cmrMhOtMNgKBBAA9raVc4QjFQ0MWaM7sco+M1+r0JZkK/WebJM5c07DJ/IWY1rM+CCQAYBaBPOqh8rhPZDwZH6jVGf774806VdZLu1ZyXMUlMXN4UhkVPJjWsz7osgMAZuFJZXpVtl6VTQiR8LkTZEMmyoZyXMWesSd+nvBWprraa1NW3JrVD/40LIfrY3oV9gy0AhghAQCztHSdiSMU+wjFPuMDR+gMmrciTk7beTFds6hKL3Hm+Xs7vfq881h5VEnMHIuJO8zssQgCCQCYqJXYkPC5/KXRPqqkDyLjtTqDtspw/r9VsemaxVmrH4jG16ZrXn3e2c9QSIUQZvZYBFN2AMA+pmk9CZ/r7+0UE+yV9qphrEPtiGUbCCGx6Zr/+3CDJsJPE+FH7YVhgpk9JsMICQDYp8X2cW8nf2+nGELIW+eMZcWat/we/GmYdvcHHvUVhBCqWw8Ne4yFQAIAtmo9NjhCsfvyeEdVlvs/Uk3desLvM/h5KRxXsSbCj8onjlCMaT2GQCABgNUy3xRDIhtCZEP0quz79657xp44r66m8kmvypY4c+8nrORJZe6R8aTl+3AxkOpuCCQAsFqtNOxNEIqJbAghRBPh5x4Zf7FmQJ0qS7hxWX1Z8Rvu0R71gYdUm77xmifh80w7OWEg1d0QSABg5Vofx1D34U6IjKfyiRCiIaQkZg4vQH5eXX3gyv0AdZWEz42sTvLk+ujrJBJ1NRVRzQ6kMIrqDAQSANg08/VeqWf0qmxjWbFn7Il9ZocpQyPUC764+FtE/Xh7PsdVTB3McRVTvXzUqZ46ikJotQSBBAA2raVpPYvDRiyN9lEdXvLb81rdLWN5sX38/Ad/GsZTZdVnlmSqq0YZCjmu4kpFnL2r57d+a8N9Rc2Ooto49WeDuYVAAgB4+u96i4GUhM/Vl1bcJ2T0P3ZTB8wnRBPhZ5j3oYEQY3nJpJzNl7411JcVm3Z7MkXLgz8N8zKLmZZ6KGzwkhUCCQDg6doykBLIo+ryUn57coE7IYSQkpg59q6eAnmUXpWdqa566dvY1UN2392UpdUZJHyuxJkn4XONT6ZsVP7DvIeiUhHHk8p4T8utNo6iqMO8H/WuVPzSymG0QyABALRV67/HW7kcRQjhCMUzAkiNt/MJVYZFkn2/YnXus0FUD8WiKr1WZ/jxdhy1urm9qyf1Wo6rmLql19Ffblp+oo2jKOowtfLaZHlkK4fRDoEEANA12jKKaja0POorxq3fONvssBplfJ0qyz0yfv+V0uHFZ4xlxR5lWTWZCo6r+Pu3p3nUl3NcxcbyYo6r+KHy+EPlcY5QLJBHtXTJ6tfBlvKaqU5mbnjYq7Gxke4auktCQkJkZCTdVQAAWDJfTE+vyhbIo8yn3QghxrLikpg55qMfvSr7fsJK07CGWhjJffmvaXfzx5se9eXaKoNnfcXNH28GeuwghEj4XEJIRsmqPWO33rV3zbl8+cb2hebnZ9ogia0jpIaGhuTk5MLCQj6fP336dLFY/PTXAAAww1OHJk8dbFELI1GjKELIiABCCHEnpCRmztCQBY3yQNORNcqoSFWSe2R8QsU505MPdq2itodnFLau9h0ZGfnBBx/U1NR8++2306dPv3XrFt0VdaWEhAS6S+ggVN7zUHnP67HKBfIo03+esScsRlHmOxlSqEtWFmnXxsOYgJUjJJVKlZGRkZiYGBgY+Pjx42nTpu3Zs2fTpk101wUA0HPaeAeV6TBv7WWqy67Zw5iAlYGUl5fH4/H8/f0JIX369AkMDPz+++/pLgoAgAZtHOgI5FHqsgSqy46xWDllp9PphEKhnd2vxbu7u+t0OnpLAgCATmJll90nn3xy8uTJ06dPUw8PHjy4a9eunBzLdhH2zlADAHQHhjces3LKrm/fvnV1daaHtbW1XC636WEM/+gBAMAcK6fsxGJxeXl5RUUF9bCwsBBt3wAAbMfKQBozZky/fv3ef//9u3fvnjx58uzZsxMnTqS7KAAA6BRWXkMihJw5c2bDhg3V1dV2dnZTp0796KOP7O1ZOf0IAAAUtgYSIaSxsbGiosLBwaHZC0gAAMAuLA4kAACwJqy8hgQAANbHCq+7sH3d1Rs3bpw7d27FihV0F9IOjx8/Tk5O/umnn1xcXF577TV3d3e6K2orvV5/4sQJjUbj5uY2a9YsoVBId0Xt9vXXX//yyy+hoaF0F9JWubm5eXl5poeBgYEDBw6ksZ52yc3NvXjxIo/HCw4OZsvvlqKiouzsbPNnnnnmmZCQELrqaYUVTtlFRERcunRp4sSJRUVFd+7cSUpKeu655+guqq0qKioiIyNra2tPnjxJdy1t1dDQMHfu3JKSEn9//+vXrz948CApKUkikdBd19M1NDTMmTNHp9P5+fldvnzZYDB88803rq6udNfVDiqVSi6XT5gw4eOPP6a7lrZavXp1fn6+l5cX9XDJkiW+vr70ltRGR48e3bRp09ixYx8+fPif//zn1KlTpnfBZGfOnNmzZ4/poVqtdnd3/+abb2gsqUWN1uWHH3544YUXzp0719jY+Msvv0ycOPHdd9+lu6i2mjVtHHXWAAAFQElEQVRrlo+PzwsvvDB16lS6a2mH06dPv/TSS2q1urGxsa6u7pVXXtmyZQvdRbXJuXPnpFLpvXv3GhsbS0pKXnrppePHj9NdVDvo9frJkycHBAS8/fbbdNfSDlOnTj1x4gTdVbRbTU3N4MGDjx07Rj0MCwvbvHkzvSV1wA8//DBixIjc3Fy6C2metU3ZsXrd1Y0bNz5+/DglJeXq1at019IOlZWVI0eOpIahPB5PJBJVVVXRXVSb9O/f/8033xSJRIQQHo9nZ2fn4OBAd1HtsHXr1hdffFEoFN6/f5/uWtrKaDTeunXLyckpLS3N2dl55MiRHA6H7qLaJDMzs2/fvrNnz75z5059ff3BgwdNy2myhV6vj4yMXLZs2bBhw+iupXnWFkisXndVKpUSQq5du8auQAoNDTVdwLh69Wp+fv7cuXPpLamN/Pz8/Pz8dDrdrl27lErlmDFjAgMDn/4yZvj+++/T09NPnjyZmJhIdy3toFarGxoaVq5c6eHhUVJSIhaLDx065OLiQnddT1dcXMzn819//fUbN248efLE29t79+7dzz77LN11tcPnn39OCFmwYAHdhbSIZQn/VE+ePDF/aGdn19DQQFcxNqWxsfHIkSOLFy+ePHnyjBkz6C6nHZ48edKnTx+RSFRYWJifn093OW1SXV29bt26jRs3Ojk50V1L+9TX18+cOTMtLe3bb79NTU2tqqpiyyLIDx8+vH379vDhw/Pz88+dO1dfX//hhx/SXVQ7VFdX79mzZ9myZX369KG7lhZZWyC1cd1V6FplZWULFizYvn37unXrtm3b1qtXL7orahO9Xq/X611cXNauXXv48OEXXnjB/Novk8XFxQ0YMODhw4fJycn//e9/7927l5aWRndRbTJ48OAtW7ZQ/WkDBw6cNm0aW/4I6N+/v52dXVRUFIfD8fT0nDt37pUrV+guqh1SU1MbGxsnT55MdyGtsbZAwrqrPU+v18+fP58QcurUqblz57IljQghmzZtMu+W9vb2vnv3Lo31tJ2jo6ODg8Phw4cPHz5cVFR0+/btY8eO0V1UmygUiq1bt5oe1tbWsuUaEnWV1DTjUl9fz67lylJTU8eNG+fo6Eh3Ia2xtkDCuqs97/jx4+Xl5evXr6+trdVoNBqNprKyku6i2mTUqFFFRUVffPFFRUXFxYsXk5OTZTIZ3UW1yZo1a776zbRp02Qy2cGDB+kuqk0cHR337t2blJT06NGjjIyM1NTUoKAguotqk1dffXXAgAEbN26sqqr64YcfDh48+Ne//pXuotqqsbGxsLBw6NChdBfyFGxK+LZwcnLasmXLhg0bAgMDqXVXw8LC6C6q3Vg0yCCEXLt2ra6ubtasWaZnFixYEB0dTWNJbTRjxowff/wxPj5+69at1L+WVatW0V1UR7Co3WvSpEk3btyIiYlZv3593759w8LCwsPD6S6qTRwcHBITE6Ojo0ePHt27d++AgIC1a9fSXVRblZaW1tXVPf/883QX8hRWeGMswbqr0B4NDQ0VFRXOzs5MvthrZerr63U6nUAg6N27N921tFt1dTWPx+vbty/dhVgh6wwkAABgHdaM9AEAwLohkAAAgBEQSAAAwAgIJAAAYAQEEgAAMAICCQAAGAGBBAAAjIBAAgAARkAgAQAAIyCQAACAERBIAADACAgkAABgBAQSAAAwAgIJAAAYAYEEAACMgEACAABGQCABAAAjIJAAAIAREEgAAMAICCQAAGAEBBIAADACAgkAABgBgQQAAIyAQAIAAEZAIAEAACMgkAAAgBH+P+cvalyPtK+XAAAAAElFTkSuQmCC\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "\n", + "f = @(x,y) 8*exp(-x)*(1 + x) - 2*y;\n", + "\n", + "dx = 0.2;\n", + "x = 0 : dx : 7;\n", + "n = length(x);\n", + "y(1) = 1;\n", + "\n", + "% midpoint method loop\n", + "for i = 1 : n - 1\n", + " y_half = y(i) + (dx/2)*f(x(i), y(i));\n", + " y(i+1) = y(i) + dx * f(x(i) + dx/2, y_half);\n", + "end\n", + "\n", + "x_exact = linspace(0, 7);\n", + "y_exact = exp(-2.*x_exact).*(8*x_exact.*exp(x_exact) + 1);\n", + "plot(x_exact, y_exact); hold on\n", + "plot(x, y, 'o--')\n", + "legend('Exact solution', \"Midpoint method solution\")\n", + "\n", + "fprintf('Maximum error: %5.3f', abs(max(y_exact) - max(y)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Fourth-order Runge–Kutta method\n", + "\n", + "Runge–Kutta methods are a family of methods that use one or more stages; the methods we have discussed so far (Forward Euler, Heun's, and midpoint) actually all fall in this family. There is also a popular fourth-order method: the **fourth-order Runge–Kutta method** (RK4). This uses four stages to get a more accurate solution:\n", + "\\begin{align}\n", + "y_{i+1} &= y_i + \\frac{\\Delta x}{6} (k_1 + 2 k_2 + 2 k_3 + k_4) \\\\\n", + "k_1 &= f(x_i, y_i) \\\\\n", + "k_2 &= f \\left( x_i + \\frac{\\Delta x}{2}, y_i + \\frac{\\Delta x}{2} k_1 \\right) \\\\\n", + "k_3 &= f \\left( x_i + \\frac{\\Delta x}{2}, y_i + \\frac{\\Delta x}{2} k_2 \\right) \\\\\n", + "k_4 &= f \\left( x_i + \\Delta x, y_i + \\Delta x \\, k_3 \\right)\n", + "\\end{align}\n", + "\n", + "This method is explicit and fourth-order accurate: error $\\sim \\mathcal{O}(\\Delta x^4)$:" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Maximum error: 0.0004" + ] + }, + { + "data": { + "image/png": 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UEbOgOmXDgj/LAj1o+BsAALAVVgZSa2vr/PnzDQaDVCrNysravXv38ePHrfdcMsfiw9f2V20c9vzIzmwZandR91lCUjLevrJ9wrnxcYpJAX1QIwAAI7AykM6dO3f9+vWzZ88KhcLbt2+PHz9erVbL5fJH/6QNZWvrFx++lvuYquUerycbWPnyxBdkcrEicquqeXFdHLUHD4BplEol3SUAiY+Pp7uEHmFlIA0cOPDVV1+lnhLg8XjOzs5ubm50F/UHZYbmsI8Lvgi63lJe4Z989NE/0CGOQOSffFSRrSrKiEq9MmHugLJePBQZoOfY/jkIDOFkfvhh6QxnMBi2bdumVqsDAwOVSmX//v3bXPDKK69Ipb8eGGzj/2DCPr76//wrnz6d3LtPtlInOFwVTZy7dQ/1jklfXqGIxPOzAPAwlsFrXl7e/v376S2mY6wcIVHu37/v6uoqFApLSkqKioqCg4PbXCCVSmn5H7fkM7qxzSVPn/6w1x81J4QMkISO1ZekrlmduOV9gudnAeBRWDR+ZeW2b6PRaDQavb29165de+DAgSeffHLnzp10F/WrbG392W+Ll+Sstj4HqLc0qFW+cVv9k4/GBAst3WL48sQGtap3/yAAANtjZSBt3LgxKirK8jIwMPD27ds01mNtwT++Sil5W5T8f72eRuS352c5AhFfnugeJqeO67+zbaX1M7YAACzFykAaO3ZsaWnpjh07ampqLl68mJmZGRLS+5/+nWR9SvcbS+KzbidYHoDtddbPz7rL5C3VFVXKhF45FBkAgHasXEOaOXPm999/n5aWtmXLFmdn52nTpq1cuZKuYiwneWdr68cem+Muk1cpE/pol0Gb52ddfPwbslWlkxV4OgkA7AArA4kQsmbNmlWrVtXU1Hh6erbb4dE2rE/yHrZv6dPPj/SLT6tSJvTpLgPrOz8WNvd2yvIdj49eOrXtng4AAHZh5ZQdpV+/fr6+vjSmESGkQa2i4iFbW09+a4hpy10GPEnIsxELXz4eTxUAAMBeLA4kJqB2GRBChu1bahm42HiXAV+eyJOE3FEm4AxWAGA1BFKPULsM0vdkNGlyHnaStw34xacZ57+7+LAD9/UCAPZj6xoSQ1C7DLxWxnB8RNRQqS9aL3dGTLDwpqE5+YwOB7ACAEshkHqBoPKq/z9OcAQiestYFCxcfLhkSH5lTLCQ3koAALoBU3Y9NfftncOfGk57GhFCxF7c3VFBe/OrsJgEAGyEQOqRPfmVo8vPusuYclCC2Iurfn1UWHoB3YUAAHQZAqlHzmnrfeO2Mu3knpgxQmxwAADWwRpSj+zJr2Rgx7yVrvnHTnxypT5c7MWl3sHZQgDAfAik7ks+o1s/kYlb2mpVqc8te3fqv7lq+WixF9ekL9fFStEzCQAYDlN23bfncuUi5u1no04zCnoxXDExgJq4s/RMors0AICOIJC6aU9+pSzQ0zInxhyW04yozd/JZ3QEPZMAgA0QSN20N7+KgcMjYnWaESFE/fqobG19trYePZMAgPkQSN1BnWQqC/Sgu5B2WPdMIoQsCvZL3/N/6JkEAMyHTQ3dsTe/clGwH91VtK9Nz6QZhLxUozL8aSYTd18AAFhBIHUHM3d7W7MeD7nL5EQRef17+fCnhtNYEgBAxzBl12V72HZYHEcgcg+TF2zfSHchAAAdQSB12Tlt/UuMXD3qgLtMPs698fSxU3QXAgDwUAikLsvW1skCPemuoms4ApFv3NanTyejsSwAMBYCqWv25FeOunWWe/BNugvpMmri7o4yge5CAADah0DqmnPa+oi7/34sbC7dhXSHu0w+uKW65HwW3YUAALQDgdQ12dq6sc0lVLdy1uEIRM8tf4t7cB0aJgEAAyGQumBPfuXy6//wi6OhQ3lv4UlCcK4dADATAqkLzmnr2Ts8suDLE42aHMtRDgAADIFA6gLPwmNBL4QxoVt5T3AEomcjFl3ZvoHuQgAA/gCB1Fl78it9K68OkITSXUgvcA+Ti714x3Z8QnchAAC/QyB11jlt/ezGf9vNmdm+cVtHXtl1/fvrdBcCAPArnGXXWZ6Fx9xldpJG5LfHkooUkV4RCy1v4kRwAKARAqmzRpd/9dgCu1p3aVCrxJ5cvXBU0IvhhBB0OgcAemHKrlP25Ffawf46a1Snc//ko9yD66h30OkcAOiFQHqoBrWqQhFZq0qtVaW6fLjAPrYzWFCdzqmJu6rfzhNCp3MAoBEC6aFqVal8eSL11+CWGr48URcrpbuoXmPpdP7f8XGWDrPodA4ANEIgtY+a0aLm6BrUqqAXwuzsjANLp3OxF5cvT7yyfYNRk4tO5wBAIwRS+6gZLerrJk0ONV9nTzNaVKfzJk1OrSrVVF0+uKWmSpngF8/iU5EAgO2wy6591IwW9QFt1OQ2z9/kbo8zWpbQ/e/4uLD0Ap0d7doAANbBCKl9lhktQoipulzsxbXvGS2xFzdmjHDx4RK6CwEAx4VAap9lRqtKmcDxEVF77ex7RmtRsLDM0IyWsgBAFwRSR/jyxEvcIJ4khC9P9E8+ak/PIT1I7MVdFOyXfEZHdyEA4KAYuoZ07969zMzMH374wdvb+y9/+Yufn1+bCwoKCgoLCy0vw8PDhwwZ0heVGDW5A2ZO7Ys7M1BMsHBvftWe/MqYYCHdtQCAw2FiILW2ti5YsKCiokImk33xxReffvppRkaGWCy2vubgwYNFRUUBAQHUyxEjRvRRIEU5LzSHhffFnZlpd1RQWHoBAgkAbI+JgZSVlVVcXHzy5MmhQ4cajcbJkyf/85//XLNmjfU1paWly5YtmzNnTp9W4oBjBbEXVxboufhwye6oILprAQDHwsQ1pNra2jFjxgwdOpQQwuPxhEJhXV2d9QUmk+nGjRseHh6nTp3Kzc01mUx9VMk5bf1LgR59dHPG2h0VlK2tw+4GALAxJo6QoqKioqKiqK8vX75cVFQ0b9486wu0Wm1ra2tCQsLgwYMrKipEItH+/fu9vb3b3Ccv7/eDq+Pj47tRSba2TjExoBs/yHaKiQHJZ3Sy10fRXQgA9JRSqaS+yMvL694noc0wMZAoZrP50KFD77333pQpU2bOnGn9rZaWllmzZq1YsUIkEt28eXPu3LlKpXL9+vVt7iCVSnvy2y8zNBNCxF7cbt+BvWSBnnvzq7K19TLHGyAC2BmGh5A1hgaSXq9PTEwsKSl58803o6KinJycrL87YsSIzZs3U18PGTJk+vTpBQUFvV5DtrZOFujZ67dlBbEXVzEpYPHha7okuzrjHACYjIlrSEajccGCBYSQ06dPz5s3r00aEUJUKtWWLVssLxsbGzkcTq+XcU5b78gL+7JAD7Enb09+Jd2FAICjYGIgHTlypLq6et26dY2NjTqdTqfT1dbWEkKSkpIOHDhACHF3d9+1a1dGRsbdu3ezsrJOnjw5ceLEXi8jW1v36Ivs2leR3i/t+gvdVQCAo2DilN2VK1eamppmz55teWfhwoVJSUnHjh1ramqKjo6ePHnytWvXFArFunXr+vfvHx0dHRMT0+tlUGtIjozqIYtTwAHANpgYSB999FG77xcXF1u+XrVq1RtvvGEwGPh8fr9+/Xq9Bgd8AqldfHlihSLy+vfXhz81nO5aAMDOMTGQOsnFxUUgEPTRzR3zCaQHUeedu6QtqP2t74a9nncOALRj4hoSEzjyFjtrVB93jkA0QBLKlye6y+T21McdABgFgdS+UbfOcg++SXcVNKP6uLuHyfnyxCplAvltVclu+rgDAKMgkNqRra2X/lJCtS13ZJY+7jxJCEcgotq321MfdwBgFARSO8oMxrHNJfbd/agzqD7u1Ne+cVupr+2vjzsAMAQCqR3ntPWDW6o5AhHdhdDMuo87NVlXoYi04z7uAEAvFu+y6ztNmhzM15Hf+rjXqlKbNDnUO02aHL84PJMEAH0CgdQOk77CZaQ/3VUwhfV4aIAk9NqFrOcxZQcAfQBTdm3tya/EjoaH4UlC/FtqqEk8AIDehUBq66ahGTsaHoYjEFm2gAMA9C4EUltldc3Y0dAB6y3gAAC9CIHUFnY0PJJlCzgAQC9CILVl0le4+GBHQ0csp4DTXQgA2BUE0h9gR0Mn8eWJRk2uSV9OdyEAYD8QSH9wTluPHQ2dQe1uwMQdAPQiBNIfZGvrsKOhk6zPcQAA6DkE0h+MunXWXYanPjsFgyQA6F0IpN9la+vpLoFl3MPkr/gm4fcGAL0CgfS7MoMROxq6SjEpYPHha3RXAQD2AIH0O+xo6AZZoIcs0DP5jI7uQgCA9RBIf4AdDd2gmBiw53JlmaGZ7kIAgN0QSL/zLDyGHQ3dIPbixowRJn+FQRIA9AjaT/xu673gD+LD6a6CfRrUqiU5qiNN4lpnIfUOOvgBQDcgkH61J78yJlhIdxWsVKtK9YtP4zeJI/Or1K+PMunLdbHSgPQ8uusCAJbBlB30SK0qlScJ4UlCZIGehJBsbT110h2eTwKArkIg/eqctv6lQA+6q2CfBrWKmqATe3EtW8D58kT0pwCArkIg/SpbW0f9Pz50ifVhDbJAD7Enb09+5Z1tK93R5hwAugiB9KsyQ7PYi0t3FezT5kS73VFBp4+dMunLsa8BALoKmxoIISRbW4/hUfdwBKKA9LxaVWqTJocQ4kbIivqsL6VrY+kuDABYB4FECCFlBiOGRz1hPR6SyuSDFZHGYCHOvACALsGUHSHY0dCrcAo4AHQPAgl6HzU2QqskAOgSBBIh2GLX26hBUpUyge5CAIBNEEiEYItdH+BJQjgCEZ5GAoDOQyCRPfmVWbdXmvTldBdib3zjtmIlCQA6D4FE/FtqxJ5cdJ3odRyByPCnmZi4A4BOQiCRJk0ONij3kWcjFv74/XXsbgCAzkAgEaMmF23L+whHIHo2YhEm7gCgM9gaSPfu3VOpVBs2bNi+fXtVVVVPbiWovIoRUt/BFnAA6CRWBlJra+uCBQvS0tKampq++OKLqVOnlpWVdftuaFvep7AFHAA6iZWBlJWVVVxc/Pnnn//v//5vRkaGm5vbP//5z+7dCvN1NoAt4ADQGawMpNra2jFjxgwdOpQQwuPxhEJhXV1dl+7QoFZVKCJrVanU8gYWOfoatoADwCOx8nDVqKioqKgo6uvLly8XFRXNmzfvwcvy8n7voh0fH2/9LarrNk8SUqVMGCAJ5UlC0HW7T1FbwHnKBL/4NLprAXAsSqWS+iIvL6/NJyHTsDKQKGaz+dChQ++9996UKVNmzpz54AVSqbTd376l6zYhxKjJ5csTLV230cWn7wx/anjVtm2EEMuKHX7bADbA8BCyxtZA0uv1iYmJJSUlb775ZlRUlJOTU+d/tkGt8k8+Sn1tqi6nPh/58sQKRSQ+IvtOrSrVXSZvqa6gBkkmfTlGpQBgjZVrSEajccGCBYSQ06dPz5s3r0tpRKy6bpv05RyfX/9vHV23+xQ1KqXyntoCbhmV0l0aADAFKwPpyJEj1dXV69ata2xs1Ol0Op2utra28z9u6bpt1ORaJu7QdbtPNahV1NTojwt3WLaA8+WJ2HoHABasDKQrV640NTXNnj178m+2b9/e+R+num43aXKaNDlGTS611w6L7X3KMiqVBXpwBCLqa4xKAcAaK9eQPvroo57fhHpaky9PxGeiDVCzc9SQ1Ddua4UikuMjMunLLYt5AACsDKTeUlbX/AzdNTgIalRaq0pt0uQQQjIGvei785M56zAqBYDfOXQg+f50lSdJorsKB2JZpZszvjksvcCfGySjtR4AYBRWriH1Fsueb7AxsRdXMTFg8eFrdBcCAAziuIGUra2nuwSHFhMsFHvy9uRX0l0IADCF4waS708FOFaVXurXRyV/paO7CgBgCscNJFVTwM7QLXRX4ehkgZ6LD5fQXQUAMILjBlJZXfMQLy7dVTg6xcSAMkMzpk8BgDhyIGVr62SBnnRX4ejEXtxFwX7JZzBxBwAOHEhlhmYxRkgMEBMsJIRgdwMAOGggIY0YZXdUEHY3AICDBhLm6xhF7MXF7gYAcNBAAqbZHRWUra3D7gYAR+aggXROW/9SoAfdVcAfFIZWDHl/Et1VAABtHDSQygzNYi8e3VXAH7iHyTkCETokATgsRw2kOqPYE5saGMc3bit6yAI4LAc97Ru77JjJqMktq2smpnLSeAAAIABJREFUsVJLkyq08QVwHI44QsrW1mOLHTPVqlKfW5bEEYgGSEL58kR3mVwXK6W7KACwEUcMpDKD8X/+HWPSl9NdCPxBrSqVJwlxD5O7y+TUxB1HIKJazdJdGgDYgiMG0k1D8+CWanRCYpoGtYqaoKPm66jdDXx5IrY5ADgIRwwkU3U5xwdpxDh8eaJlMETtbjDpy+9sW2lZTwIA++aIgdSkyeVJQuiuAtriSUKMmlyjJpcQwhGI3MPkd7atNOnLsa8BwEE46C47YCCOQBSQnlerSm3S5BBCTPry699ff25ZEt11AYCNOGIgtejLOaMC6K4C2mc9HnosLLdKmYApOwAH4XBTdtna+sGtNVhDYgWeJIQnCalSJtBdCADYgsMFUpnBOLa5BGtIbMGXJ/74/XVqYQkA7JvDBdJNQzPdJUAXcASiZyMWYZAE4AgcLpDK6vAQEstQh67i8VgAu+dwgWTS4yEk9vGN29qgVuFwDQD75nCBJKi8igUk1uEIRP+dEHdn20q6CwGAPuRwgVSGNSR2CnohjPx2nhAA2CXHCqQyQ7P0l5IBklC6C4Eu4whExvnvYiUJwI45ViBla+voLgG6b/hTww1/mokddwD2yrECiRCCh5BYjS9PnOe8iO4qAKBPOFYgofEE24m9uIuC/QI25tBdCAD0PscKpLI67GhgvZhgoSzQM/mMju5CAKCXOdbhqk2aHOxosAMf+hYf2/nJFV242ItLvYMWFQB2wLECaXBLjctgf7qrgJ6qVaX+d/za1U0BavkoQohJX66LlQak59FdFwD0iGNN2W29F+wXn0Z3FdAjtapUniQkNiaCEEJN3HEEIp4kBDvCAdiO0SOka9euffPNNytWrHjwWwUFBYWFhZaX4eHhQ4YM6fhuZYZmywwPsFeDWuWffJQQsjsqKCy9YFGwUOzF5csTKxSRmLgDYDXmjpBqamreeeedM2fOtPvdgwcPHjp06Nvf6PX6R94wW1snC/Ts7TLB1vjyRGowJPbi6pJCFx8uIYTc2bYSffwA2I6hI6SIiIjS0tLW1tbhw4e3e0FpaemyZcvmzJlj48KAdtTsnFGTa3meLH1PxmR9OTVsAgD2YmggpaSk3Lt379ixY5cvX37wuyaT6caNGx4eHqdOnfL09BwzZgyHw3nkPc9p618K9OiDYsGmOAJRQHperSq1SZNDCHnfubnk66zmZe/SXRcA9BRDA0kikRBCrly50m4gabXa1tbWhISEwYMHV1RUiESi/fv3e3t7t7ksL+/3bVfx8fF9WjDYmGW5iE9I8XN/fT2/Sv0ivRUBMJRSqaS+yMvLY/gnIUMDqWMtLS2zZs1asWKFSCS6efPm3LlzlUrl+vXr21wmlUqtf/vZ2jrFxACbFgo2ERMs3JtflXxGp5iEf74AbTE8hKwxd1NDB0aMGLF582aRSEQIGTJkyPTp04uKih75U9hlZ8e+ivR++US8UZNLdyEA0H2sDCSVSrVlyxbLy8bGxs6sIYEd4whEfHkiDgIHYDU2BVJSUtKBAwcIIe7u7rt27crIyLh7925WVtbJkycnTpzY8c9ma+ux59u+Bb0YXvDERGQSAHsxPZCcnJwsX1s23U2ePPnVV19VKBSjR49OSEiIjo6OiYnp+D5lBiPm6+zezCWvldU1Y+IOgKUYvalh6dKlS5cutbwsLi62fL1q1ao33njDYDDw+fx+/fo98lY3Dc1iTwSSneMIRI3j46qUS3GuHQAbMX2E1AEXFxeBQNCZNCKElNU1z/tsXF+XBLQbHzISXWUBWIrFgdQlJn05xwd9+RzCsxELf/z+OibuAFjHUQJJUHkVncsdBEcg8o1Pq1ImmPTldNcCAF3gKIFUZkCvWAcy/Knh/50Qh4YUAOzC6E0Nvci/pZojwGP8DkTsya06qGqprrCMjNGcAoDhHGKElK2tH9xagzUkh1KrSvWLSzPpy91lcr480V0m18VK6S4KADriEIFUZjCObS7BGpLjoLrKuofJ+fLEO9tWEnSVBWADhwikm1hAcjANahU1QUd17aNyiC9PbFCraK4MAB7OIQKprK55cEs1R4ApO0dh6SpLCPGN22rU5Bo1uegqC8BwDhFIg1uqsYDkUHiSECqEiNW5qyZ9OfY1ADCZQ+yya7Jqdw2OoE1XWcubNJYEAI/kEIEEjsl6POQuk9/ZtrJWlYpBEgBjOcSUnW/l1QGSULqrADpxBCLfuK0l57NwpBAAY9l/IOGMBqBYLybRXQsAtMP+AylbW4eHkIAS9GK4e5icejIJAJjG/gOJULvssKANhBBC+PLEV3yTsrX1dBcCAG3ZfyDhqVhoY3dU0OLD1zCXC8A09h9IAuxogD8Se3EVEwPC0gvoLgQA/sD+A8lUXe7i4093FcAsMcHCmDHCxYdL6C4EAH7nAIGkL8cCEjxoUbCwzNC8J7+S7kIA4Ff2/2Ds6vtTE+XhdFcBjCP24n4RdP3Yh6uvvBAu9uJSb+KxWQAa2X8gATxMrSp1zrq0ZzLNuuWhhBCTvlwXKw1Iz6O7LgAHZedTdmWGZsv//AJYo3om8SQhlsUk9EwCoJedB1K2tk4W6El3FcBElp5JikkBhJDkMzqCnkkAtLLzQAJ4GOueSYqJAWV1zdnaevRMAqCRnQfSTUOz2BNTdtAO655J1JNJ6Xv+Dz2TAGhk55sayuqaXwr0oLsKYKI2PZPcCEl1zk30fu0I3YUBOCx7DyRD86JgHt1VAHO1GQ99rErFRjsAutj5lF1ZnRFTdtB5/x0fV/DExCplAt2FADgiew8kbPuGrhB7cWcuee3H769j8zeA7dl5IAF0FUcgkr69vShjH/Z/A9iYPQdS9o94CAm6gyMQ+ScfrVWlot85gC3ZcyCZqst3FS+nuwpgpeFPDS+dokC/cwBbsudAQudy6IkpM6e6h8krFJF0FwLgKOw5kNASFHoIm+4AbMmen0Pydb2PTkjQE2Iv7uAlr93ZtlIXK7U+UginOQD0BXsOJIGrmeODQIIe4QhE1DISx0dEZRK6VAD0EXueshvtpMcaEvQQ1aXCetMdulQA9BF2B9K1a9c++uijh33XNIBvy2LALlFdKjgC0bm/Zlo23aFLBUBfYHEg1dTUvPPOO2fOnHnYBYNbqrGGBD1k6VIREyzkyxOpTXfoUgHQF9i6hhQREVFaWtra2jp8+PA232pQqxqyVdRkHfVRgiVo6DZqds6oyeVJQtzD5Kbqcl2slBDin3yU7tIA7A1bAyklJeXevXvHjh27fPlym2/VqlL94tN4khCqtw2WoKEn2nSpsLxJY0kA9srJbDbTXUP37dix44svvjh+/LjlnVpVqklf7hefRgg5tjD01tj5hJAnLh0cN30OxknQW65s3zi4pZr61wyA4ZRKJfVFXl7e/v376S2mY2wdIT1Mg1plmUu5NXZ+fHw8IcSkn1WhiEQgQW/hyxMvfrJRpkrFv1TAfNTHICuweFNDuyxL0NawBA29i+pScR1dKgB6lb0FEk8SYtTkWh/SbNTkUotJNFYF9ocjED23LKnkfBYyCaC32NuUnfUSdGDZpVrVL0ZNLub6oS9QnZPyUpYHYScnQG9gfSA5OTk9+Cb16aDVK6fIWTN5CmzEEYj8k/8v8nCJQlsvC/SguxwAdmN3IC1dunTp0qV0VwEOTezF3R0VtGrde8P8iq2PqsKYCaCr2B1IAEwg9uKmOp9KvP/ae+OjxV5cggNYAbrF3jY1ANgedQDreyujw9ILqC5cOIAVoBsQSAA9RR3AKvbi6pJCLZmEA1gBugqBBNBT1k+/KSYGUJmEp98AugqBBNBT1k+/xQQLFRMD1m49gKffALoKmxoAeqrNAawzCBnvnJvo/dp7hmZqjwMAdAYCCaB3tBkPTcmvDEsvUMeORiYBdBICCaBPxAQLCSHIJIDOQyAB9BVLJu2OegbnOAA8EjY1APShmGChOnb04sPX8EwSwCMhkAD6ltiLq44djXPBAR4JU3YAfU7sxR389vYKRSS1NRxH3gG0C4EEYAscgcg/+eiFN6YHvRBmCSEceQdgDVN2ADbSkK0a/tTwH636zOLIOwBrCCQAG2lQq3zjtkrf3m69noQj7wAsEEgANkIdeUf19COEVCgiCSE48g7AAoEEYCOWI+/EXtz/jo/jSUJ0sVIceQdggU0NADZifeSdm9X7Jn05RyCirSwAxkAgAdiU9XiIL0/8Orf4dsryoBfDMU4CQCAB0Gl8yEhT4HZqPQmZBA4Oa0gANKMeUfo6txj7v8HBIZAA6McRiCLWpZQZmqltDnSXA0APTNkBMAJHIOLLE928uLrXpQMkoTheCBwQRkgATCH24vLliRwfkUlfzvER8eWJfHmiu0yui5XSXRqALSCQABikVpXKk4T4Jx9tyFYln9ERHC8EjgSBBMAgDWoVX57IEYh847YuzIyicgjHC4GDQCABMAh1vBD5besdIUQXK8XxQuAgEEgADGI5Xoj8ts2BJwm5/v115WMRdJcG0Oewyw6AQayPF6Leaamu8IvbuufflYQQxaQAWqsD6FsIJADGeXCft1rSvDe/MmBjjjp2tNiLS0tVAH0NU3YALCD24iomBcSMEYalF1C77wDsD0ZIAKyhmBSwKFgYll5ACFnpmt+QrcLzs2BPEEgAbCL24qpjR+/Nr7zyyUa/uK3+L4ZT75v05bpYaUB6Hr3lAfQEpuwAWEbsxY3/OUMvHDX131zL9B2enwU7gEACYJ8GtSpiXYo6dnS2tj5gY06ZoZng+VlgP0zZAbAP9fysOD5td1TQ3vzKsPSCy8NzjZpcPD8LrIYREgD7WJ6fpXbf6ZJCv84tbtLkuMsQSMBiGCEBsM+Dz8+Oc290kckrFJHuYXJstwOWYmggtba2ZmZmlpSUeHl5zZgxQyQStbmgoKCgsLDQ8jI8PHzIkCG2rRGAZg8Gj0lffmznp6Njpf7JRzmCtv/VADAcQwMpPj7+22+/nTBhwqVLlz799NOMjIyhQ4daX3Dw4MGioqKAgF9PUhkxYgQCCYAjEM1Zl1KUMTwvZXnQi+EcHxGeVQIWYWIgaTSarKys9PT08PDwe/fuTZ8+fefOnRs3brS+prS0dNmyZXPmzKGrSADGejZioemFsKKMfW6qBKrLH/U+nlUChmPipobCwkIejyeTyQghrq6u4eHh1rNzhBCTyXTjxg0PD49Tp07l5uaaTCZ6CgVgKo5AJPbiDpCEGjW5loeT8KwSMBwTR0gGg0EgEDg7/xqWfn5+BoPB+gKtVtva2pqQkDB48OCKigqRSLR//35vb+8298nL+/3/BOPj4/u6bABGaVCrqI5KDdkqXayU2uzAlydWKCIxcedQlEol9UVeXh7DPwmZGEj379+3funs7Nza2mr9TktLy6xZs1asWCESiW7evDl37lylUrl+/fo295FKpQz/7QP0HepZJb/4NL480V0mv7NtpS5WyhGI8KySo2HRxyATp+z69+/f1NRkednY2Mjl/uG8/REjRmzevJnaejdkyJDp06cXFRXZukoAZmvT66/lb5/zJCFNmhyTvtykL6e7OoB2MHGEJBKJqqura2pqqFm4kpKSNtu+VSrVzZs316xZQ71sbGzkcDg0FArAYG2eVXIjxFhd4ReXZqoux+NKwExMDKTQ0NABAwa88847a9euvXr16tdff/3mm28SQpKSkp555pno6Gh3d/ddu3YFBgZOmjQpLy/v5MmTCQkJdFcNwETtpo67TE49ruQeJsfWcGAOJgaSh4fH5s2b33rrrfDwcGdn52nTpkVHRxNCjh071tTUFB0dPXny5GvXrikUinXr1vXv3z86OjomJobuqgFYg3pcyaR/7c62lbWa1Na/fY42FsAETAwkQsiECRPGjx9fU1Pj5uZmWUAqLi62XLBq1ao33njDYDDw+fx+/frRVCYAi1G7wAkhrxR7ypp0ikkBxGprOMZJYHtM3NRAcXJy8vHxabOdwZqLi4tAIEAaAXRbg1rlG7d1d1QQISTs46tUdyW0sQC6MDeQAKCv/drGwourmBRAxVLAxpy8lOXYGg60QCABOC7rreFULJ36c/Pglpox10MsvWgBbIaha0gAYAMPtrFw0+Ty49PUwlF78ysDNubEjBFSa0sNahU240FfQyABOLoHo0VMiGJSwKJgoSWWFmam+sWnWQIJm/GgL2DKDgDaR03iqWNHjyjYRQj5WX3EcsQDzmmFvoBAAoCOiL24o2995Refpn989J1tK6uUCdSaEzbjQa/DlB0APAJfnviz+siz8WkkYmGDWlWrSjXpy3FOK/Q6jJAA4BGsN+O5h8n9k4/y5YkmfbklnOguEOwERkgA8AgPbsYzanL94tNcfPwbslV5KcuHPTX8sbC5Jn05duJBTyCQAKBT2k0XvjxRKpOXXFD/uH2DoPKqu0yOjunQbZiyA4Ae4QhEz0YsDHoxfIAklBCii5VSu++wEw+6CiMkAOgFVMd0jkBETdxdWDFtcEsNTxLSoFZh4g46CYEEAL3A0jGdIxDx5YkvyBNN+vILb0wXe3IXHy5ZFCyUBXpQV+LQB3gYBBIA9AJqds6oybUkTUt1hdiT65989KWbLslndIvrjNRBRFRu4dAHeBACCQB6wcN24nEEohgBiQkWlhma9+ZXpq5Z7c8NKq3wWyRsFntxCTowgRUEEgD0mg5ChTqISJd5viXh89IyEpZeQAjRJYVSP1WhiEQgAQIJAGyHL09sOrtNEZ+mmBRQZmhefLgkW1uXUvK2LFBkPd2HdSbHhEACANuxXmoSe3F3RwUZNblVFY08ydQr2zdQG/MeC5uLdSbHhEACANt52FITTxJCbcwruaA2KhMIIT+rjxBCqEzCOpODQCABgK09LFeoZ2x1Z7f5xaeZ9OXUQXk8ScgASegASWibQMK0nv1BIAEAs1CHi/vFp7mHyU36cqMmt0mT05Ct4viIqpQJAySh1CnjmNazPzg6CACYxfpwcarJxWNhczk+Iv/kowMkodcuZP0Q+fgPkY9zBCLrn8JJRXYAIyQAYJYOHmniCETPh8nLDGn3Xhvq4uNPzen9deR2WaDHS8M8x7W3fRwzeyyCQAIAJur4kaaGuLQmTY5/8tEyQ7Oirvncj3XJZ3RLclbfEb7YeEb30jBPy0lFmNljEQQSALCPZfu4WBIi9uLKAj3+nya3qrjx5vK3qHBaXGckhLzvfHrw46OchKPEv/0gNuwxGQIJANjnYdN6AYEeskAPBSGEkDJDs+71WGXolovpBVkVK3mSEI5ANEAS+uDBEJjWYwgEEgCwVcexIfbiei1LCtKc8ktKKzOcG1B47Pr3131VqU2aHGrDHpVP1IAJ03pMgEACALtlPbNHwuTPhRGjJtekLPdPPmrU5Frn08/qI02aHCrhHjath4FUX0MgAYDd6njD3nNhhBCii5VSz+Fe//66URFp0pdbruT4iKhwot7BQKqvIZAAwM51PI6xPIdL5RMhxKQvr1BE8iQhTZqcluwKUePrYi9ufH2GPzfI2CQWa+upLXztDqQwiuoJBBIAOLR2WwsSQvzi06iXZkIIIeqoWO3CHRe19Xvzq8K0dWIvrtiTN/j+1BT123x5ouXHOzOKQmg9DAIJABzaw6b12lz23LKkIM2Bpb+9X2ZoLqtrdkl7q+CJie8cLkkp+fXHLctRxwPmxwQL2x1FdXLqzwFzC4EEAPDoz/o2AymxF9e38mpVS80L61LmEELIUULID5GPN89/l99SY6oun5z3XsWJCpO+3FRdzvERGTW5Lj7+HIHIpC/nCEQuPv6WOz9sD4UDLlkhkAAAHq0zAym/uLSmwmPuf3yzQhHp4uP/WNhcat+EW7aK4yOqUERSQcURiN4OSjHdn5qifvt4wHyxF49aoKpVpfIkIdbDo54sWVGXBd7tV6v6pYPLaIdAAgDorI4/xx9cjjJqck36cv/ko9R3nwsjDU8Nb9LkUElm0pe3VFfslgRdWLG64ImJ57T1e/OrFtcZywzNN5pUr/glcT6+Kvbiij25K13zCSEP9uDo5CiKukyrvjJFHt/BZbRDIAEA9I7OjKKsQ4vafW7U5A7+fervVw3qxKOaLL/X0/bkV940NFM3pHpw/BD5OCGE4yOixlg/q4/8rD5CdeV42JLVr4Mt9RVLncw8P8nJbDbTXUNfUSqV8fHxdFcBANCWdZsMoyaXL0+0nnYjv209tx79GDW5VcoEy7DGpC/XvS71i/s17ajEovYHWuKQ4/Nrh46d47bcdvHJu3Tp2geLrO/PtEESW0dIra2tmZmZJSUlXl5eM2bMEIlEj/4ZAABmeOTQ5JGDLY5A5BeXZpn9o5oWEkKoY/rarDPFazL84tOUNd9Y3ryzbaXlR5iDrQ364uPjN2zY0NDQ8OWXX86YMePGjRt0V9SblEol3SV0Eyq3PVRuezarnIoW6i//5KNtRlHWnQwp1JJVm7Tr5GVMwMoRkkajycrKSk9PDw8Pv3fv3vTp03fu3Llx40a66wIAsJ1OPkFluSyw7BK1y67dy5iAlYFUWFjI4/FkMhkhxNXVNTw8/MKFC3QXBQBAg04OdPjyRK1eSe2yYyxWTtkZDAaBQODs/Gvxfn5+BoOB3pIAAKCHWLnL7h//+MeJEye++uor6uW+ffu2bduWl9d2uwh7Z6gBAPoCwzces3LKrn///k1NTZaXjY2NXC73wcsY/qsHAABrrJyyE4lE1dXVNTU11MuSkhJs+wYAYDtWBlJoaOiAAQPeeeed27dvnzhx4uuvv54wYQLdRQEAQI+wcg2JEHL27Nm33nqrvr7e2dl52rRpmzZtcnFh5fQjAABQ2BpIhBCz2VxTU+Pm5tbuAhIAALALiwMJAADsCSvXkAAAwP7Y4boL289dvXbt2jfffLNixQq6C+mCe/fuZWZm/vDDD97e3n/5y1/8/PzorqizjEbj0aNHdTqdr6/v7NmzBQIB3RV12b/+9a9ffvklKiqK7kI6q6CgoLCw0PIyPDx8yJAhNNbTJQUFBRcvXuTxeJMmTWLLZ0tpaWlubq71O4899lhERARd9XTADqfsYmNjv/322wkTJpSWlt66dSsjI2Po0KF0F9VZNTU18fHxjY2NJ06coLuWzmptbZ03b15FRYVMJrt69eqdO3cyMjLEYjHddT1aa2trZGSkwWCQSqWXLl1qbm4+fvy4j48P3XV1gUajkcvl48eP//DDD+mupbNWr15dVFQUEBBAvVy6dGlwcDC9JXXSoUOHNm7cOG7cuJ9//vk///nP6dOnLX8XTHb27NmdO3daXmq1Wj8/v+PHj9NY0kOZ7ct333335JNPfvPNN2az+ZdffpkwYcLf//53uovqrNmzZwcFBT355JPTpk2ju5Yu+Oqrr55++mmtVms2m5uamv785z9v3ryZ7qI65ZtvvpFIJD/99JPZbK6oqHj66aePHDlCd1FdYDQap0yZEhYW9sYbb9BdSxdMmzbt6NGjdFfRZQ0NDSNGjDh8+DD1Mjo6+r333qO3pG747rvvnnvuuYKCAroLaZ+9Tdmx+tzVlJSUe/fuHTt27PLly3TX0gW1tbVjxoyhhqE8Hk8oFNbV1dFdVKcMHDjw1VdfFQqFhBAej+fs7Ozm5kZ3UV2wZcuWp556SiAQVFVV0V1LZ5lMphs3bnh4eJw6dcrT03PMmDEcDofuojolOzu7f//+c+bMuXXrVktLy759+yzHabKF0WiMj49fvnz5qFGj6K6lffYWSKw+d1UikRBCrly5wq5AioqKsixgXL58uaioaN68efSW1ElSqVQqlRoMhm3btqnV6tDQ0PDwcLqL6qwLFy6cOXPmxIkT6enpdNfSBVqttrW1NSEhYfDgwRUVFSKRaP/+/d7e3nTX9Wjl5eVeXl5z5869du3a/fv3AwMDt2/f/sQTT9BdVxd8+umnhJCFCxfSXchDsSzhH+n+/fvWL52dnVtbW+kqxqGYzeaDBw8uWbJkypQpM2fOpLucLrh//76rq6tQKCwpKSkqKqK7nE6pr69/8803U1JSPDw86K6la1paWmbNmnXq1Kkvv/zy5MmTdXV1bDkE+eeff7558+bo0aOLioq++eablpaWd999l+6iuqC+vn7nzp3Lly93dXWlu5aHsrdA6uS5q9C79Hr9woULP/jggzfffPP99993cnKiu6JOMRqNRqPR29t77dq1Bw4cePLJJ63XfpksNTV10KBBP//8c2Zm5o8//vjTTz+dOnWK7qI6ZcSIEZs3b6b2pw0ZMmT69Ols+Z+AgQMHOjs7JyYmcjgcf3//efPm5efn011UF5w8edJsNk+ZMoXuQjpib4GEc1dtz2g0LliwgBBy+vTpefPmsSWNCCEbN2603i0dGBh4+/ZtGuvpPHd3dzc3twMHDhw4cKC0tPTmzZuHDx+mu6hOUalUW7ZssbxsbGxkyxoStUpqmXFpaWlh13FlJ0+efOGFF9zd3ekupCP2Fkg4d9X2jhw5Ul1dvW7dusbGRp1Op9Ppamtr6S6qU8aOHVtaWrpjx46ampqLFy9mZmaGhITQXVSnrFmz5p+/mT59ekhIyL59++guqlPc3d137dqVkZFx9+7drKyskydPTpw4ke6iOuWll14aNGhQSkpKXV3dd999t2/fvpdffpnuojrLbDaXlJQ8++yzdBfyCGxK+M7w8PDYvHnzW2+9FR4eTp27Gh0dTXdRXcaiQQYh5MqVK01NTbNnz7a8s3DhwqSkJBpL6qSZM2d+//33aWlpW7Zsof5tWblyJd1FdQeLtntNnjz52rVrCoVi3bp1/fv3j46OjomJobuoTnFzc0tPT09KSnr++ef79esXFha2du1auovqrMrKyqampmHDhtFdyCPY4YOxBOeuQle0trbW1NR4enoyebHXzrS0tBgMBj6f369fP7pr6bL6+noej9e/f3+6C7FD9hlIAADAOqwZ6QMAgH1DIAEAACMgkAAAgBFSSaXbAAAAjUlEQVQQSAAAwAgIJAAAYAQEEgAAMAICCQAAGAGBBAAAjIBAAgAARkAgAQAAIyCQAACAERBIAADACAgkAABgBAQSAAAwAgIJAAAYAYEEAACMgEACAABGQCABAAAjIJAAAIAREEgAAMAICCQAAGAEBBIAADACAgkAABgBgQQAAIyAQAIAAEZAIAEAACP8f60AD9M98Qi1AAAAAElFTkSuQmCC\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "\n", + "f = @(x,y) 8*exp(-x)*(1 + x) - 2*y;\n", + "\n", + "dx = 0.2;\n", + "x = 0 : dx : 7;\n", + "n = length(x);\n", + "y(1) = 1;\n", + "\n", + "% 4th-order Runge-Kutta method loop\n", + "for i = 1 : n - 1\n", + " k1 = f(x(i), y(i));\n", + " k2 = f(x(i) + dx/2, y(i) + dx*k1/2);\n", + " k3 = f(x(i) + dx/2, y(i) + dx*k2/2);\n", + " k4 = f(x(i) + dx, y(i) + dx*k3);\n", + " y(i+1) = y(i) + (dx/6) * (k1 + 2*k2 + 2*k3 + k4);\n", + "end\n", + "\n", + "x_exact = linspace(0, 7);\n", + "y_exact = @(x) exp(-2.*x).*(8*x.*exp(x) + 1);\n", + "plot(x_exact, y_exact(x_exact)); hold on\n", + "plot(x, y, 'o--')\n", + "legend('Exact solution', \"RK4 solution\")\n", + "\n", + "fprintf('Maximum error: %6.4f', max(abs(y_exact(x) - y)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The maximum error (0.0004) is actually a bit smaller than $\\Delta x^4 = 0.0016$, but approximately the same order of magnitude.\n", + "\n", + "Matlab also offers a built-in RK4 integrator: `ode45`. (It is actually slightly more complicated than the equations shown just now, because it automatically adjusts the step size $\\Delta x$ to control error.) You can call this function with the syntax:\n", + " \n", + " [X, Y] = ode45(function_name, [x_start x_end], [IC]);\n", + "\n", + "where `function_name` is the name of a function that provides the derivative (this can be a regular function given in a file, or an anonymous function); `[x_start x_end]` provides the domain of integration ($x_{\\text{start}} \\leq x \\leq x_{\\text{end}}$), and `[IC]` provides the initial condition $y(x=x_{\\text{start}})$.\n", + "\n", + "For example, let's use this and compare with our exact solution:" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Maximum error: 0.0007" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "\n", + "f = @(x,y) 8*exp(-x)*(1 + x) - 2*y;\n", + "\n", + "[X, Y] = ode45(f, [0 7], [1]);\n", + "\n", + "x_exact = linspace(0, 7);\n", + "y_exact = @(x) exp(-2.*x).*(8*x.*exp(x) + 1);\n", + "plot(x_exact, y_exact(x_exact)); hold on\n", + "plot(X, Y, 'o--')\n", + "legend('Exact solution', \"ode45 solution\")\n", + "\n", + "fprintf('Maximum error: %6.4f', max(abs(y_exact(X) - Y)))" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.7" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/docs/_sources/content/numerical-methods/integrals.ipynb b/docs/_sources/content/numerical-methods/integrals.ipynb new file mode 100644 index 0000000..271427f --- /dev/null +++ b/docs/_sources/content/numerical-methods/integrals.ipynb @@ -0,0 +1,310 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Numerical integrals\n", + "\n", + "What about when we cannot integrate a function analytically? In other words, when there is no (obvious) closed-form solution. In these cases, we can use **numerical methods** to solve the problem.\n", + "\n", + "Let's use this problem:\n", + "\\begin{align}\n", + "\\frac{dy}{dx} &= e^{-x^2} \\\\\n", + "y(x) &= \\int e^{-x^2} dx + C\n", + "\\end{align}\n", + "\n", + "(You may recognize this as leading to the error function, $\\text{erf}$:\n", + "$\\frac{1}{2} \\sqrt{\\pi} \\text{erf}(x) + C$,\n", + "so the exact solution to the integral over the range $[0,1]$ is 0.7468.)" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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MZvNnn31mMpmsVuvmzZsrKytvuOEG72dPnjy5e/du72FbW1t8fDyFhHFu3V3JD85PWF1+7M2i0/sfm5ukjVQ7EXB5ZCyklpaW+vr6TZs2GY3G8+fPL1u2rLS0tKioyDvAZDKZTCbv4AcffLCgoEClsIBEmCohqMm4htTU1BQVFZWWliaECA8PNxqNTU1Nw47s7e01m82PPvro7bffHtCIgMS4MCuClIwzJKfTqdPpNJqLZRkfH+90OocduXnzZiHEqlWrhv2s1Wpdu3at8njFihUGg2EMwgIyUjY7CCGYKo1zVqu1srLS+zgzM1PdPJcmYyENDAz4Hmo0GrfbPXTY2bNnS0tL8/Pzw8PDf+pLLViwQHmg1+v9GxKQn7KqtPXQ6eSig6wqjU96vd77Mig/GQspIiLC5XJ5D3t6eiIjh/mHtHPnTo/Hs3Tp0p/6OgaDQfI/B4Cxxj0sxjm9Xu99GTx16pS6YUYk4xpSYmJiZ2enw+FQDm0226Bt34qdO3f+4he/mDx5cmDTAcHH9x4WrCpBWjIWUmpqanR0dEFBgd1ur62t3bt3r7KnLj8/v6ysTBnj8XhsNtttt92malIgaChTpXVLkleXH+OyDpCTjIUUGxtrsVgaGhqMRmNeXl56enp2drYQoqamprGxURlz+vRpl8t14403qpoUCDJMlSAzGdeQhBAmk2nx4sUOhyMmJsa7gHT06FHvgOuvv/7Pf/6zSumAIMaqEqQl4wxJERYWNm3atGG3MwC4SkyVICFJZ0gAxhpTJchG3hkSgABgqgR5MEMCxjumSpAEMyQAQjBVggSYIQG4iKkS1MUMCcCPMFWCWigkAIMpU6WceQmLNh3hsg4IGAoJwPC4rxICjEIC8JOYKiGQKCQAI2CqhMCgkACMjKkSAoBCAjBaylTpQFsXUyWMBQoJwGVI0kbuf3wuUyWMBQoJwGXzTpUWvXqEqRL8hUICcCWStJFbsm5NS4ljqgR/oZAAXCFlpwNTJfgLhQTgqjBVgr9QSACulneq9Gbj6dXlNqZKuDIUEgD/SNJG7n9sblJcJFMlXBkKCYDf+E6V6CRcLgoJgJ8pUyUhRHLRQWoJo0chAfA/pkq4ApLeMdbtdldXV9tsNq1Wm5GRkZiYOHTMkSNHPvnkk6ioqLvuumvYAQDUpUyVth46nVx0cEvWLWkpcWongtQknSGZzebCwsLu7u66urqMjIz29vZBA7Zv375q1aovvvhi7969JpOpo4M/wQAZeadKq8ttTJVwaTIWUktLS319fXFxscViqaio0Ol0paWlvgP+93//d8OGDf/1X//1xz/+sby8fN68eTt27FArLYAR+a4qHWjrUjsOJCXjKbumpqaoqKi0tDQhRHh4uNFo/Pjjj30HHDhwICIi4t577z1x4kR/f/+2bds0GhmbFYCXMlW688bY53Z3fJRydt1dyWongnRkLCSn06nT6bwdEx8f73Q6fQecPHlSq9X+5je/OXbs2MDAQEpKyh/+8Ief//zng76O3W63Wq3KY71er9frAxAewCWkpcQlZUUpq0r7H5ubpI1UO1GIs9vtdrvd+1jyl0EZC2lgYMD3UKPRuN1u34+cO3fu66+/XrVqVXl5+XfffffQQw89//zzmzZtGvR1rFar9yeRm5sr+U8CGCeUqdIMbeTq8mNpKXFMlcaU3W4vKSnxPs7MzFQ3z6XJWEgREREul8t72NPTExn5oz+jrr32Wo1G89RTT02cOPFnP/vZypUrX3nllaFfJzMzMzc3d8zjArh8OfMT0lLimCqNNYPB8NZbbymPvc0kLRmXXhITEzs7Ox0Oh3Jos9kG7eq+4YYbhBDeaVN/f/+ECTI2K4BLUKZK65Ykc6khKGQspNTU1Ojo6IKCArvdXltbq2zsFkLk5+eXlZUJIe68885JkyatX7++q6vriy++2LZt269+9Su1UwO4EjnzE7wb8Lgq6zgnYyHFxsZaLJaGhgaj0ZiXl5eenp6dnS2EqKmpaWxsFELExMRs2rTp888//4d/+If777//tttuW7NmjdqpAVwhZarEbdER5vF41M4wPI/H43A4YmJiBi0g+Tp79mxUVFRERMTQTylnS1lDAoLIcWff6vJjx7v6WFUaC/K/Kso4Q1KEhYVNmzbtEm0khIiNjR22jQAEoyRt5P7H5zJVGrfkLSQA4xO3RR+3KCQA0uG26OMThQRARt6rsjJVGj8oJADyYqo0rlBIAKTme6+/1eU2pkohjEICEASUG1gkxUUyVQphFBKA4MBt0UMehQQgmPje649aCjEUEoAgw1QpVFFIAIISU6XQQyEBCFZMlUIMhQQguPlOlQ60dakdB1eOQgIQ9LxTpdXlNqZKwYtCAhAimCoFOwoJQOhgqhTUKCQAoYapUpCikACEIKZKwYhCAhCymCoFFwoJQChjqhREKCQAoY+pUlCgkACMC0yV5EchARhHmCrJjEICML4wVZLWBLUDDM/tdldXV9tsNq1Wm5GRkZiYOGjAkSNHmpqavIdGo3HGjBmBzQggiClTpa2HTicXHdySdUtaSpzaiSBrIZnN5s8++8xkMlmt1s2bN1dWVt5www2+A955553m5ubk5GTlcM6cORQSgMuiTJUenJ+waNORnHkJ6+5KVjvReCdjIbW0tNTX12/atMloNJ4/f37ZsmWlpaVFRUW+Y1pbW//t3/7t3nvvVSskgNDgO1WiltQl4xpSU1NTVFRUWlqaECI8PNxoNPqenRNCXLhwob29PTY2dteuXZ9++umFCxfUCQogJHBfJUnIWEhOp1On02k0F7PFx8c7nU7fAW1tbW63+8knn3zppZceeeSRe+65x+FwDP06L7/88qwfVFVVBSI6gKAVkregraqq8r4Mvvzyy2rHGYGMhTQwMOB7qNFo3G6370f6+/t//etf79q1q66ubufOnV1dXcN+o81m859/kJmZObahAQQ/36nS6nLbcWef2omuVmZmpvdl0Gw2qx1nBDIWUkREhMvl8h729PRERkb6DpgzZ47FYlG23s2YMWPZsmXNzc2BTgkgRClTpaS4yEWbjoTMVCkoyFhIiYmJnZ2d3rNwNptt0LbvP/3pTxs3bvQe9vT0TJw4MaARAYQ036nSolePhMBUKSjIWEipqanR0dEFBQV2u722tnbv3r0mk0kIkZ+fX1ZWJoSYPHnyG2+8UVlZ+be//a2+vn7nzp1LlixROzWAUKNMldJS4pgqBYaMhRQbG2uxWBoaGoxGY15eXnp6enZ2thCipqamsbFRCPFP//RPjzzyyLp16+bOnfvkk09mZ2fn5OSoHBpAKPJOlQ60dSUXHWSqNKbCPB6P2hmG5/F4HA5HTEzMoAUkr/7+fqfTOWXKlGuuuWboZ0tKSoQQubm5Y5sSwPhw3Nm39dDpNxtPB+97leR/VZRxhqQICwubNm3aT7WREGLChAk6nW7YNgIA/2KqFADyFhIAyCZJG7n/8bk58xJYVRoLFBIAXB5lqiS4h4W/yXgtOwCQnHIGb4Y28rndHR+lnA3SVSXZMEMCgCuUMz9hS9atgqmSn1BIAHDluN2fH1FIAHC1QvLCrIFHIQGAH4TehVkDj0ICAL/hwqxXg0ICAH/iLbRXjEICAP/jLbRXgEICgLHCW2gvC4UEAGNIOYO3JesW9oWPiEICgDGXlhLHvvARUUgAEAjchXZEFBIABA53ob0ECgkAAso7VRJsdvgxrvYNACrgeuFDMUMCANX4Xi+cM3gUEgCoic0OXhQSAKiPzQ6CQgIASbDZgUICAIkMurLDuDqDJ2khud3u9957r7Cw8NVXXz158uQlRlZVVZWXlwcsGAAEgPfKDuPqDJ6khWQ2mwsLC7u7u+vq6jIyMtrb24cd1tLS8uyzz3766acBjgcAY20c3sZCxkJqaWmpr68vLi62WCwVFRU6na60tHTosL6+vry8vOnTpwc+IQAEhnIbi3VLkleXHwv5qZKMhdTU1BQVFZWWliaECA8PNxqNTU1NQ4dt3Lhx1qxZJpMp0PkAILDGyduVZLxSg9Pp1Ol0Gs3FsoyPj3c6nYPGfPzxx7t3766trd20adNPfZ2qqiqr1ao8zs3NNRgMYxQYAMaacgbvwfkJq8uPvVl0ev9jc5O0kSP+V1artaSkRHlst9szMzPHOOZVkbGQBgYGfA81Go3b7fb9yNmzZ9euXbt+/frY2NhLfB2DwbBixQrlsV6v93tOAAiwJG3klqxbtx46vWjTkZx5CSNecEiv1+fm5iqPKysrxz7gVZGxkCIiIlwul/ewp6cnMvJHfwgUFxdPmjTp3Llz1dXVX331VXd3965du+6+++5BX0ev1zMrAhBivFOlrYdOJxcdvHQt6fV675/jDQ0Ngcp4hWRcQ0pMTOzs7HQ4HMqhzWZLTEz0HTB58uSYmJiysrKysrLW1tavv/6and8AxpWQ3IMn4wwpNTU1Ojq6oKBgzZo1n3/++d69e9euXSuEyM/Pv/XWW7Ozs/Py8ryDn3/++W+//fb3v/+9enkBQB2XewZPcjLOkGJjYy0WS0NDg9FozMvLS09Pz87OFkLU1NQ0NjYOHe/d/gAA482gCw4F9R68MI/Ho3aG4Xk8HofDERMTM2gBaZSUjSXe1TwACHnHnX2ry48d7+obdg+e/K+K8s4twsLCpk2bdmVtBADjkHIGL2deQpC+i1beQgIAXK4frs0alO+ipZAAINQM2oMXLHeykHGXHQDg6inXwXvz0OnV5ba0lLgpXX0z4qReBGGGBAChLGd+wv7H5ibFRb50Oqk/eoracS6FGRIAhDjlDJ72L7XCNfJgFTFDAgBIgUICAEiBQgIASIFCAgBIgUICAEiBQgIASIFCAgBIgUICAEiBQgIASIFCAgBIgUICAEiBQgIASIFCAgBIgUICAEiBQgIASIFCAgBIgUICAEhB0jvGut3u6upqm82m1WozMjISExMHDejt7a2oqOjo6Jg+fXpmZqZOp1MlJwDAXySdIZnN5sLCwu7u7rq6uoyMjPb2dt/Put3uf/mXf3n99dd7enq2b99+zz33dHZ2qhUVAOAXMhZSS0tLfX19cXGxxWKpqKjQ6XSlpaW+Az766KMvv/yyvLzcYrGUlZWdO3du//79aqUFAPiFjIXU1NQUFRWVlpYmhAgPDzcajU1NTb4Drr322kceeSQhIUEIERUVpdFoYmJiVIkKAPAXGdeQnE6nTqfTaC6WZXx8vNPp9B2wYMGCBQsWOJ3OV155Zf/+/ampqUajcejXsdvtVVVVymODwaDX68c6OQBIxW63W61W5bHVajUYDOrmuTQZC2lgYMD3UKPRuN3uYYeFh4cnJCTYbLbm5ub58+cPGuD9MQgh9Ho9hQRgvLHb7Q0NDWqnGC0ZCykiIsLlcnkPe3p6IiMjfQf09vYKIaZOnbpmzRohxEMPPVRaWjq0kDIzM3Nzc8c+LwBIymAweGdFJSUl6oYZkYxrSImJiZ2dnQ6HQzm02WyDtn0XFRVlZWV5D1NSUux2e0AjAgD8TcZCSk1NjY6OLigosNvttbW1e/fuNZlMQoj8/PyysjIhhMFgaG1tff311x0OxyeffFJdXb1w4UK1UwMAroqMp+xiY2MtFsuzzz5rNBo1Gk16enp2drYQoqamxuVyZWdn33PPPX/+859ffPHFjRs3KgP+/d//Xe3UAICrImMhCSFMJtPixYsdDkdMTIx3Aeno0aPeAXl5ef/xH//hcDji4uLCw8NVigkA8BtJC0kIERYWNm3atEsMuOaaa6ZPnx6wPACAMSXjGhIAYByikAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFKYoHaA4bnd7urqapvNptVqMzIyEhMTBw04f/58dXX1X/7yl6lTpy5fvjw+Pl6VnAAAf5F0hmQ2mwsLC7u7u+vq6jIyMtrb230/63a7H3jggRdffNHlcv33f//33Xffffz4cZWSAgD8Q8ZCamlpqa+vLy4utlgsFRUVOp2utLTUd0B9ff3Ro0fffvvtDRs2VFZWxsTE7NixQ620AAC/kLGQmpqaoqKi0tLShBDh4eFGo7GpTiqKOAAADzNJREFUqcl3wJkzZ+bNm3fDDTcIIaKiohISErq6ulSJCgDwFxnXkJxOp06n02gulmV8fLzT6fQdkJWVlZWVpTxubGxsbm5euXLl0K9jt9vtdrvyWK/Xj2VkAJCR9zUwKMhYSAMDA76HGo3G7XYPHebxeLZv3/7CCy8sXbr0nnvuGTrAarX+67/+q/I4Nzc3MzNzLNICgLTsdvvatWu9j81ms7p5Lk3GQoqIiHC5XN7Dnp6eyMjIQWP++te/PvXUUzabbe3atVlZWWFhYUO/TmZmZm5u7thmBQCJGQyG+vp65XFJSYm6YUYkYyElJiZ2dnY6HI6pU6cKIWw226Bt3729vQ888MD06dM/+OCDadOmqRQTAOBPMm5qSE1NjY6OLigosNvttbW1e/fuNZlMQoj8/PyysjIhxLvvvtvZ2fnMM8/09PR0dHR0dHScOXNG7dQAgKsi4wwpNjbWYrE8++yzRqNRo9Gkp6dnZ2cLIWpqalwuV3Z29uHDh10ul++a0KpVq/Lz89WLDAC4WjIWkhDCZDItXrzY4XDExMR4F5COHj2qPJD/TCgA4HJJWkhCiLCwMNaHAGD8kHENCQAwDlFIAAApUEgAAClQSAAAKVBIAAApUEgAAClQSAAAKVBIAAApUEgAAClQSAAAKVBIAAApUEgAAClQSAAAKVBIAAApUEgAAClQSAAAKVBIAAApUEgAAClQSAAAKVBIAAApUEgAAClQSAAAKVBIAAApTFA7wPDcbnd1dbXNZtNqtRkZGYmJicMOO3bs2L59+3JzcwMcDwDgd5LOkMxmc2FhYXd3d11dXUZGRnt7+9AxDoejoKBg9+7dgY8HAPA7GQuppaWlvr6+uLjYYrFUVFTodLrS0tJBY1asWPHLX/7y888/VyUhAMDvZDxl19TUFBUVlZaWJoQIDw83Go0ff/zxoDHr168/f/58TU1NY2OjChEBAP4mYyE5nU6dTqfRXJy9xcfHO53OQWNmz54thDh8+PAlCqmqqspqtSqPc3NzDQbD2OQFAElZrdaSkhLlsd1uz8zMVDfPpclYSAMDA76HGo3G7XZfwdcxGAwrVqxQHuv1ej8kA4Cgotfrvdu+Kisr1Q0zIhkLKSIiwuVyeQ97enoiIyOv4Ovo9XpmRQDGM71e7/1zvKGhQd0wI5JxU0NiYmJnZ6fD4VAObTbbT237BgCEDBkLKTU1NTo6uqCgwG6319bW7t2712QyCSHy8/PLysrUTgcAGBMyFlJsbKzFYmloaDAajXl5eenp6dnZ2UKIYffUhYWFqZERAOBnMq4hCSFMJtPixYsdDkdMTIx3Aeno0aODhj388MMPP/xwwNMBAPxP0kISQoSFhU2bNk3tFACAAJHxlB0AYByikAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFKgkAAAUqCQAABSoJAAAFII1kJyu93vvfdeYWHhq6++evLkSbXjDK+qqkrdAFar1W63q5tB9W+C6gHsdrvValU3g+rfBH4VZQggv2AtJLPZXFhY2N3dXVdXl5GR0d7ernaiwex2e0lJiboZSkpK1H0VsFqtlZWVKgYQEnwTKisrGxoaVAzAr6LgVzFIBGUhtbS01NfXFxcXWyyWiooKnU5XWlqqdigAwFUJykJqamqKiopKS0sTQoSHhxuNxqamJrVDAQCuygS1A1wJp9Op0+k0mottGh8f73Q6hw6zWq0qnqlQ5ubqniqx2+3qni+y2+2SnC/S6/VqPbuygMSvIr+KQoJfRYPBoNazj0aYx+NRO8Nle+mll2pra/fs2aMcbtu27ZVXXhn6u676Lx8ASCU3N1ftCJcSlDOkiIgIl8vlPezp6YmMjBw6TPJvPQDAV1CuISUmJnZ2djocDuXQZrMlJiaqGwkAcJWCspBSU1Ojo6MLCgrsdnttbe3evXtNJpPaoQAAVyUo15CEEB9++OGzzz579uxZjUaTnp7+u9/9bsKEoDz9CABQBGshCSE8Ho/D4YiJiRl2AQkAEFyCuJAAAKEkKNeQAAChJwTXXdxud3V1tc1m02q1GRkZAdiAN8pnPHbs2L59+8ZoM/qIGc6fP19dXf2Xv/xl6tSpy5cvj4+PD3CA3t7eioqKjo6O6dOnZ2Zm6nQ6/wYYTQavqqqq77//PisrK8ABjhw54ntVEaPROGPGjEAGUDJ88sknUVFRd91111j867h0htbW1k8//dT3I9ddd92KFSsCFkD88G+htbV1ypQpy5Yt8++PYJQZent7Kysr29vbExIS7rvvvtjYWL9nGGpMX4L8IgRP2T322GOfffaZyWRqbW09ceJEZWXlDTfcoPozOhwOs9nc09NTW1sb+Axut3vlypWnTp1KS0v7/PPPv/vuu8rKyqSkpEAGuO+++5xO54IFC6xWa19f3/vvvz9t2jQ/Bhgxg1dLS8v999+/ePHi3//+9wEO8PTTTzc3NycnJyuHDz/88Pz58wMZYPv27UVFRf/4j/947ty5//mf//nggw+8YQKT4cMPP/S98mRbW1t8fPz7778fsABCiFWrVtlsNpPJ1NTUdPLkyffee++mm27yY4ARM7hcrhUrVnz//fcLFy60Wq3ff//9zp07J0+e7N8Mg4z1S5B/eELLF198MXPmzH379nk8nu+//95kMv3nf/6n6s+YmZl5yy23zJw5Mz09XZUMe/bsufnmm9va2jwej8vl+uUvf2mxWAIZYN++fbNnz/7mm288Hs+pU6duvvnmd999148BRpNB0dvbu3Tp0kWLFj3xxBOBD5Cenl5RUeHf5x19gO7u7jlz5pSXlyuH2dnZL7zwQoAzDBp8xx13HDlyJJABTpw4MXPmzPr6eo/H43K57rjjjhdffNGPAUaTYevWrX//93/f2dnp8Xi6u7sXLlz40ksv+TfDIGP9EuQvoXbKbuh1Vz/++GPVn3H9+vXnz5+vqalpbGxUJcOZM2fmzZun/I0WFRWVkJDQ1dUVyADXXnvtI488kpCQoATQaDQxMTF+DDCaDIqNGzfOmjVLp9N9++23AQ5w4cKF9vb22NjYXbt2xcXFzZs3b+LEiYEMcODAgYiIiHvvvffEiRP9/f3btm3zXhAyYBm8ent7zWbzo48+evvttwcywHXXXXfNNdf09fUJIS5cuHDhwoWpU6f6McBoMrS1tc2aNUt53piYmNtvv725udm/GQYZ65cgfwm1QhrldVcD/IyzZ88WQhw+fHiMfhtGzJCVleVdL2lsbGxubl65cmUgAyxYsGDBggVOp/OVV17Zv39/amqq0Wj0Y4DRZBBCfPzxx7t3766trd20aZN/n300Adra2txu95NPPqnX60+dOpWYmPjWW2/58dVwxAAnT57UarW/+c1vjh07NjAwkJKS8oc//OHnP/+5vwKMJoPX5s2bhRCrVq3y47OPJsDkyZPXrFnz9NNPV1VVHT16dPbs2cuXLw9whunTp9fV1blcrujoaLfb/eWXX4aHh/s3wyBj/RLkL6G2y25gYMD3UKPRuN3uEHvGK87g8Xjeeeed3/72t0uXLr3nnnsCH2BgYCA8PDwhIcFms/n9T8IRM5w9e3bt2rXr168fowXkEQP09/f/+te/3rVrV11d3c6dO7u6ul5++eVABjh37tzXX389d+7c5ubmffv29ff3P//8834MMJoMirNnz5aWlj766KN+fyEeMUB3d/eOHTvi4uL0ev2NN95os9kOHjwY4AzLly/v6+vLycnZunXrb3/72++++27QfzJuhdoMaZTXXQ3qZ7yyDH/961+feuopm822du3arKyssLCwQAbo7e0VQkydOnXNmjVCiIceeqi0tNS/6/kjZiguLp40adK5c+eqq6u/+uqr7u7uXbt23X333QELMGfOHIvFojyeMWPGsmXLjhw54q9nH02Aa6+9VqPRPPXUUxMnTvzZz362cuXKV155xY8BRpNBsXPnTo/Hs3TpUv8++2gCHDhw4MSJEx9++OH06dOFEE888cRrr73m32uPjZjh+uuv37FjxxtvvLFv3z6DwXDTTTe1trb6MUDwCrUZUuCvuyrDlV5HzNDb2/vAAw8IIT744IOVK1f6t41GE6CoqMh3j3VKSorf7+U8YobJkyfHxMSUlZWVlZW1trZ+/fXX5eXlgQzwpz/9aePGjd7Dnp4e/64hjRhAWUT0/rXe39/v9wtujfKfw86dO3/xi1+Mxb6yEQOcOnVq4sSJcXFxyuGtt94a+F/Fr7766vDhwxs2bNi2bZvZbG5qavL7Nr8gFWqFFPjrrv7UM+bn55eVlY3pU48+w7vvvtvZ2fnMM8/09PR0dHR0dHScOXMmkAEMBkNra+vrr7/ucDg++eST6urqhQsX+jHAaDLk5eXt+MGyZcsWLly4bdu2QAaYPHnyG2+8UVlZ+be//a2+vn7nzp1LliwJZIA777xz0qRJ69ev7+rq+uKLL7Zt2/arX/3KjwFGk0EI4fF4bDbbbbfd5t+nHmWAefPm9fT0FBcXf/PNN83Nzdu3bw/8r6JWq33hhRc2b9585syZt99+u7m52e/rWMFK5V1+Y2DPnj0Gg2HmzJk333zzU089deHCBVWecc6cOU8++aTvsNdeey0jI0OVDGazeeaPFRYWBjKAx+OxWCyzZ8/2DnC5XP4NMJoMXhs2bBj6wQAEKC4unjNnzsyZM//u7/7u+eef7+/vD3CAQ4cOLVmyZObMmbfccsvjjz9+7tw5/wYYTQa73T5z5sw9e/b4/alHGWDLli1z585V/iE88sgjTqcz8BneeeedxYsXz5w584477njnnXf8HmBYY/oS5Bch+MZYocZ1VwP/jBJmGDGA2+12OBxxcXFjt6dI/m9Cf3+/0+mcMmXKNddco0oAIcTZs2ejoqIiIiLGIsAoM4ypEQMMDAw4HI5JkyZFR0erlUEIcebMGa1W6/dT6MErNAsJABB0Qm0NCQAQpCgkAIAUKCQAgBQoJACAFCgkAIAUKCQAgBQoJACAFCgkAIAUKCQAgBQoJACAFCgkAIAUKCQAgBQoJACAFCgkAIAUKCQAgBQoJACAFCgkAIAUKCQAgBQoJACAFCgkAIAUKCQAgBQoJACAFCgkAIAUKCQAgBQoJACAFCgkAIAU/g+77wsY7feeeQAAAABJRU5ErkJggg==\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x = linspace(0, 1);\n", + "f = @(x) exp(-x.^2);\n", + "plot(x, f(x))\n", + "axis([0 1 0 1])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Numerical integration: Trapezoidal rule\n", + "\n", + "In such cases, we can find the integral by using the **trapezoidal rule**, which finds the area under the curve by creating trapezoids and summing their areas:\n", + "\\begin{equation}\n", + "\\text{area under curve} = \\sum \\left( \\frac{f(x_{i+1}) + f(x_i)}{2} \\right) \\Delta x\n", + "\\end{equation}\n", + "\n", + "Let's see what this looks like with four trapezoids ($\\Delta x = 0.25$):" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "hold off\n", + "x = linspace(0, 1);\n", + "plot(x, f(x)); hold on\n", + "axis([0 1 0 1])\n", + "\n", + "x = 0 : 0.25 : 1;\n", + "\n", + "% plot the trapezoids\n", + "for i = 1 : length(x)-1\n", + " xline = [x(i), x(i)];\n", + " yline = [0, f(x(i))];\n", + " line(xline, yline, 'Color','red','LineStyle','--')\n", + " xline = [x(i+1), x(i+1)];\n", + " yline = [0, f(x(i+1))];\n", + " line(xline, yline, 'Color','red','LineStyle','--')\n", + " xline = [x(i), x(i+1)];\n", + " yline = [f(x(i)), f(x(i+1))];\n", + " line(xline, yline, 'Color','red','LineStyle','--')\n", + "end\n", + "hold off" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now, let's integrate using the trapezoid formula given above:" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Numerical integral: 0.746211\n", + "Exact integral: 0.746824\n", + "Error: 0.082126 %\n" + ] + } + ], + "source": [ + "dx = 0.1;\n", + "x = 0.0 : dx : 1.0;\n", + "\n", + "area = 0.0;\n", + "for i = 1 : length(x)-1\n", + " area = area + (dx/2)*(f(x(i)) + f(x(i+1)));\n", + "end\n", + "\n", + "fprintf('Numerical integral: %f\\n', area)\n", + "exact = 0.5*sqrt(pi)*erf(1);\n", + "fprintf('Exact integral: %f\\n', exact)\n", + "fprintf('Error: %f %%\\n', 100.*abs(exact-area)/exact)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can see that using the trapezoidal rule, a numerical integration method, with an internal size of $\\Delta x = 0.1$ leads to an approximation of the exact integral with an error of 0.08%.\n", + "\n", + "You can make the trapezoidal rule more accurate by:\n", + "\n", + "- using more segments (that is, a smaller value of $\\Delta x$, or\n", + "- using higher-order polynomials (such as with Simpson's rules) over the simpler trapezoids.\n", + "\n", + "First, how does reducing the segment size (step size) by a factor of 10 affect the error?" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Numerical integral: 0.746818\n", + "Exact integral: 0.746824\n", + "Error: 0.000821 %\n" + ] + } + ], + "source": [ + "dx = 0.01;\n", + "x = 0.0 : dx : 1.0;\n", + "\n", + "area = 0.0;\n", + "for i = 1 : length(x)-1\n", + " area = area + (dx/2)*(f(x(i)) + f(x(i+1)));\n", + "end\n", + "\n", + "fprintf('Numerical integral: %f\\n', area)\n", + "exact = 0.5*sqrt(pi)*erf(1);\n", + "fprintf('Exact integral: %f\\n', exact)\n", + "fprintf('Error: %f %%\\n', 100.*abs(exact-area)/exact)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "So, reducing our step size by a factor of 10 (using 100 segments instead of 10) reduced our error by a factor of 100!" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Numerical integration: Simpson's rule\n", + "\n", + "We can increase the accuracy of our numerical integration approach by using a more sophisticated interpolation scheme with each segment. In other words, instead of using a straight line, we can use a polynomial. **Simpson's rule**, also known as Simpson's 1/3 rule, refers to using a quadratic polynomial to approximate the line in each segment.\n", + "\n", + "Simpson's rule defines the definite integral for our function $f(x)$ from point $a$ to point $b$ as\n", + "\\begin{equation}\n", + "\\int_a^b f(x) \\approx \\frac{1}{6} \\Delta x \\left( f(a) + 4 f \\left(\\frac{a+b}{2}\\right) + f(b) \\right)\n", + "\\end{equation}\n", + "where $\\Delta x = b - a$.\n", + "\n", + "That equation comes from interpolating between points $a$ and $b$ with a third-degree polynomial, then integrating by parts." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "hold off\n", + "x = linspace(0, 1);\n", + "plot(x, f(x)); hold on\n", + "axis([-0.1 1.1 0.2 1.1])\n", + "\n", + "plot([0 1], [f(0) f(1)], 'Color','black','LineStyle',':');\n", + "\n", + "% quadratic polynomial\n", + "a = 0; b = 1; m = (b-a)/2;\n", + "p = @(z) (f(a).*(z-m).*(z-b)/((a-m)*(a-b))+f(m).*(z-a).*(z-b)/((m-a)*(m-b))+f(b).*(z-a).*(z-m)/((b-a)*(b-m)));\n", + "plot(x, p(x), 'Color','red','LineStyle','--');\n", + "\n", + "xp = [0 0.5 1];\n", + "yp = [f(0) f(m) f(1)];\n", + "plot(xp, yp, 'ok')\n", + "hold off\n", + "legend('exact', 'trapezoid fit', 'polynomial fit', 'points used')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can see that the polynomial fit, used by Simpson's rule, does a better job of of approximating the exact function, and as a result Simpson's rule will be more accurate than the trapezoidal rule.\n", + "\n", + "Next let's apply Simpson's rule to perform the same integration as above:" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Simpson rule integral: 0.746824\n", + "Exact integral: 0.746824\n", + "Error: 0.000007 %\n" + ] + } + ], + "source": [ + "dx = 0.1;\n", + "x = 0.0 : dx : 1.0;\n", + "\n", + "area = 0.0;\n", + "for i = 1 : length(x)-1\n", + " area = area + (dx/6.)*(f(x(i)) + 4*f((x(i)+x(i+1))/2.) + f(x(i+1)));\n", + "end\n", + "\n", + "fprintf('Simpson rule integral: %f\\n', area)\n", + "exact = 0.5*sqrt(pi)*erf(1);\n", + "fprintf('Exact integral: %f\\n', exact)\n", + "fprintf('Error: %f %%\\n', 100.*abs(exact-area)/exact)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Simpson's rule is about three orders of magnitude (~1000x) more accurate than the trapezoidal rule.\n", + "\n", + "In this case, using a more-accurate method allows us to significantly reduce the error while still using the same number of segments/steps." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.7" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/docs/_sources/content/numerical-methods/numerical-methods.md b/docs/_sources/content/numerical-methods/numerical-methods.md new file mode 100644 index 0000000..82c8308 --- /dev/null +++ b/docs/_sources/content/numerical-methods/numerical-methods.md @@ -0,0 +1,3 @@ +# Numerical Methods + +This chapter describes numerical methods used to solve integrals and first-order ordinary differential equations, along with concepts related to these such as error and stability. diff --git a/docs/_sources/content/numerical-methods/stability.ipynb b/docs/_sources/content/numerical-methods/stability.ipynb new file mode 100644 index 0000000..bcc5ffc --- /dev/null +++ b/docs/_sources/content/numerical-methods/stability.ipynb @@ -0,0 +1,404 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Stability and Stiffness\n", + "\n", + "In the past when you've talked about **stability**, it has likely been regarding the stability of a *system*. Stable systems are those will well-behaved exact solutions, meaning they do not grow unbounded.\n", + "In engineering we mostly focus (or want!) stable systems, although there are some interesting unstable systems such as those involving resonance, nonlinear dynamics, or chaos—generally we want to know when that happens so we can prevent it.\n", + "\n", + "We can also define the stability of a *numerical scheme*, which is when the numerical solution exhibits unphysical behavior. In other words, it blows up.\n", + "\n", + "For example, let's consider the relatively simple 1st-order ODE\n", + "\\begin{equation}\n", + "\\frac{dy}{dt} = -3 y\n", + "\\end{equation}\n", + "with the initial condition $y(0) = 1$. As we will see, this ODE can cause explicit numerical schemes to become unstable, and thus it is a **stiff** ODE. (Note that we can easily obtain the exact solution for this problem, which is $y(t) = e^{-3 t}$.)\n", + "\n", + "Let's try solving this with the Forward Euler method, integrating over $0 \\leq t \\leq 10$, for a range of time-step size values: $\\Delta t = 0.1, 0.25, 0.5, 0.75$:" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "f = @(t,y) -3*y;\n", + "\n", + "dt = 0.1;\n", + "t = 0 : dt : 20;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) + dt*f(t(i), y(i));\n", + "end\n", + "subplot(4,1,1);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));\n", + "\n", + "dt = 0.25;\n", + "t = 0 : dt : 20;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) + dt*f(t(i), y(i));\n", + "end\n", + "subplot(4,1,2);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));\n", + "\n", + "dt = 0.5;\n", + "t = 0 : dt : 20;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) + dt*f(t(i), y(i));\n", + "end\n", + "subplot(4,1,3);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));\n", + "\n", + "dt = 0.75;\n", + "t = 0 : dt : 20;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) + dt*f(t(i), y(i));\n", + "end\n", + "subplot(4,1,4);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "At the smaller step sizes, $\\Delta t = 0.1$ and $\\Delta t = 0.25$, we see that the solution is well-behaved. But, when we increase $\\Delta t$ to 0.5, we see some instability that goes away with time. Then, when we increase $\\Delta t$ to 0.75, the solution eventually blows up, leading to error **much** larger than what we should expect based on the method's order of accuracy (first) and the step size value.\n", + "\n", + "Compare this behavior to that for the ODE\n", + "\\begin{equation}\n", + "\\frac{dy}{dt} = e^{-t}\n", + "\\end{equation}\n", + "which is **non-stiff**:" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "f = @(t,y) exp(-t);\n", + "\n", + "dt = 0.1;\n", + "t = 0 : dt : 10;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) + dt*f(t(i), y(i));\n", + "end\n", + "subplot(4,1,1);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));\n", + "\n", + "dt = 0.25;\n", + "t = 0 : dt : 10;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) + dt*f(t(i), y(i));\n", + "end\n", + "subplot(4,1,2);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));\n", + "\n", + "dt = 0.5;\n", + "t = 0 : dt : 10;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) + dt*f(t(i), y(i));\n", + "end\n", + "subplot(4,1,3);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));\n", + "\n", + "dt = 0.75;\n", + "t = 0 : dt : 10;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) + dt*f(t(i), y(i));\n", + "end\n", + "subplot(4,1,4);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this case, we see that the solution remains well-behaved even for larger time-step sizes, and the error matches the expected order based on the method and step-size value.\n", + "\n", + "In general numerical schemes can be:\n", + "\n", + "- **unstable**: the scheme blows up for any choice of parameters\n", + "- **conditionally stable**: the scheme is stable for a particular choice of parameters (for example, $\\Delta t$ is less than some threshold\n", + "- **unconditionally stable**: the scheme is always stable\n", + "\n", + "Schemes may be stable for some problem/system and not for another, and vice versa.\n", + "\n", + "Stability is related to robustness of a method, which is generally a tradeoff between complexity and computational cost. The choice of method and solution strategy depends on what you want, and how long you can wait for it. In general, we almost always want to use the largest $\\Delta t$ allowable.\n", + "\n", + "Rather than reducing $\\Delta t$ to avoid stability problems, we can also use a method that is unconditionally stable, such as the **Backward Euler** method." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Backward Euler method\n", + "\n", + "The Backward Euler method is very similar to the Forward Euler method, except in one way: it uses the slope at the *next* time step: \n", + "\\begin{equation}\n", + " \\left(\\frac{dy}{dx}\\right)_{i+1} \\approx \\frac{y_{i+1} - y_i}{\\Delta x}\n", + "\\end{equation}\n", + "Then, the resulting recursion formula is\n", + "\\begin{equation}\n", + "y_{i+1} = y_i + \\Delta x \\left(\\frac{dy}{dx}\\right)_{i+1}, \\text{ or} \\\\\n", + "y_{i+1} = y_i + \\Delta x \\, f(x_{i+1}, y_{i+1})\n", + "\\end{equation}\n", + "where $f(x,y) = dy/dx$.\n", + "\n", + "Notice that this recursion formula cannot be directly solved, because $y_{i+1}$ shows up on both sides. This is an **implicit** method, where all the other methods we have covered (Forward Euler, Heun's, Midpoint, and 4th-order Runge-Kutta) are **explicit**. Implicit methods require more work to actually implement.\n", + "\n", + "### Backward Euler example\n", + "For example, consider the problem \n", + "\\begin{equation} \n", + "\\frac{dy}{dx} = f(x,y) = 8 e^{-x} (1+x) - 2y\n", + "\\end{equation}\n", + "To actually solve this problem with the Backward Euler method, we need to incorporate the derivative function $f(x,y)$ into the recursion formula and solve for $y_{i+1}$:\n", + "\\begin{align}\n", + "y_{i+1} &= y_i + \\Delta x \\, f(x_{i+1}, y_{i+1}) \\\\\n", + "y_{i+1} &= y_i + \\Delta x \\left[ 8 e^{-x_{i+1}} (1 + x_{i+1}) - 2 y_{i+1} \\right] \\\\\n", + "y_{i+1} &= y_i + 8 e^{-x_{i+1}} (1 + x_{i+1}) \\Delta x - 2 \\Delta x \\, y_{i+1} \\\\\n", + "y_{i+1} + 2 \\Delta x \\, y_{i+1} &= y_i + 8 e^{-x_{i+1}} (1 + x_{i+1}) \\Delta x \\\\\n", + "y_{i+1} &= \\frac{ y_i + 8 e^{-x_{i+1}} (1 + x_{i+1}) \\Delta x }{ 1 + 2 \\Delta x }\n", + "\\end{align}\n", + "Now we have a useable recursion formula that we can use to solve this problem. Let's use the initial condition $y(0) = 1$, the domain $0 \\leq x \\leq 7$, and $\\Delta x = 0.2$." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "\n", + "dx = 0.2;\n", + "x = 0 : dx : 7;\n", + "y = zeros(length(x), 1);\n", + "y(1) = 1;\n", + "\n", + "% Backward Euler loop\n", + "for i = 1 : length(x) - 1\n", + " y(i+1) = (y(i) + 8*exp(-x(i+1))*(1 + x(i+1))*dx) / (1 + 2*dx);\n", + "end\n", + "\n", + "x_exact = linspace(0, 7);\n", + "y_exact = exp(-2.*x_exact).*(8*x_exact.*exp(x_exact) + 1);\n", + "plot(x_exact, y_exact); hold on\n", + "plot(x, y, 'o--')\n", + "legend('Exact solution', 'Backward Euler solution')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This matches nearly what we saw with the Forward Euler method before—Backward Euler is also a **first-order** method, so the global error should be proportional to $\\Delta x$.\n", + "\n", + "Let's now return to the stiff ODE $y^{\\prime} = -3 y$, and see how the Backward Euler method does. First, we need to obtain our useable recursion formula:\n", + "\\begin{align}\n", + "y_{i+1} &= y_i + \\Delta t \\, f(t_{i+1}, y_{i+1}) \\\\\n", + "y_{i+1} &= y_i + \\Delta t \\, \\left( -3 y_{i+1} \\right) \\\\\n", + "y_{i+1} + 3 y_{i+1} \\Delta t &= y_i \\\\\n", + "y_{i+1} &= \\frac{y_i}{1 + 3 \\Delta t}\n", + "\\end{align}" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "\n", + "dt = 0.1;\n", + "t = 0 : dt : 10;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) / (1 + 3*dt);\n", + "end\n", + "subplot(4,1,1);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));\n", + "\n", + "dt = 0.25;\n", + "t = 0 : dt : 10;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) / (1 + 3*dt);\n", + "end\n", + "subplot(4,1,2);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));\n", + "\n", + "dt = 0.5;\n", + "t = 0 : dt : 10;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) / (1 + 3*dt);\n", + "end\n", + "subplot(4,1,3);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));\n", + "\n", + "dt = 0.75;\n", + "t = 0 : dt : 10;\n", + "y = zeros(length(t), 1);\n", + "y(1) = 1;\n", + "for i = 1 : length(t) - 1\n", + " y(i+1) = y(i) / (1 + 3*dt);\n", + "end\n", + "subplot(4,1,4);\n", + "plot(t, y); title(sprintf('dt = %4.2f', dt));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this case, we see that the solution remains well-behaved for all the step sizes, not showing any of the instability we saw with the Forward Euler method. This is because the Backward Euler method is **unconditionally stable**." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Stability analysis\n", + "\n", + "### Stability analysis of Forward Euler\n", + "\n", + "We can perform a stability analysis of the stiff problem to identify when the Forward Euler method becomes unstable. Let's apply the method to the ODE at hand:\n", + "\\begin{align}\n", + "\\frac{dy}{dt} &= -3 y \\\\\n", + "y_{i+1} &= y_i + \\Delta t f(t_i, y_i) \\\\\n", + "y_{i+1} &= y_i + \\Delta t (-3 y_i) \\\\\n", + " &= y_i (1 - 3 \\Delta t) \\\\\n", + "\\frac{y_{i+1}}{y_i} &= \\sigma = 1 - 3 \\Delta t \n", + "\\end{align}\n", + "where $\\sigma$ is the **amplification factor**. This defines whether the solution grows or decays each step—for a stable physical system, we expect the solution to get smaller or remain contant with each step.\n", + "\n", + "Therefore, for the method to remain stable, we must have $\\sigma | \\leq 1$. We can use this stability criterion to find conditions on $\\Delta t$ for stability:\n", + "\\begin{gather}\n", + "| \\sigma | = | 1 - 3 \\Delta t | \\leq 1 \\\\\n", + "-1 \\leq 1 - 3 \\Delta t \\leq 1 \\\\\n", + "-1 \\leq 1 - 3 \\Delta t \\quad \\text{or} \\quad 1 - 3 \\Delta t \\leq 1 \\\\\n", + "\\frac{-2}{3} \\leq -\\Delta t \\quad \\quad -\\Delta t \\leq 0 \\\\\n", + "\\rightarrow \\Delta t \\leq \\frac{2}{3} \\quad \\text{and} \\quad \\Delta t \\geq 0 \\\\\n", + "\\therefore 0 \\leq \\Delta t \\leq \\frac{2}{3}\n", + "\\end{gather}\n", + "for stability. (For safety, we might use $\\Delta t < 1/2$ for safety, to stay away from the absolute stability limit.)\n", + "\n", + "The Forward Euler method is then *conditionally stable*. \n", + "\n", + "As a general rule of thumb, all **explicit** methods are conditionally stable; these are methods where the recursion formula for $y_{i+1}$ can be written and calculated explicitly in terms of known quantities.\n", + "\n", + "### Stability analysis of Backward Euler\n", + "\n", + "We can also perform a stability analysis on the Backward Euler method to show that its stability does not depend on the step size:\n", + "\\begin{align}\n", + "\\frac{dy}{dt} &= -3 y \\\\\n", + "y_{i+1} &= y_i + \\Delta t f(t_{i+1}, y_{i+1}) \\\\\n", + "y_{i+1} &= y_i + \\Delta t (-3 y_{i+1}) \\\\\n", + "y_{i+1} (1 + 3 \\Delta t) &= y_i \\\\\n", + "\\sigma &= \\frac{y_{i+1}}{y_i} = \\frac{1}{1 + 3 \\Delta t}\n", + "\\end{align}\n", + "For stability, we need $| \\sigma | \\leq 1$:\n", + "\\begin{align}\n", + "| \\sigma | &= \\left| \\frac{1}{1 + 3 \\Delta t} \\right| \\leq 1 \\\\\n", + "\\rightarrow \\Delta t &> 0\n", + "\\end{align}\n", + "Therefore the Backward Euler method is *unconditionally stable*." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.7" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/docs/_sources/content/pdes/elliptic.ipynb b/docs/_sources/content/pdes/elliptic.ipynb new file mode 100644 index 0000000..1c0cbf5 --- /dev/null +++ b/docs/_sources/content/pdes/elliptic.ipynb @@ -0,0 +1,1089 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "tags": [] + }, + "source": [ + "# Elliptic PDEs\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The classic example of an elliptic PDE is **Laplace's equation** (yep, the same Laplace that gave us the Laplace transform), which in two dimensions for a variable $u(x,y)$ is\n", + "\\begin{equation}\n", + "\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} = \\nabla^2 u = 0 \\;,\n", + "\\end{equation}\n", + "where $\\nabla$ is del, or nabla, and represents the gradient operator: $\\nabla = \\frac{\\partial}{\\partial x} + \\frac{\\partial}{\\partial y}$.\n", + "\n", + "Laplace's equation shows up in a number of physical problems, including heat transfer, fluid dynamics, and electrostatics. For example, the heat equation for conduction in two dimensions is\n", + "\\begin{equation}\n", + "\\frac{\\partial u}{\\partial t} = \\alpha \\left( \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} \\right) \\;,\n", + "\\end{equation}\n", + "where $u(x,y,t)$ is temperature and $\\alpha$ is thermal diffusivity. Steady-state heat transfer (meaning after any initial transient period) is then described by Laplace's equation.\n", + "\n", + "A related elliptic PDE is **Poisson's equation**:\n", + "\\begin{equation}\n", + "\\nabla^2 u = f(x,y) \\;,\n", + "\\end{equation}\n", + "which also appears in multiple physical problems—most notably, when solving for pressure in the Navier–Stokes equations.\n", + "\n", + "To numerically solve these equations, and any elliptic PDE, we can use finite differences, where we replace the continuous $x,y$ domain with a discrete grid of points. This is similar to what we did with boundary-value problems in one dimension—but now we have two dimensions.\n", + "\n", + "To approximate the second derivatives in Laplace's equation, we can use central differences in both the $x$ and $y$ directions, applied around the $u_{i,j}$ point:\n", + "\\begin{align}\n", + "\\frac{\\partial^2 u}{\\partial x^2} &\\approx \\frac{u_{i-1,j} - 2u_{i,j} + u_{i+1,j}}{\\Delta x^2} \\\\\n", + "\\frac{\\partial^2 u}{\\partial y^2} &\\approx \\frac{u_{i,j-1} - 2u_{i,j} + u_{i,j+1}}{\\Delta y^2}\n", + "\\end{align}\n", + "where $i$ is the index used in the $x$ direction, $j$ is the index in the $y$ direction, and $\\Delta x$ and $\\Delta y$ are the step sizes in the $x$ and $y$ directions.\n", + "In other words, $x_i = (i-1) \\Delta x$ and $y_j = (j-1) \\Delta y$.\n", + "\n", + "The following figure shows the points necessary to approximate the partial derivatives in the PDE at a location $(x_i, y_j)$, for a general 2D region. This is known as a **five-point stencil**:\n", + "\n", + ":::{figure-md} fig-stencil\n", + "\"five-point\n", + "\n", + "Five-point finite difference stencil\n", + ":::\n", + "\n", + "Applying these finite differences gives us an approximation for Laplace's equation:\n", + "\\begin{equation}\n", + "\\frac{u_{i-1,j} - 2u_{i,j} + u_{i+1,j}}{\\Delta x^2} + \\frac{u_{i,j-1} - 2u_{i,j} + u_{i,j+1}}{\\Delta y^2} = 0 \\;.\n", + "\\end{equation}\n", + "If we use a uniform grid where $\\Delta x = \\Delta y = h$, then we can simplify to \n", + "\\begin{equation}\n", + "u_{i+1,j} + u_{i,j+1} + u_{i-1,j} + u_{i,j-1} - 4 u_{i,j} = 0 \\;.\n", + "\\end{equation}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: heat transfer in a square plate\n", + "\n", + "As an example, let's consider the problem of steady-state heat transfer in a square solid object. If $u(x,y)$ is temperature, then this is described by Laplace's equation:\n", + "\\begin{equation}\n", + "\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} = \\nabla^2 u = 0 \\;,\n", + "\\end{equation}\n", + "and we can solve this using finite differences. Using a uniform grid where $\\Delta x = \\Delta y = h$, Laplace's equation gives us a recursion formula that relates the values at neighboring points:\n", + "\\begin{equation}\n", + "u_{i+1,j} + u_{i,j+1} + u_{i-1,j} + u_{i,j-1} - 4 u_{i,j} = 0 \\;.\n", + "\\end{equation}\n", + "\n", + "Consider a case where the square has sides of length $L$, and the boundary conditions are that the temperature is fixed at 100 on the left, right, and bottom sides, and fixed at 0 on the top.\n", + "For now, we'll use two segments to discretize the domain in each directions, giving us nine total points in the grid.\n", + "The following figures show the example problem, and the grid of points we'll use.\n", + "\n", + ":::{figure-md} fig-heat-transfer-square\n", + "\"Heat\n", + "\n", + "Heat transfer in a square object\n", + ":::\n", + "\n", + ":::{figure-md} fig-grid-three\n", + "\"3x3\n", + "\n", + "Simple 3x3 grid of points\n", + ":::\n", + "\n", + "Using the above recursion formula, we can write an equation for each of the nine unknown points (in the interior, not the boundary points):\n", + "\\begin{align}\n", + "u_{1,1} &= 100 \\\\\n", + "u_{2,1} &= 100 \\\\\n", + "u_{3,1} &= 100 \\\\\n", + "u_{1,2} &= 100 \\\\\n", + "\\text{for } u_{2,2}: \\quad u_{3,2} + u_{2,3} + u_{1,2} + u_{2,1} - 4u_{2,2} &= 0 \\\\\n", + "u_{3,2} &= 100 \\\\\n", + "u_{1,3} &= 100 \\\\\n", + "u_{2,3} &= 0 \\\\\n", + "u_{3,3} &= 100\n", + "\\end{align}\n", + "where $u_{i,j}$ are the unknowns. Note that in this we used the side boundary condition values for the corner points $u_{1,3}$ and $u_{3,3}$, rather than the top value. (In reality this would represent a discontinuity in temperature, so these aren't very realistic boundary conditions.)\n", + "\n", + "This is a system of linear equations, that we can represent as a matrix-vector product:\n", + "\\begin{align}\n", + "\\begin{bmatrix} \n", + "1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 1 & 0 & 1 & -4 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\end{bmatrix}\n", + "\\begin{bmatrix} u_{1,1} \\\\ u_{2,1} \\\\ u_{3,1} \\\\ u_{1,2} \\\\ u_{2,2} \\\\ u_{3,2} \\\\ u_{1,3} \\\\ u_{2,3} \\\\ u_{3,3} \\end{bmatrix} &= \n", + "\\begin{bmatrix} 100 \\\\ 100 \\\\ 100 \\\\ \n", + "100 \\\\ 0 \\\\ 100 \\\\\n", + "100 \\\\ 0 \\\\ 100 \\end{bmatrix} \\\\\n", + "\\text{or} \\quad A \\mathbf{u} &= \\mathbf{b}\n", + "\\end{align}\n", + "where $A$ is a $9\\times 9$ coefficient matrix, $\\mathbf{u}$ is a nine-element vector of unknown variables, and $\\mathbf{b}$ is a nine-element right-hand side vector.\n", + "For $\\mathbf{u}$, we had to take variables that physically represent points in a two-dimensional space and combine them in some order to form a one-dimensional column vector. Here, we used a **row-major** mapping, where we started with the point in the first row and first column, then added the remaining points in that row, before moving to the next row and repeating. We'll discuss this a bit more later.\n", + "\n", + "If we set this up in Matlab, we can solve with `u = A \\ b`:" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " 100\n", + " 100\n", + " 100\n", + " 100\n", + " 75\n", + " 100\n", + " 100\n", + " 0\n", + " 100\n", + "\n" + ] + } + ], + "source": [ + "clear all; clc\n", + "\n", + "A = [\n", + "1 0 0 0 0 0 0 0 0;\n", + "0 1 0 0 0 0 0 0 0;\n", + "0 0 1 0 0 0 0 0 0;\n", + "0 0 0 1 0 0 0 0 0;\n", + "0 1 0 1 -4 1 0 1 0;\n", + "0 0 0 0 0 1 0 0 0;\n", + "0 0 0 0 0 0 1 0 0;\n", + "0 0 0 0 0 0 0 1 0;\n", + "0 0 0 0 0 0 0 0 1];\n", + "b = [100; 100; 100; 100; 0; 100; 100; 0; 100];\n", + "\n", + "% solve system of linear equations\n", + "u = A \\ b;\n", + "\n", + "disp(u)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This gives us the values for temperature at each of the nine points. In this example, we really only have one unknown temperature: $u_{2,2}$, located in the middle. Does the value given make sense? We can check by rearranging the recursion formula for Laplace's equation:\n", + "\\begin{equation}\n", + "u_{i,j} = \\frac{u_{i+1,j} + u_{i,j+1} + u_{i-1,j} + u_{i,j-1}}{4} \\;,\n", + "\\end{equation}\n", + "which shows that in such problems the value of the middle point should be the average of the four surrounding points. This matches the value of 75 found above.\n", + "\n", + "We can use a contour plot to visualize the results, though we'll need to convert the one-dimensional solution array into a two-dimensional matrix to plot. The Matlab `reshape()` function can help us here: it reshapes an array into a matrix, by specifying the target number of desired columns and rows:" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "b array:\n", + " 1\n", + " 2\n", + " 3\n", + " 4\n", + " 5\n", + " 6\n", + " 7\n", + " 8\n", + " 9\n", + " 10\n", + "\n", + "A matrix:\n", + " 1 6\n", + " 2 7\n", + " 3 8\n", + " 4 9\n", + " 5 10\n", + "\n" + ] + } + ], + "source": [ + "% Example of using the reshape function, with a simple array going from 1 to 10\n", + "\n", + "% We want to convert it into a matrix with 5 columns and 2 rows.\n", + "% The expected output is:\n", + "% [1 2 3 4 5; \n", + "% 6 7 8 9 10]\n", + "\n", + "b = (1 : 10)';\n", + "A = reshape(b, [5, 2]);\n", + "disp('b array:')\n", + "disp(b)\n", + "disp('A matrix:')\n", + "disp(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This behavior may be a bit unexpected, because `reshape()` uses a column-major mapping. We can fix this by taking the transpose of the resulting matrix:" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "transpose of output matrix:\n", + " 1 2 3 4 5\n", + " 6 7 8 9 10\n", + "\n" + ] + } + ], + "source": [ + "disp('transpose of output matrix:')\n", + "disp(A')" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% We can use the reshape function to convert the calculated temperatures\n", + "% into a 3x3 matrix:\n", + "\n", + "n = 3; m = 3;\n", + "u_square = reshape(u, [n, m]);\n", + "\n", + "contourf(u_square')\n", + "colorbar" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Overall that looks correct: the boundary conditions are right, and we see that the center is the average of the boundaries.\n", + "\n", + "But, clearly only using nine points (with eight of those being boundary conditions) doesn't give us a very good solution. To make this more accurate, we'll need to use more points, which also means we need to automate the construction of the system of equations." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Row-major mapping\n", + "\n", + "For a two-dimensional elliptic PDE like Laplace's equation, we can generate a general recursion formula, but we need a way to take a grid of points where location is defined by row and column index and map these into a one-dimensional column vector, which has its own index.\n", + "\n", + "The following figure shows a general 2D grid of points, with $n$ number of columns in the $x$ direction (using index $i$) and $m$ number of rows in the $y$ direction (using index $j$):\n", + "\n", + ":::{figure-md} fig-twodim-grid\n", + "\"2D\n", + "\n", + "2D grid of points with *n* columns and *m* columns.\n", + ":::\n", + "\n", + "We want to convert the rows and columns of $u_{i,j}$ points defined by column and row index into a single column array using a different index, $k$ (this choice is arbitrary):\n", + "\\begin{equation}\n", + "\\begin{bmatrix} u_{1,1} \\\\ u_{2,1} \\\\ u_{3,1} \\\\ \\vdots \\\\ u_{n,1} \\\\\n", + "u_{1,2} \\\\ u_{2,2} \\\\ u_{3,2} \\\\ \\vdots \\\\ u_{n, 2} \\\\ u_{1,3} \\\\ \\vdots \\\\\n", + "u_{1,m} \\\\ u_{2,m} \\\\ \\vdots \\\\ u_{n,m}\n", + "\\end{bmatrix}\n", + "\\end{equation}\n", + "where $k$ refers to the index used in that array.\n", + "\n", + "To do this mapping, we can use this formula:\n", + "\\begin{equation}\n", + "k_{i,j} = (j-1)n + i\n", + "\\end{equation}\n", + "where $k_{i,j}$ refers to the 1D index $k$ mapped from the 2D indices $i$ and $j$.\n", + "\n", + ":::{figure-md} fig-grid-three\n", + "\"3x3\n", + "\n", + "Simple 3x3 grid of points\n", + ":::\n", + "\n", + "For example, in this $3\\times 3$ grid, where $n=3$ and $m=3$, consider the point where $i=2$ and $j=2$ (the point right in the center). Using our formula, \n", + "\\begin{equation}\n", + "k_{2,2} = (2-1)3 + 2 = 5\n", + "\\end{equation}\n", + "which matches what we can visually confirm.\n", + "\n", + "Using that mapping, we can also identify the 1D indices associated with the points surrounding location $(i,j)$:\n", + "\\begin{align}\n", + "k_{i-1,j} &= (j-1)n + i - 1 \\\\\n", + "k_{i+1,j} &= (j-1)n + i + 1 \\\\\n", + "k_{i,j-1} &= (j-2)n + i \\\\\n", + "k_{i,j+1} &= j n + i\n", + "\\end{align}\n", + "which we can use to determine the appropriate locations to place values in the coefficient and right-hand side matrices." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: heat transfer in a square plate (redux)\n", + "\n", + "Let's return to the example of steady-state heat transfer in a square plate—but this time we'll set the solution up more generally so we can vary the step size $h = \\Delta x = \\Delta y$." + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear; clc; close all\n", + "\n", + "h = 0.1;\n", + "x = [0 : h : 1]; n = length(x);\n", + "y = [0 : h : 1]; m = length(y);\n", + "\n", + "% The coefficient matrix A is now m*n by m*n, since that is the total number of points.\n", + "% The right-hand side vector b is m*n by 1.\n", + "A = zeros(m*n, m*n);\n", + "b = zeros(m*n, 1);\n", + "\n", + "u_left = 100;\n", + "u_right = 100;\n", + "u_bottom = 100;\n", + "u_top = 0;\n", + "\n", + "for j = 1 : m\n", + " for i = 1 : n\n", + " % for convenience we calculate all the indices once\n", + " kij = (j-1)*n + i;\n", + " kim1j = (j-1)*n + i - 1;\n", + " kip1j = (j-1)*n + i + 1;\n", + " kijm1 = (j-2)*n + i;\n", + " kijp1 = j*n + i;\n", + " \n", + " if i == 1 \n", + " % this is the left boundary\n", + " A(kij, kij) = 1;\n", + " b(kij) = u_left;\n", + " elseif i == n \n", + " % right boundary\n", + " A(kij, kij) = 1;\n", + " b(kij) = u_right;\n", + " elseif j == 1 \n", + " % bottom boundary\n", + " A(kij, kij) = 1;\n", + " b(kij) = u_bottom;\n", + " elseif j == m \n", + " % top boundary\n", + " A(kij, kij) = 1;\n", + " b(kij) = u_top;\n", + " else\n", + " % these are the coefficients for the interior points,\n", + " % based on the recursion formula\n", + " A(kij, kim1j) = 1;\n", + " A(kij, kip1j) = 1;\n", + " A(kij, kijm1) = 1;\n", + " A(kij, kijp1) = 1;\n", + " A(kij, kij) = -4;\n", + " end\n", + " end\n", + "end\n", + "u = A \\ b;\n", + "\n", + "u_square = reshape(u, [n, m]);\n", + "contourf(x, y, u_square')\n", + "c = colorbar;\n", + "c.Label.String = 'Temperature';" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Neumann (derivative) boundary conditions\n", + "\n", + "So far, we have only discussed cases where we have Dirichlet boundary conditions; in other words, when we have all fixed values at the boundary. Frequently we also encounter Neumann-style boundary conditions, where we have the *derivative* specified at the boundary.\n", + "\n", + "We can handle this in the same way we do for one-dimensional boundary value problems: either with a forward or backward difference (both of which are first-order accurate), or with a central difference using an imaginary point/ghost node (which is second-order accurate). Let's focus on using the central difference, since it is more accurate.\n", + "\n", + ":::{figure-md} fig-ghost-node\n", + "\"ghost\n", + "\n", + "Ghost/imaginary node beyond an upper boundary\n", + ":::\n", + "\n", + "For example, let's say that at the upper boundary, the derivative of temperature is zero:\n", + "\\begin{equation}\n", + "\\left. \\frac{\\partial u}{\\partial y} \\right|_{\\text{boundary}} = 0\n", + "\\end{equation}\n", + "\n", + "Let's consider this boundary condition applied at the point shown, $u_{2,3}$.\n", + "We can approximate this derivative using a central difference:\n", + "\\begin{align}\n", + "\\frac{u_{2,3}}{\\partial y} \\approx \\frac{u_{2,4} - u_{2,2}}{\\Delta x} &= 0 \\\\\n", + "u_{2,4} &= u_{2,2}\n", + "\\end{align}\n", + "This tells us the value of the point above the boundary, $u_{2,4}$; however, this point is a \"ghost\" or imaginary point located outside the boundary, so we don't really care about its value. Instead, we can use this relationship to give us a usable equation for the boundary point, by incorporating it into the normal recursion formula for Laplace's equation:\n", + "\\begin{align}\n", + "u_{1,3} + u_{3,3} + u_{2,4} + u_{2,2} - 4u_{2,3} &= 0 \\\\\n", + "u_{1,3} + u_{3,3} + u_{2,2} + u_{2,2} - 4u_{2,3} &= 0 \\\\\n", + "\\rightarrow u_{1,3} + u_{3,3} + 2 u_{2,2} - 4u_{2,3} &= 0\n", + "\\end{align}\n", + "\n", + "The recursion formula for points along the upper boundary would then become\n", + "\\begin{equation}\n", + "u_{i+1,j} + u_{i-1,j} + 2 u_{i,j-1} - 4 u_{i,j} = 0 \\;.\n", + "\\end{equation}\n", + "\n", + "Now let's try solving the above example, but with $\\frac{\\partial u}{\\partial y} = 0$ at the top boundary and $u = 0$ at the bottom boundary:" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear; clc; close all\n", + "\n", + "h = 0.1;\n", + "x = [0 : h : 1]; n = length(x);\n", + "y = [0 : h : 1]; m = length(y);\n", + "\n", + "% The coefficient matrix A is now m*n by m*n, since that is the total number of points.\n", + "% The right-hand side vector b is m*n by 1.\n", + "A = zeros(m*n, m*n);\n", + "b = zeros(m*n, 1);\n", + "\n", + "u_left = 100;\n", + "u_right = 100;\n", + "u_bottom = 0;\n", + "\n", + "for j = 1 : m\n", + " for i = 1 : n\n", + " % for convenience we calculate all the indices once\n", + " kij = (j-1)*n + i;\n", + " kim1j = (j-1)*n + i - 1;\n", + " kip1j = (j-1)*n + i + 1;\n", + " kijm1 = (j-2)*n + i;\n", + " kijp1 = j*n + i;\n", + " \n", + " if i == 1 \n", + " % this is the left boundary\n", + " A(kij, kij) = 1;\n", + " b(kij) = u_left;\n", + " elseif i == n \n", + " % right boundary\n", + " A(kij, kij) = 1;\n", + " b(kij) = u_right;\n", + " elseif j == 1 \n", + " % bottom boundary\n", + " A(kij, kij) = 1;\n", + " b(kij) = u_bottom;\n", + " elseif j == m \n", + " % top boundary, using the ghost node + recursion formula\n", + " A(kij, kim1j) = 1;\n", + " A(kij, kip1j) = 1;\n", + " A(kij, kijm1) = 2;\n", + " A(kij, kij) = -4;\n", + " else\n", + " % these are the coefficients for the interior points,\n", + " % based on the recursion formula\n", + " A(kij, kim1j) = 1;\n", + " A(kij, kip1j) = 1;\n", + " A(kij, kijm1) = 1;\n", + " A(kij, kijp1) = 1;\n", + " A(kij, kij) = -4;\n", + " end\n", + " end\n", + "end\n", + "u = A \\ b;\n", + "\n", + "u_square = reshape(u, [n, m]);\n", + "% the \"20\" indicates the number of levels for the contour plot\n", + "contourf(x, y, u_square', 20);\n", + "c = colorbar;\n", + "c.Label.String = 'Temperature';" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Iterative solutions for (very) large problems\n", + "\n", + "So far, we've been able to solve our systems of linear equations in Matlab by using `y = A \\ b`, which directly finds the solution to the equation $A \\mathbf{y} = \\mathbf{b}$.\n", + "\n", + "However, this approach will become very slow as the grid resolution ($h = \\Delta x = \\Delta y$) becomes smaller, and eventually unfeasable due to the associated computational requirements. First, let's create a function that takes as input the segment size $h$, then returns the time ittakes to solve the problem for different sizes." + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Created file '/Users/kyle/projects/ME373-book/content/pdes/heat_equation.m'.\n" + ] + } + ], + "source": [ + "%%file heat_equation.m\n", + "function [time, num] = heat_equation(h)\n", + "\n", + "x = [0 : h : 1]; n = length(x);\n", + "y = [0 : h : 1]; m = length(y);\n", + "\n", + "% The coefficient matrix A is now m*n by m*n, since that is the total number of points.\n", + "% The right-hand side vector b is m*n by 1.\n", + "A = zeros(m*n, m*n);\n", + "b = zeros(m*n, 1);\n", + "\n", + "num = m*n; % number of points\n", + "\n", + "tic;\n", + "\n", + "u_left = 100;\n", + "u_right = 100;\n", + "u_bottom = 100;\n", + "u_top = 0;\n", + "\n", + "for j = 1 : m\n", + " for i = 1 : n\n", + " % for convenience we calculate all the indices once\n", + " kij = (j-1)*n + i;\n", + " kim1j = (j-1)*n + i - 1;\n", + " kip1j = (j-1)*n + i + 1;\n", + " kijm1 = (j-2)*n + i;\n", + " kijp1 = j*n + i;\n", + " \n", + " if i == 1 \n", + " % this is the left boundary\n", + " A(kij, kij) = 1;\n", + " b(kij) = u_left;\n", + " elseif i == n \n", + " % right boundary\n", + " A(kij, kij) = 1;\n", + " b(kij) = u_right;\n", + " elseif j == 1 \n", + " % bottom boundary\n", + " A(kij, kij) = 1;\n", + " b(kij) = u_bottom;\n", + " elseif j == m \n", + " % top boundary\n", + " A(kij, kij) = 1;\n", + " b(kij) = u_top;\n", + " else\n", + " % these are the coefficients for the interior points,\n", + " % based on the recursion formula\n", + " A(kij, kim1j) = 1;\n", + " A(kij, kip1j) = 1;\n", + " A(kij, kijm1) = 1;\n", + " A(kij, kijp1) = 1;\n", + " A(kij, kij) = -4;\n", + " end\n", + " end\n", + "end\n", + "u = A \\ b;\n", + "\n", + "u_square = reshape(u, [n, m]);\n", + "% the \"20\" indicates the number of levels for the contour plot\n", + "%contourf(x, y, u_square', 20);\n", + "%c = colorbar;\n", + "%c.Label.String = 'Temperature';\n", + "\n", + "time = toc;" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now, we can see how long it takes to solve as we increase the resolution, and get an idea about the relationship between time-to-solution and number of unknowns.\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear all; clc;\n", + "\n", + "step_sizes = [0.1, 0.05, 0.025, 0.02, 0.0125, 0.01];\n", + "\n", + "n = length(step_sizes);\n", + "nums = zeros(n,1); times = zeros(n,1);\n", + "\n", + "for i = 1 : n\n", + " [times(i), nums(i)] = heat_equation(step_sizes(i));\n", + "end\n", + "\n", + "loglog(nums, times, '-o')\n", + "xlabel('Number of unknowns'); \n", + "ylabel('Time for direct solution (sec)')\n", + "hold on\n", + "\n", + "x = nums(3:end);\n", + "n2 = x.^2 * (times(3) / x(1)^2);\n", + "n3 = x.^3 * (times(3) / x(1)^3);\n", + "plot(x, n2, '-x');\n", + "plot(x, n3, '-s');\n", + "legend('Actual cost', 'Quadratic cost', 'Cubic cost', 'Location', 'northwest')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Interestingly, we see that the slope in this log-log plot that after about 400 unknowns (so a coefficient matrix of about 160,000), the cost begins to increase exponentially, somewhere between quadratic ($\\mathcal{O}(n^2)$) and cubic ($\\mathcal{O}(n^3)$).\n", + "\n", + "If we try to reduce the step size further, for example to 0.005, we'll see that we cannot get a solution in a reasonable amount of time. But, clearly we want to get solutions for large numbers of unknowns, so what can we do?\n", + "\n", + "We can solve larger systems of linear equations using *iterative* methods. There are a number of these, and we'll focus on two:\n", + "\n", + "- Jacobi method\n", + "- Gauss-Seidel method\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Jacobi method\n", + "\n", + "The Jacobi method essentially works by starting with an initial guess to the solution, then using the recursion formula to solve for values at each point, then repeating this until the values converge (i.e., stop changing). \n", + "\n", + "An algorithm we can use to solve Laplace's equation:\n", + "\n", + "1. Set some initial guess for all unknowns: $u_{i,j}^{\\text{old}}$\n", + "2. Set the boundary values\n", + "3. For each point in the interior, use the recursion formula to solve for new values based on old values at the surrounding points: $u_{i,j} = \\left( u_{i+1,j}^{\\text{old}} + u_{i-1,j}^{\\text{old}} + u_{i,j+1}^{\\text{old}} + u_{i,j-1}^{\\text{old}} \\right)/4$.\n", + "4. Check for convergence: is $\\epsilon$ less than some tolerance, such as $10^{-6}$? Where $\\epsilon = \\max \\left| u_{i,j} - u_{i,j}^{\\text{old}} \\right|$. If no, then return to step 2 and repeat.\n", + "\n", + "More formally, if we have a system $A \\mathbf{x} = \\mathbf{b}$, where\n", + "\\begin{equation}\n", + "A = \\begin{bmatrix}\n", + "a_{11} & a_{12} & \\cdots & a_{1n} \\\\\n", + "a_{21} & a_{22} & \\cdots & a_{2n} \\\\\n", + "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n", + "a_{n1} & a_{n2} & \\cdots & a_{nn} \\end{bmatrix}\n", + "\\quad \\mathbf{x} = \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix}\n", + "\\quad \\mathbf{b} = \\begin{bmatrix} b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_n \\end{bmatrix}\n", + "\\end{equation}\n", + "then we can solve iterative for $\\mathbf{x}$ using\n", + "\\begin{equation}\n", + "x_i^{(k+1)} = \\frac{1}{a_{ii}} \\left( b_i - \\sum_{j \\neq i} a_{ij} x_j^{(k)} \\right) , \\quad i = 1,2,\\ldots, n \n", + "\\end{equation}\n", + "where $x_i^{(k)}$ is a value of the solution at iteration $k$ and $x_i^{(k+1)}$ is at the next iteration." + ] + }, + { + "cell_type": "code", + "execution_count": 54, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Created file '/Users/kyle/projects/ME373-book/content/pdes/heat_equation_jacobi.m'.\n" + ] + } + ], + "source": [ + "%%file heat_equation_jacobi.m\n", + "function [time, num_point, num_iter] = heat_equation_jacobi(h)\n", + "\n", + "x = [0 : h : 1]; n = length(x);\n", + "y = [0 : h : 1]; m = length(y);\n", + "\n", + "% The coefficient matrix A is now m*n by m*n, since that is the total number of points.\n", + "% The right-hand side vector b is m*n by 1.\n", + "A = zeros(m*n, m*n);\n", + "b = zeros(m*n, 1);\n", + "num_point = m*n;\n", + "\n", + "tic;\n", + "\n", + "u_left = 100;\n", + "u_right = 100;\n", + "u_bottom = 100;\n", + "u_top = 0;\n", + "\n", + "% initial guess\n", + "u = 100*ones(m*n, 1);\n", + "\n", + "% dummy value for residual variable\n", + "epsilon = 1.0; \n", + "\n", + "num_iter = 0;\n", + "while epsilon > 1e-6\n", + " u_old = u;\n", + " \n", + " epsilon = 0;\n", + " for j = 1 : m\n", + " for i = 1 : n\n", + " kij = (j-1)*n + i;\n", + " kim1j = (j-1)*n + i - 1;\n", + " kip1j = (j-1)*n + i + 1;\n", + " kijm1 = (j-2)*n + i;\n", + " kijp1 = j*n + i;\n", + "\n", + " if i == 1 \n", + " % this is the left boundary\n", + " u(kij) = u_left;\n", + " elseif i == n \n", + " % right boundary\n", + " u(kij) = u_right;\n", + " elseif j == 1 \n", + " % bottom boundary\n", + " u(kij) = u_bottom;\n", + " elseif j == m \n", + " % top boundary\n", + " u(kij) = u_top;\n", + " else\n", + " % interior points\n", + " u(kij) = (u_old(kip1j) + u_old(kim1j) + u_old(kijm1) + u_old(kijp1))/4.0;\n", + " end\n", + " end\n", + " end\n", + " \n", + " epsilon = max(abs(u - u_old));\n", + " num_iter = num_iter + 1;\n", + "end\n", + "\n", + "u_square = reshape(u, [n, m]);\n", + "% the \"20\" indicates the number of levels for the contour plot\n", + "contourf(x, y, u_square', 20);\n", + "c = colorbar;\n", + "c.Label.String = 'Temperature';\n", + "\n", + "time = toc;\n", + "fprintf('Number of iterations: %d\\n', num_iter)" + ] + }, + { + "cell_type": "code", + "execution_count": 55, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Number of iterations: 291\n", + "Number of iterations: 1061\n", + "Number of iterations: 3803\n", + "Number of iterations: 5717\n", + "Number of iterations: 13421\n", + "Number of iterations: 20067\n", + "Number of iterations: 69037\n" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "step_sizes = [0.1, 0.05, 0.025, 0.02, 0.0125, 0.01, 0.005];\n", + "n = length(step_sizes);\n", + "\n", + "nums_jac = zeros(n,1); times_jac = zeros(n,1); num_iter_jac = zeros(n,1);\n", + "\n", + "for i = 1 : n\n", + " [times_jac(i), nums_jac(i), num_iter_jac(i)] = heat_equation_jacobi(step_sizes(i));\n", + "end" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Gauss-Seidel method\n", + "\n", + "The Gauss-Seidel method is very similar to the Jacobi method, but with one important difference: rather than using all old values to calculate the new values, incorporate updated values as they are available. Because the method incorporates newer information more quickly, it tends to converge faster (meaning, with fewer iterations) than the Jacobi method.\n", + "\n", + "Formally, if we have a system $A \\mathbf{x} = \\mathbf{b}$, where\n", + "\\begin{equation}\n", + "A = \\begin{bmatrix}\n", + "a_{11} & a_{12} & \\cdots & a_{1n} \\\\\n", + "a_{21} & a_{22} & \\cdots & a_{2n} \\\\\n", + "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n", + "a_{n1} & a_{n2} & \\cdots & a_{nn} \\end{bmatrix}\n", + "\\quad \\mathbf{x} = \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix}\n", + "\\quad \\mathbf{b} = \\begin{bmatrix} b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_n \\end{bmatrix}\n", + "\\end{equation}\n", + "then we can solve iterative for $\\mathbf{x}$ using\n", + "\\begin{equation}\n", + "x_i^{(k+1)} = \\frac{1}{a_{ii}} \\left( b_i - \\sum_{j=1}^{i-1} a_{ij} x_j^{(k+1)} - \\sum_{j =i+1}^n a_{ij} x_j^{(k)} \\right) , \\quad i = 1,2,\\ldots, n \n", + "\\end{equation}\n", + "where $x_i^{(k)}$ is a value of the solution at iteration $k$ and $x_i^{(k+1)}$ is at the next iteration." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Created file '/Users/kyle/projects/ME373-book/content/pdes/heat_equation_gaussseidel.m'.\n" + ] + } + ], + "source": [ + "%%file heat_equation_gaussseidel.m\n", + "function [time, num_point, num_iter] = heat_equation_gaussseidel(h)\n", + "\n", + "x = [0 : h : 1]; n = length(x);\n", + "y = [0 : h : 1]; m = length(y);\n", + "\n", + "% The coefficient matrix A is now m*n by m*n, since that is the total number of points.\n", + "% The right-hand side vector b is m*n by 1.\n", + "A = zeros(m*n, m*n);\n", + "b = zeros(m*n, 1);\n", + "num_point = m*n;\n", + "\n", + "tic;\n", + "\n", + "u_left = 100;\n", + "u_right = 100;\n", + "u_bottom = 100;\n", + "u_top = 0;\n", + "\n", + "% initial guess\n", + "u = 100*ones(m*n, 1);\n", + "\n", + "% dummy value for residual variable\n", + "epsilon = 1.0; \n", + "\n", + "num_iter = 0;\n", + "while epsilon > 1e-6\n", + " u_old = u;\n", + " \n", + " epsilon = 0;\n", + " for j = 1 : m\n", + " for i = 1 : n\n", + " kij = (j-1)*n + i;\n", + " kim1j = (j-1)*n + i - 1;\n", + " kip1j = (j-1)*n + i + 1;\n", + " kijm1 = (j-2)*n + i;\n", + " kijp1 = j*n + i;\n", + "\n", + " if i == 1 \n", + " % this is the left boundary\n", + " u(kij) = u_left;\n", + " elseif i == n \n", + " % right boundary\n", + " u(kij) = u_right;\n", + " elseif j == 1 \n", + " % bottom boundary\n", + " u(kij) = u_bottom;\n", + " elseif j == m \n", + " % top boundary\n", + " u(kij) = u_top;\n", + " else\n", + " % interior points\n", + " u(kij) = (u(kip1j) + u(kim1j) + u(kijm1) + u(kijp1))/4.0;\n", + " end\n", + " end\n", + " end\n", + " \n", + " epsilon = max(abs(u - u_old));\n", + " num_iter = num_iter + 1;\n", + "end\n", + "\n", + "u_square = reshape(u, [n, m]);\n", + "%% the \"20\" indicates the number of levels for the contour plot\n", + "contourf(x, y, u_square', 20);\n", + "c = colorbar;\n", + "c.Label.String = 'Temperature';\n", + "\n", + "time = toc;\n", + "fprintf('Number of iterations: %d\\n', num_iter)" + ] + }, + { + "cell_type": "code", + "execution_count": 58, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Number of iterations: 156\n", + "Number of iterations: 564\n", + "Number of iterations: 2025\n", + "Number of iterations: 3048\n", + "Number of iterations: 7181\n", + "Number of iterations: 10762\n", + "Number of iterations: 37378\n" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "step_sizes = [0.1, 0.05, 0.025, 0.02, 0.0125, 0.01, 0.005];\n", + "n = length(step_sizes);\n", + "\n", + "nums_gs = zeros(n,1); times_gs = zeros(n,1); num_iter_gs = zeros(n,1);\n", + "\n", + "for i = 1 : n\n", + " [times_gs(i), nums_gs(i), num_iter_gs(i)] = heat_equation_gaussseidel(step_sizes(i));\n", + "end\n", + "\n", + "loglog(nums, times, '-o'); hold on\n", + "loglog(nums_jac, times_jac, '-^')\n", + "loglog(nums_gs, times_gs, '-x')\n", + "xlabel('Number of unknowns'); \n", + "ylabel('Time for solution (sec)')\n", + "legend('Direct solution', 'Jacobi solution', 'Gauss-Seidel solution', 'Location', 'northwest')" + ] + }, + { + "cell_type": "code", + "execution_count": 60, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "loglog(nums_jac, num_iter_jac, '-^')\n", + "hold on;\n", + "loglog(nums_gs, num_iter_gs, '-x')\n", + "xlabel('Number of unknowns')\n", + "ylabel('Number of iterations required')\n", + "legend('Jacobi method', 'Gauss-Seidel method', 'Location', 'northwest')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "These results show us a few things:\n", + "\n", + "- For very small problems, the direct solution method is faster.\n", + "- For the heat equation, once we get to around 1000 unknowns, the methods perform similarly. Beyond this, the direct solution method becomes unreasonably slow, and even fails to solve in a reasonable time for a step size of 0.005.\n", + "- The Gauss-Seidel method converges with around half the number of iterations than the Jacobi method.\n", + "\n", + "For larger, more-realistic problems, iterative solution methods like Jacobi and Gauss-Seidel are essential." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.11" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/docs/_sources/content/pdes/parabolic.ipynb b/docs/_sources/content/pdes/parabolic.ipynb new file mode 100644 index 0000000..27c90ef --- /dev/null +++ b/docs/_sources/content/pdes/parabolic.ipynb @@ -0,0 +1,345 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Parabolic PDEs\n", + "\n", + "A classic example of a parabolic partial differential equation (PDE) is the one-dimensional unsteady heat equation:\n", + "\\begin{equation}\n", + "\\frac{\\partial T}{\\partial t} = \\alpha \\frac{\\partial^2 T}{\\partial t^2} \n", + "\\end{equation}\n", + "where $T(x,t)$ is the temperature varying in space and time, and $\\alpha$ is the thermal diffusivity: $\\alpha = k / (\\rho c_p)$, which is a constant.\n", + "\n", + "We can solve this using finite differences to represent the spatial derivatives and time derivatives separately.\n", + "First, let's rearrange the PDE slightly:\n", + "\\begin{equation}\n", + "\\frac{\\partial^2 T}{\\partial x^2} = \\frac{1}{\\alpha} \\frac{\\partial T}{\\partial t}\n", + "\\end{equation}\n", + "\n", + "## Explicit scheme\n", + "\n", + "Let's use a *central difference* for the spatial derivative with a spacing of $\\Delta x$, and a *forward difference* for the time derivative with a time-step size of $\\Delta t$. With these choices, we can obtain an approximation to the PDE that applies at time $t^k$ and spatial location $x_i$:\n", + "\\begin{equation}\n", + "\\frac{T_{i-1}^k - 2 T_i^k + T_{i+1}^k}{\\Delta x^2} = \\frac{1}{\\alpha} \\left( \\frac{T_i^{k+1} - T_i^k}{\\Delta t} \\right)\n", + "\\end{equation}\n", + "where $T_i^k$ is the temperature at time $t^k$ and spatial location $x_i$. The following figure shows the stencil of points involved in the PDE, for a domain with five points in the $x$-direction.\n", + "\n", + ":::{figure-md} fig-stencil-explicit\n", + "\"stencil\n", + "\n", + "Stencil for explicit solution to heat equation\n", + ":::\n", + "\n", + "To solve the heat equation for a one-dimensional domain over $0 \\leq x \\leq L$, we will need both initial conditions at $t = 0$ and boundary conditions at $x=0$ and $x=L$ (for all time). In terms of our nodal values, this means we need $T_i^{k=1}$ for $i = 1 \\ldots n$, where $n$ is the number of points, as well as information about $T_1^k$ and $T_n^k$ for all times $k$.\n", + "\n", + "We can rearrange the above equation to obtain our recursion formula:\n", + "\\begin{equation}\n", + "T_i^{k+1} = \\left( T_{i+1}^k + T_{i-1}^k \\right) \\frac{\\alpha \\Delta t}{\\Delta x^2} + T_i^k \\left( 1 - 2 \\frac{\\alpha \\Delta t}{\\Delta x^2} \\right) \\;.\n", + "\\end{equation}\n", + "This is an **explicit** scheme in time, similar to the Forward Euler method we used for ordinary differential equations, and like that method it may have stability issues. The combination of terms we see repeated is also known as the Fourier number: $\\text{Fo} = \\frac{\\alpha \\Delta t}{\\Delta x^2}$, and governs the stability.\n", + "We can rewrite the recursion formula using this:\n", + "\\begin{equation}\n", + "T_i^{k+1} = \\left( T_{i+1}^k + T_{i-1}^k \\right) \\text{Fo} + T_i^k \\left( 1 - 2 \\text{Fo} \\right) \\;.\n", + "\\end{equation}\n", + "\n", + "The term in parentheses there must be greater than or equal to zero for stability ($1 - 2 \\text{Fo} \\geq 0$); if not, the solution may become unstable and blow up. This gives us some conditions on our choice of step sizes:\n", + "\\begin{align}\n", + "1 - 2 \\text{Fo} &\\geq 0 \\\\\n", + "1 & \\geq 2 \\text{Fo} \\\\\n", + "\\text{Fo} &\\leq \\frac{1}{2} \\\\\n", + "\\frac{\\alpha \\Delta t}{\\Delta x^2} &\\leq \\frac{1}{2}\n", + "\\end{align}\n", + "This is the stability criterion for the explicit method: the Fourier number must be smaller than 0.5.\n", + "For a given thermal diffusivity and chosen spatial step size, this also gives us the limit on the time-step size: $\\Delta t \\leq \\Delta x^2 / (2 \\alpha)$.\n", + "\n", + "Let's look at an example where the initial temperature is 200, the temperature at the boundaries are 50, the thermal diffusivity is $\\alpha = 2.3 \\times 10^{-1}$ m$^2 /$ s, and $L = 1$. \n", + "In other words,\n", + "\\begin{align}\n", + "T(x, t=0) &= 200 \\\\\n", + "T(x=0, t) &= 50 \\\\\n", + "T(x=L, t) &= 50\n", + "\\end{align}\n", + "We'll integrate out to $t = 1$, using a Fourier number of 0.25 to be comfortably below the stability limit (`Fo = 0.25`):" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [], + "source": [ + "clear all\n", + "\n", + "dx = 0.1;\n", + "alpha = 2.3e-1;\n", + "\n", + "% for stability, set the Fourier number at 0.25 (half the stability limit of 0.5)\n", + "Fo = 0.25;\n", + "% then choose the time step based on the Fourier number\n", + "dt = Fo * dx^2 / alpha;\n", + "\n", + "x = [0 : dx : 1]; n = length(x);\n", + "t = [0 : dt : 1]; m = length(t);\n", + "\n", + "T = zeros(m, n);\n", + "\n", + "% initial conditions\n", + "T(1,:) = 200;\n", + "\n", + "plot(x, T(1,:))\n", + "axis([0 1 50 200]);\n", + "xlabel('Distance'); ylabel('Temperature');\n", + "F(1) = getframe(gcf);\n", + "\n", + "for k = 1 : m - 1\n", + " for i = 1 : n\n", + " if i == 1\n", + " T(k+1, 1) = 50;\n", + " elseif i == n\n", + " T(k+1, n) = 50;\n", + " else\n", + " T(k+1, i) = (T(k,i+1) + T(k,i-1))*Fo + T(k,i)*(1 - 2*Fo);\n", + " end\n", + " end\n", + " plot(x, T(k+1,:))\n", + " axis([0 1 50 200]);\n", + " xlabel('Distance'); ylabel('Temperature');\n", + " F(k+1) = getframe(gcf);\n", + "end\n", + "close\n", + "\n", + "%% If you are working interactively, you can use this to make a movie in Matlab\n", + "%fig = figure;\n", + "%movie(fig, F, 2)\n", + "\n", + "%% This generates a GIF of the results (for use in Jupyter Notebook)\n", + "filename = 'parabolic_animated.gif';\n", + "for i = 1 : length(F)\n", + " im = frame2im(F(i)); \n", + " [imind,cm] = rgb2ind(im,256); \n", + " % Write to GIF\n", + " if i == 1 \n", + " imwrite(imind,cm,filename,'gif', 'Loopcount',inf, 'DelayTime',1e-3); \n", + " else \n", + " imwrite(imind,cm,filename,'gif','WriteMode','append', 'DelayTime',1e-3); \n", + " end\n", + "end" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + ":::{figure-md} fig-transient-heat\n", + "\"movie\n", + "\n", + "Animated solution to 1D transient heat transfer PDE\n", + ":::" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This shows the temperature decaying exponentially from the initial conditions, constrained by the boundary conditions.\n", + "\n", + "What happens if we tried to use a Fourier number larger than 0.5, or arbitrarily chose a time-step size that was too large (and resulted in $\\text{Fo} > 0.5$)?" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [], + "source": [ + "clear all\n", + "\n", + "dx = 0.1;\n", + "alpha = 2.3e-1;\n", + "\n", + "%% Purposely choose a Fourier number that is past the stability limit:\n", + "Fo = 0.75;\n", + "dt = Fo * dx^2 / alpha;\n", + "\n", + "x = [0 : dx : 1]; n = length(x);\n", + "t = [0 : dt : 1]; m = length(t);\n", + "\n", + "T = zeros(m, n);\n", + "\n", + "% initial conditions\n", + "T(1,:) = 200;\n", + "\n", + "plot(x, T(1,:))\n", + "axis([0 1 50 200]);\n", + "xlabel('Distance'); ylabel('Temperature');\n", + "F(1) = getframe(gcf);\n", + "\n", + "for k = 1 : m - 1\n", + " for i = 1 : n\n", + " if i == 1\n", + " T(k+1, 1) = 50;\n", + " elseif i == n\n", + " T(k+1, n) = 50;\n", + " else\n", + " T(k+1, i) = (T(k,i+1) + T(k,i-1))*Fo + T(k,i)*(1 - 2*Fo);\n", + " end\n", + " end\n", + " plot(x, T(k+1,:))\n", + " xlabel('Distance'); ylabel('Temperature');\n", + " F(k+1) = getframe(gcf);\n", + "end\n", + "close\n", + "\n", + "%% If you are working interactively, you can use this to make a movie in Matlab\n", + "%fig = figure;\n", + "%movie(fig, F, 2)\n", + "\n", + "%% This generates a GIF of the results (for use in Jupyter Notebook)\n", + "filename = 'parabolic_unstable_animated.gif';\n", + "for i = 1 : length(F)\n", + " im = frame2im(F(i)); \n", + " [imind,cm] = rgb2ind(im,256); \n", + " % Write to GIF\n", + " if i == 1 \n", + " imwrite(imind,cm,filename,'gif', 'Loopcount',2, 'DelayTime',1e-3); \n", + " else \n", + " imwrite(imind,cm,filename,'gif','WriteMode','append', 'DelayTime',1e-3); \n", + " end\n", + "end" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + ":::{figure-md} fig-unstable-solution\n", + "\"movie\n", + "\n", + "Animated unstable solution\n", + ":::" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this case, the solution becomes unstable and blows up, leading to unphysical results.\n", + "\n", + "For this explicit scheme, the choice of $\\Delta t$ is limited by the stability criterion. This means that we may be stuck using a small time-step size.\n", + "\n", + "Rather than being forced to use a very small time-step size, we can also explore *implicit schemes* that are unconditionally stable." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Implicit scheme\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "clear all\n", + "\n", + "alpha = 2.3e-1;\n", + "dx = 0.1;\n", + "x = [0 : dx : 1]; n = length(x);\n", + "\n", + "% choose a Fourier number that is deliberately past the explicit method stability limit\n", + "Fo = 0.75;\n", + "dt = Fo * dx^2 / alpha;\n", + "\n", + "t = [0 : dt : 1]; m = length(t);\n", + "\n", + "T = zeros(m, n);\n", + "\n", + "% Initial conditions\n", + "T(1,:) = 200;\n", + "\n", + "plot(x, T(1,:))\n", + "axis([0 1 50 200]);\n", + "xlabel('Distance'); ylabel('Temperature');\n", + "F(1) = getframe(gcf);\n", + "\n", + "for k = 1 : m - 1\n", + " A = zeros(n,n);\n", + " b = zeros(n,1);\n", + " for i = 1 : n\n", + " if i == 1\n", + " A(1,1) = 1;\n", + " b(1) = 50;\n", + " elseif i == n\n", + " A(n,n) = 1;\n", + " b(n) = 50;\n", + " else\n", + " A(i,i-1) = Fo;\n", + " A(i,i) = -2*Fo - 1;\n", + " A(i,i+1) = Fo;\n", + " b(i) = -T(k,i);\n", + " end\n", + " end\n", + " \n", + " T(k+1, :) = A \\ b;\n", + " plot(x, T(k+1,:))\n", + " axis([0 1 50 200]);\n", + " xlabel('Distance'); ylabel('Temperature');\n", + " F(k+1) = getframe(gcf);\n", + "end\n", + "close\n", + "\n", + "%% This generates a GIF of the results (for use in Jupyter Notebook)\n", + "filename = 'parabolic_implicit_animated.gif';\n", + "for i = 1 : length(F)\n", + " im = frame2im(F(i)); \n", + " [imind,cm] = rgb2ind(im,256); \n", + " % Write to GIF\n", + " if i == 1 \n", + " imwrite(imind,cm,filename,'gif', 'Loopcount',inf, 'DelayTime',1e-3); \n", + " else \n", + " imwrite(imind,cm,filename,'gif','WriteMode','append', 'DelayTime',1e-3); \n", + " end\n", + "end" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + ":::{figure-md} fig-implicit-solution\n", + "\"movie\n", + "\n", + "Solution to 1D heat equation with implicit method. Fo = 0.75\n", + ":::" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.11" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/docs/_sources/content/pdes/partial-differential-equations.md b/docs/_sources/content/pdes/partial-differential-equations.md new file mode 100644 index 0000000..47282ea --- /dev/null +++ b/docs/_sources/content/pdes/partial-differential-equations.md @@ -0,0 +1,15 @@ +# Partial Differential Equations + +This chapter focuses on numerical methods for solving partial differential equations (PDEs), which involve derivatives in multiple dimensions. + +We can write a general, linear 2nd-order PDE for a variable $u(x,y)$ as +\begin{equation} +A \frac{\partial^2 u}{\partial x^2} + 2 B \frac{\partial^2 u}{\partial x \, \partial y} + C \frac{\partial^2 u}{\partial y^2} = F \left( x, y, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y} \right) +\end{equation} +where $A$, $B$, and $C$ are constants. Depending on their value, we can categorize a PDE into one of three categories: + +- $B^2 - AC < 0$: elliptic +- $B^2 - AC = 0$: parabolic +- $B^2 - AC > 0$: hyperbolic + +The different PDE types will exhibit different characteristics and will also require slightly different solution approaches. diff --git a/docs/_sources/content/quizzes/quiz2-IVPs.md b/docs/_sources/content/quizzes/quiz2-IVPs.md new file mode 100644 index 0000000..45559cc --- /dev/null +++ b/docs/_sources/content/quizzes/quiz2-IVPs.md @@ -0,0 +1,213 @@ +# Sample Quiz 2 problems: IVPs + +## Problem 1: Stability analysis + +Given the first-order ODE +\begin{equation} +\frac{dy}{dt} = -8y \;, +\end{equation} + +a.) Perform a linear stability analysis to find the amplification factor ($\sigma$) for the **midpoint method** (also known as the modified Euler method) applied to this ODE. + +b.) Using your result from part (a), show whether the numerical solution would be stable or unstable for time-step sizes of $\Delta t = 0.2$ and $\Delta t = 0.4$. (Show for each value) + +c.) Based on your results for parts (a) and (b), is the midpoint method unstable, conditionally stable, or unconditionally stable for this ODE? + +d.) What is the order of accuracy for the midpoint method? Based on this, what (approximate) global errors would you expect in the solution when using time-step sizes of $\Delta t = 0.2$ and 0.4? + +e.) What are your two options for reducing error in the solution? + +### Solution + +a.) +\begin{align} +y_{i+1/2} &= y_i + \frac{\Delta t}{2} \, f(t_i, y_i) \\ +y_{i+1} &= y_i + \Delta t \, f(t_i + \frac{\Delta t}{2}, y_{i+1/2}) +\end{align} + +For this ODE: +\begin{align} +y_{i+1/2} &= y_i + \frac{\Delta t}{2} \, (-8 y_i) = y_i - 4 \Delta t \, y_i \\ +y_{i+1} &= y_i + \Delta t \, \left[ -8 \left( y_i - 4 \Delta t \, y_i \right) \right] \\ +&= y_i - 8 \Delta t \, y_i + 32 \Delta t^2 \, y_i \\ +&= y_i (1 - 8 \Delta t + 32 \Delta t^2) +\end{align} +So, +\begin{equation} +\sigma = \frac{y_{i+1}}{y_i} = 1 - 8 \Delta t + 32 \Delta t^2 +\end{equation} + +b.) For stability, $|\sigma| \leq 1$. + +$\Delta t = 0.2$: +\begin{equation} +\sigma = 1 - 8(0.2) + 32 (0.2)^2 = 0.68 +\end{equation} +so **stable**. + +$\Delta t = 0.4$: +\begin{equation} +\sigma = 1 - 8(0.4) + 32 (0.4)^2 = 2.92 +\end{equation} +so **unstable**. + +c.) Conditionally stable; the method is stable for some values of $\Delta t$ and unstable for other values. + +d.) The midpoint method is 2nd-order accurate. So, for $\Delta t = 0.2$, we should expect global errors on the order of 0.04, and for $\Delta t = 0.4$ we should expect errors on the order of 0.16. + +e.) + + - reduce the step size + - choose a higher order method, such as 4th-order Runge-Kutta + + +## Problem 2: Second-order Backward Euler + +Given the second-order ODE +\begin{equation} +2 y^{\prime\prime} + y^{\prime} + 4y = 3x +\end{equation} + +a.) Find the recursion formulas (i.e., $y_{i+1} = \ldots$) for numerically solving this ODE using the backward Euler method. Clearly define/state any variable or function you use. + + +b.) What is the order of accuracy for the backward Euler method? Given a step size $\Delta x = 0.15$, approximately what local error and what global error would you expect in your solution? What is the difference between these two errors? + +c.) Why would you want to use this method to solve an ODE over a simpler method like forward Euler? + +### Solution + +a.) System of Backward Euler recursion formulas, for $y(x)$ and $u(x) = y^{\prime}$, where $f(x,y,u) = dy/dx$ and $g(x,y,u) = du/dx = y^{\prime\prime}$: +\begin{align} +y_{i+1} &= y_i + \Delta x \, f(x_{i+1}, y_{i+1}, u_{i+1}) = y_i + \Delta x \, u_{i+1} \\ +u_{i+1} &= u_i + \Delta x \, g(x_{i+1}, y_{i+1}, u_{i+1}) = u_i + \Delta x \, \left( \frac{3}{2}x_{i+1} - \frac{1}{2} u_{i+1} - 2 y_{i+1} \right) +\end{align} +This form is implicit and cannot be used directly, so we need to solve the system of equations. + +*There are two ways to approach this, that give an equivalent solution* + +**Option 1:** Use substitution and elimination to solve: +\begin{align} +u_{i+1} &= u_i + \frac{3}{2} \Delta x \, x_{i+1} - \frac{1}{2} \Delta x \, u_{i+1} - 2 \Delta x \left( y_i + \Delta x \, u_{i+1} \right) \\ +&= u_i + \frac{3}{2} \Delta x \, x_{i+1} - \frac{1}{2} \Delta x \, u_{i+1} - 2 \Delta x \, y_i - 2 \Delta x^2 \, u_{i+1} \\ +u_{i+1} \left( 1 + 2 \Delta x^2 + \frac{1}{2} \Delta x \right) &= u_i + \frac{3}{2} \Delta x \, x_{i+1} - 2 \Delta x \, y_i +\end{align} +Thus, +\begin{align} +u_{i+1} &= \frac{u_i + \frac{3}{2} \Delta x \, x_{i+1} - 2 \Delta x \, y_i}{1 + 2 \Delta x^2 + \frac{1}{2} \Delta x} \\ +y_{i+1} &= y_i + \Delta x \frac{u_i + \frac{3}{2} \Delta x \, x_{i+1} - 2 \Delta x \, y_i}{1 + 2 \Delta x^2 + \frac{1}{2} \Delta x} +\end{align} + +**Option 2:** Or, use Cramer's rule: +\begin{align} +y_{i+1} - \Delta x u_{i+1} &= y_i \\ +2 \Delta x \, y_{i+1} + \left( 1 + \frac{1}{2} \Delta x \right) u_{i+1} &= u_i + \frac{3}{2} \Delta x \, x_{i+1} \\ +\rightarrow \begin{bmatrix} 1 & -\Delta x \\ 2 \Delta x & \left( 1 + \frac{1}{2} \Delta x \right)\end{bmatrix} \begin{bmatrix} y_{i+1} \\ u_{i+1} \end{bmatrix} &= +\begin{bmatrix} y_i \\ u_i + \frac{3}{2} \Delta x \, x_{i+1} \end{bmatrix} +\end{align} +Then, +\begin{align} +y_{i+1} &= \frac{y_i \left(1 + \frac{\Delta x}{2} \right) + \Delta x \left( u_i + \frac{3}{2} \Delta x \, x_{i+1} \right)}{1 + \frac{\Delta x}{2} + 2 \Delta x^2} \\ +u_{i+1} &= \frac{u_i + \frac{3}{2} \Delta x \, x_{i+1} - 2 \Delta x \, y_i}{1 + \frac{\Delta x}{2} + 2 \Delta x^2} +\end{align} + +b.) Backward Euler is 1st-order accurate, so the global error is on the order of the step size ($\Delta x$). + +The global error should then be on the order of 0.15, and the local error on the order of $0.15^2 = 0.0225$. + +The difference: the local (or truncation) error is the error at each step of the solution, while the global error is the overall error that accumulates over the whole solution. + +c.) Backward Euler is unconditionally stable, while Forward Euler is conditionally stable. + +## Problem 3: Fourier series + +Given the input waveform $R(t)$ shown here, + +![Increasing square wave form](../../images/wave.png "Waveform") + +a.) What is the period and fundamental frequency of the input forcing function? + + +b.) Is the periodic function $R(t)$ odd, even, or neither? + +c.) Find the coefficients $a_0$ and $a_n$ of the Fourier series representation of $R(t)$. (For purposes of time, you do not need to find $b_n$): +\begin{equation} +R(t) = a_0 + \sum_{n=1}^{\infty} a_n \cos(n \omega t) + \sum_{n=1}^{\infty} b_n \sin(n \omega t) +\end{equation} + + +### Solution + +a.) +\begin{equation} +T = 4\pi \quad \omega = \frac{2\pi}{T} = \frac{1}{2} +\end{equation} + +b.) Neither. + +c.) +\begin{align} +a_0 &= \frac{1}{T} \int_0^T f(t) dt = \frac{1}{4\pi} \left[ \int_0^{\pi} 0 dt + \int_{\pi}^{2\pi} 3 dt + \int_{2\pi}^{3\pi} 2 dt + \int_{3\pi}^{4\pi} 1 dt \right] \\ +&= \frac{1}{4\pi} \left[ 3 \pi + 2 \pi + 1 \pi \right] \\ +a_0 &= \frac{3}{2} +\end{align} + +\begin{align} +a_n &= \frac{2}{T} \int_0^T f(t) \cos(n \omega t) dt \\ +&= \frac{2}{4\pi} \left[ \int_0^{\pi} 0 \cos \left(\frac{n t}{2}\right) dt + \int_{\pi}^{2\pi} 3 \cos \left(\frac{n t}{2}\right) dt + \int_{2\pi}^{3\pi} 2 \cos \left(\frac{n t}{2}\right) dt + \int_{3\pi}^{4\pi} 1 \cos \left(\frac{n t}{2}\right) dt \right] \\ +&\cdots \\ +a_n &= \frac{1}{n\pi} \left[ \sin\left(\frac{3 n \pi}{2}\right) - 3 \sin\left(\frac{n \pi}{2}\right) \right] +\end{align} + + +## Problem 4: Second-order analytical + +The displacement $y(t)$ of a harmonically forced mass-spring system is given by: +\begin{equation} +y^{\prime\prime} + 8y^{\prime} + 16y = 6 e^{-4t} +\end{equation} + +a.) For initial conditions $y(0)=0$ and $y^{\prime}(0) = 2$, find the response of the system $y(t)$. + +b.) Given a specified time increment $\Delta t$ and a domain, write the recursion formulas for solving this equation with the forward Euler method. Clearly define any variables or functions used. You do not need to write Matlab code. + +### Solution + +a.) First, get the homogeneous solution: +\begin{align} +y_H^{\prime\prime} + 8y_H^{\prime} + 16y_H &= 0 \\ +\rightarrow \lambda^2 + 8 \lambda + 16 &= 0 = (\lambda + 4)^2 \\ +\lambda &= 4 \text{ (repeated)} \\ +\text{so } y_H &= c_1 e^{-4t} + c_2 t e^{-4t} +\end{align} + +Next, use the method of undetermined coefficients to get the inhomogeneous solution: +\begin{align} +y_{IH} &= K t^2 e^{-4t} \\ +y_{IH}^{\prime} &= K e^{-4t} \left( 2t - 4t^2 \right) \\ +y_{IH}^{\prime\prime} &= K e^{-4t} \left( -8t + 2 - 8t + 16t^2 \right) \\ +\rightarrow K &= 3 +\end{align} + +Then, the general solution is +\begin{equation} +y(t) = c_1 e^{-4t} + c_2 t e^{-4t} + 3 t^2 e^{-4t} +\end{equation} + +Applying the initial conditions: +\begin{equation} +y(t) = 2 t e^{-4t} + 3 t^2 e^{-4t} +\end{equation} + +b.) If $z_1 = y$ and $z_2 = y^{\prime}$, then +\begin{align} +z_1^{\prime} &= z_2 \\ +z_2^{\prime} &= y^{\prime\prime} = 6e^{-4t} - 8z_2 - 16z_1 = f(t, z_1, z_2) +\end{align} + +Then, the recursion formulas are: +\begin{align} +z_{1, i+1} &= z_{1,i} + \Delta t \, z_{2,i} \\ +z_{2, i+1} &= z_{2, i} + \Delta t \left( 6 e^{-4t} - 8 z_{2,i} - 16 z_{1,i} \right) +\end{align} +where $z_{1,1} = 0$ and $z_{2,1} = 4$. diff --git a/docs/_sources/content/quizzes/quiz3-BVPs.md b/docs/_sources/content/quizzes/quiz3-BVPs.md new file mode 100644 index 0000000..f1504a9 --- /dev/null +++ b/docs/_sources/content/quizzes/quiz3-BVPs.md @@ -0,0 +1,144 @@ +# Sample Quiz 3 problems: BVPs + +## Problem 1: Finite difference method + +The temperature distribution $T(r)$ in an annular fin of inner radius $r_1$ and outer radius $r_2$ is described by the equation +\begin{equation} +r \frac{d^2 T}{dr^2} + \frac{dT}{dr} - rm^2 (T - T_{\infty}) = 0 \;, +\end{equation} +where $r$ is the radial distance from the centerline (the independent +variable) and $m^2$ is a constant that depends on the heat transfer coefficient, thermal conductivity, and thickness of the annulus. Assuming we choose a spatial step size $\Delta r$, + +
+
+ Annular fin +
Figure: Annular fin
+
+
+ +a.) Write the finite-difference representation of the ODE (that applies at a location $r_i$), using central differences. + +b.) Based on the last part, write the recursion formula. + +c.) The boundary condition at the outer radius $r = r_2$ is described by convection heat transfer: +\begin{equation} +-k \left. \frac{dT}{dr} \right|_{r=r_2} = h \left[ T(r=r_2) - T_{\infty} \right] \;. +\end{equation} +Write the boundary condition at $r = r_2$ in recursion form (i.e., the equation you would implement into your system of equations to solve for temperature). + + +### Solution + +a.) Replace the derivatives in the given ODE with finite differences, and replace any locations with the $i$ location: +\begin{equation} +r_i \frac{T_{i-1} - 2T_i + T_{i+1}}{\Delta r^2} + \frac{T_{i+1} - T_{i-1}}{2\Delta r} - r_i m^2 (T_i - T_{\infty}) = 0 +\end{equation} +or +\begin{equation} +r_i (T_{i-1} - 2T_i + T_{i+1}) + \frac{\Delta r}{2} (T_{i+1} - T_{i-1}) - r_i m^2 \Delta r^2 (T_i - T_{\infty}) = 0 +\end{equation} + +b.) Rearrange and combine terms: +\begin{align} +r_i (T_{i-1} - 2T_i + T_{i+1}) + \frac{\Delta r}{2} (T_{i+1} - T_{i-1}) - r_i m^2 \Delta r^2 (T_i - T_{\infty}) &= 0 \\ +r_i (T_{i-1} - 2T_i + T_{i+1}) + \frac{\Delta r}{2} (T_{i+1} - T_{i-1}) - r_i m^2 \Delta r^2 T_i &= -r_i m^2 \Delta r^2 T_{\infty} \\ +\left(r_i - \frac{\Delta r}{2}\right) T_{i-1} + \left( -2 r_i - r_i m^2 \Delta r^2 \right) T_i + \left( r_i + \frac{\Delta r}{2} \right) T_{i+1} &= -r_i m^2 \Delta r^2 T_{\infty} +\end{align} + +c.) We can use a backward difference to approximate the $dT/dr$ term. $T_n$ represents the temperature at node $n$ where $r_n = r_2$: +\begin{align} +-k \frac{T_n - T_{n-1}}{\Delta r} &= h (T_n - T_{\infty}) \\ +-k (T_n - T_{n-1}) &= h \Delta r (T_n - T_{\infty}) \\ +k T_{n-1} - (k + h\Delta r) T_n &= -h \Delta r T_{\infty} +\end{align} + + +## Problem 2: eigenvalue + +Given the equation $y^{\prime\prime} + 9 \lambda^2 y = 0$ with $y(0) = 0$ and $y(2) = 0$, + +a.) Find the expression that gives all eigenvalues ($\lambda$). What is the eigenfunction? + +b.) Calculate the principal eigenvalue. + + +### Solution + +a.) +\begin{gather} +y(x) = A \sin (3 \lambda x) + B \cos (3 \lambda x) \\ +\text{Apply BCs: } y(x=0) = 0 = A \sin(0) + B \cos(0) = B \\ +\therefore B = 0 \\ +y(x) = A \sin (3 \lambda x) \\ +y(x=2) = 0 = A \sin (3 \lambda 2) \\ +A \neq 0 \text{ so } \sin(3 \lambda 2) = \sin(6 \lambda) = 0 \therefore 6 \lambda = n \pi \quad n=1,2,3,\ldots \\ +\lambda = \frac{n \pi}{6} \quad n=1,2,3,\ldots,\infty +\end{gather} + +The eigenfunction is then the solution function associated with an eigenvalue: +\begin{equation} +y_n = A_n \sin \left( \frac{n \pi x}{2} \right) \quad n = 1, 2, 3, \ldots, \infty +\end{equation} + +b.) The principal eigenvalue is just that associated with $n = 1$: +\begin{equation} +\lambda_p = \lambda_1 = \frac{\pi}{6} +\end{equation} + + + +## Problem 3: shooting method + +Use the shooting method to solve the boundary value problem +\begin{equation} +y^{\prime\prime} - 4y = 0 +\end{equation} +where $y(0) = 0$ and $y(1) = 3$. Find the initial value of $y'$ (meaning, $y'(0)$) that satisfies the given boundary conditions. Use the forward Euler method with a step size of $\Delta x = 0.5$. + + +### Solution + +First decompose into two 1st-order ODEs: +\begin{align} +z_1' &= y' = z_2 \\ +z_2' &= y'' = 4 z_1 +\end{align} +with BCs $z_1 (x=0) = z_{1,1} = 0$ and $z_1(x=1) = z_{1,3} = 3$, we do not know $y'(0) = z_2(x=0) = z_{2,1} = ?$ + +Try some guess \#1: $y' (0) = 0 = z_2 (0)$, with the forward Euler method: +\begin{align} +z_{1,2} = z_1 (0.5) &= z_1 (0) + z_2(0) 0.5 = 0 \\ +z_{2,2} = z_2 (0.5) &= z_2 (0) + \left( 4z_1(0) \right) 0.5 = 0 \\ +z_{1,3} = z_1 (1.0) &= z_1 (0.5) + z_2(0.5) 0.5 = 0 \leftarrow \text{solution 1} \\ +z_{2,3} = z_2 (1.0) &= z_2 (0.5) + \left( 4z_1(0.5) \right) 0.5 = 0 +\end{align} +so for solution 1: $y(1) = 0 \neq 3$. + +For guess \#2: $y' (0) = 2 = z_2 (0)$, with the forward Euler method: +\begin{align} +z_1 (0.5) &= z_1 (0) + z_2(0) 0.5 = 1.0 \\ +z_2 (0.5) &= z_2 (0) + \left( 4z_1(0) \right) 0.5 = 2.0 \\ +z_1 (1.0) &= z_1 (0.5) + z_2(0.5) 0.5 = 2.0 \leftarrow \text{solution 2} \\ +z_2 (1.0) &= z_2 (0.5) + \left( 4z_1(0.5) \right) 0.5 = 4.0 +\end{align} +so for solution 1: $y(1) = 2 \neq 3$. + +For guess \#3, we can interpolate: +\begin{align} +m &= \frac{\text{guess 1} - \text{guess 2}}{\text{solution 1} - \text{solution 2}} = \frac{0 - 2}{0 - 2} = 1 \\ +\text{guess 3} &= \text{guess 2} + m (\text{target} - \text{solution 2}) = 2 + 1(3-2) = 3 +\end{align} +then, use this guess: +\begin{align} +z_1 (0.5) &= z_1 (0) + z_2(0) 0.5 = 1.5 \\ +z_2 (0.5) &= z_2 (0) + \left( 4z_1(0) \right) 0.5 = 3.0 \\ +z_1 (1.0) &= z_1 (0.5) + z_2(0.5) 0.5 = 3.0 \leftarrow \text{solution 3} \\ +z_2 (1.0) &= z_2 (0.5) + \left( 4z_1(0.5) \right) 0.5 = 6.0 +\end{align} +so for solution 3: $y(1) = 3$ which is the target. + +So our answer is $y'(0) = 3$. + +```python + +``` diff --git a/docs/_sources/content/second-order/analytical.ipynb b/docs/_sources/content/second-order/analytical.ipynb new file mode 100644 index 0000000..02c5ca4 --- /dev/null +++ b/docs/_sources/content/second-order/analytical.ipynb @@ -0,0 +1,305 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Analytical Solutions to 2nd-order ODEs\n", + "\n", + "Let's first focus on simple analytical solutions for 2nd-order ODEs. As before, let's categorize problems based on their solution approach." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Solution by direct integration\n", + "\n", + "If you have a 2nd-order ODE of this form:\n", + "\\begin{equation}\n", + "\\frac{d^2 y}{dx^2} = f(x)\n", + "\\end{equation}\n", + "then you can solve by **direct integration**.\n", + "\n", + "For example, let's say we are trying to solve for the deflection of a cantilever beam $y(x)$ with a force $P$ at the end, where $E$ is the modulus and $I$ is the moment of intertia, and the initial conditions are $y(0)=0$ and $y^{\\prime}(0) = 0$:\n", + "\n", + "![Cantilever beam with force at end](../../images/cantilever.png \"Cantilever beam\")\n", + "\n", + "\\begin{align}\n", + "\\frac{d^2 y}{dx^2} &= \\frac{-P (L-x)}{EI} \\\\\n", + "\\frac{d}{dx} \\left(\\frac{dy}{dx}\\right) &= \\frac{-P}{EI} (L-x) \\\\\n", + "\\int d \\left(\\frac{dy}{dx}\\right) &= \\frac{-P}{EI} \\int (L-x) dx \\\\\n", + "y^{\\prime} = \\frac{dy}{dx} &= \\frac{-P}{EI} \\left(Lx - \\frac{x^2}{2}\\right) + C_1 \\\\\n", + "\\int dy &= \\int \\left( \\frac{-P}{EI} \\left(Lx - \\frac{x^2}{2}\\right) + C_1 \\right) dx \\\\\n", + "y(x) &= \\frac{-P}{EI} \\left(\\frac{L}{2} x^2 - \\frac{1}{6} x^3\\right) + C_1 x + C_2\n", + "\\end{align}\n", + "That is our general solution; we can obtain the specific solution by applying our two initial conditions:\n", + "\\begin{align}\n", + "y(0) &= 0 = C_2 \\\\\n", + "y^{\\prime}(0) &= 0 = C_1 \\\\\n", + "\\therefore y(x) &= \\frac{P}{EI} \\left( \\frac{x^3}{6} - \\frac{L x^2}{2} \\right)\n", + "\\end{align}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Solution by substitution\n", + "\n", + "If we have a 2nd-order ODE of this form:\n", + "\\begin{equation}\n", + "\\frac{d^2 y}{dx^2} = f(x, y^{\\prime})\n", + "\\end{equation}\n", + "then we can solve by **substitution**, meaning by substituting a new variable for $y^{\\prime}$. (Notice that $y$ itself does not show up in the ODE.)\n", + "\n", + "Let's substitute $u$ for $y^{\\prime}$ in the above ODE:\n", + "\\begin{align}\n", + "u &= y^{\\prime} \\\\\n", + "u^{\\prime} &= y^{\\prime\\prime} \\\\\n", + "\\rightarrow u^{\\prime} &= f(f, u)\n", + "\\end{align}\n", + "Now we have a 1st-order ODE! Then, we can apply the methods previously discussed to solve this; once we find $u(x)$, we can integrate that once more to get $y(x)$.\n", + "\n", + "### Example: falling object\n", + "\n", + "For example, consider a falling mass where we are solving for the downward distance as a function of time, $y(t)$, that is experiencing the force of gravity downward and a drag force upward. It starts at some reference point so $y(0) = 0$, and has a zero initial downward velocity: $y^{\\prime}(0) = 0$. The governing equation is:\n", + "\\begin{equation}\n", + "m \\frac{d^2 y}{dt^2} = mg - c \\left( \\frac{dy}{dt} \\right)^2\n", + "\\end{equation}\n", + "where $m$ is the mass, $g$ is acceleration due to gravity, and $c$ is the drag proportionality constant.\n", + "We can substitute $V$ for $y^{\\prime}$, which gives us a first-order ODE:\n", + "\\begin{align}\n", + "\\text{let} \\quad \\frac{dy}{dt} = V \\\\\n", + "\\rightarrow m \\frac{dV}{dt} &= mg - c V^2 \\\\\n", + "\\end{align}\n", + "Then, we can solve this for $V(t)$ using our initial condition for velocity $V(0) = 0$. Once we have that, we can integrate once more:\n", + "\\begin{equation}\n", + "y(t) = \\int V(t) dt\n", + "\\end{equation}\n", + "and apply our initial condition for position, $y(0) = 0$, to obtain $y(t)$.\n", + "\n", + "Here is the full process:\n", + "\\begin{align}\n", + "\\frac{dV}{dt} &= g - \\frac{c}{m} V^2 \\\\\n", + "\\frac{dV}{g - \\frac{c}{m} V^2} &= dt \\\\\n", + "\\frac{m}{c} \\int \\frac{dV}{a^2 - V^2} &= \\int dt = t + \\bar{c}, \\quad \\text{where} \\quad a = \\sqrt{\\frac{mg}{c}} \\\\\n", + "\\frac{m}{c} \\frac{1}{a} \\tanh^{-1} \\left(\\frac{V}{a}\\right) &= t + c_1 \\\\\n", + "V &= a \\tanh \\left( \\frac{a c}{m} t + c_1 \\right) \\\\\n", + "\\therefore V(t) &= \\sqrt{\\frac{mg}{c}} \\tanh \\left(\\sqrt{\\frac{gc}{m}} t + c_1\\right)\n", + "\\end{align}\n", + "Applying the initial condition for velocity, $V(0) = 0$:\n", + "\\begin{align}\n", + "V(0) &= 0 = \\sqrt{\\frac{mg}{c}} \\tanh \\left(0 + c_1\\right) \\\\\n", + "\\therefore c_1 &= 0 \\\\\n", + "V(t) &= \\sqrt{\\frac{mg}{c}} \\tanh \\left(\\sqrt{\\frac{gc}{m}} t\\right)\n", + "\\end{align}\n", + "Then, to get $y(t)$, we just need to integrate once more:\n", + "\\begin{align}\n", + "\\frac{dy}{dt} = V(t) &= \\sqrt{\\frac{mg}{c}} \\tanh \\left(\\sqrt{\\frac{gc}{m}} t\\right) \\\\\n", + "\\int dy &= \\sqrt{\\frac{mg}{c}} \\int \\tanh \\left(\\sqrt{\\frac{gc}{m}} t\\right) dt \\\\\n", + "y(t) &= \\sqrt{\\frac{mg}{c}} \\sqrt{\\frac{m}{gc}} \\log\\left(\\cosh\\left(\\sqrt{\\frac{gc}{m}} t\\right)\\right) + c_2 \\\\\n", + "\\rightarrow y(t) &= \\frac{m}{c} \\log\\left(\\cosh\\left(\\sqrt{\\frac{gc}{m}} t\\right)\\right) + c_2\n", + "\\end{align}\n", + "Finally, we can apply the initial condition for position, $y(0) = 0$, to get our solution:\n", + "\\begin{align}\n", + "y(0) &= 0 = \\frac{m}{c} \\log\\left(\\cosh\\left(0\\right)\\right) + c_2 = c_2 \\\\\n", + "\\rightarrow c_2 &= 0 \\\\\n", + "y(t) &= \\frac{m}{c} \\log\\left(\\cosh\\left(\\sqrt{\\frac{gc}{m}} t\\right)\\right)\n", + "\\end{align}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Example: catenary problem\n", + "\n", + "The catenary problem describes the shape of a hanging chain or rope fixed between two points. (It was also a favorite of one of my professors, [Joe Prahl](https://en.wikipedia.org/wiki/Joseph_M._Prahl), and I like to teach it as an example in his honor.) The downward displacement of the hanging string/chain/rope as a function of horizontal position, $y(x)$, is governed by the equation\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} = \\sqrt{1 + (y^{\\prime})^2}\n", + "\\end{equation}\n", + "\n", + "![Catenary problem (hanging rope/chain)](../../images/catenary.png \"Catenary problem\")\n", + "\n", + "This is actually a **boundary value problem**, with the boundary conditions for the displacement at one side $y(0) = 0$ and that the slope is zero in the middle: $\\frac{dy}{dx}\\left(\\frac{L}{2}\\right) = 0$. (Please note that I have skipped the derivation of the governing equation, and left some details out.)\n", + "\n", + "We can solve this via substitution, by letting a new variable $u = y^{\\prime}$; then, $u^{\\prime} = \\frac{du}{dx} = y^{\\prime\\prime}$. This gives is a first-order ODE, which we can integrate:\n", + "\\begin{align}\n", + "\\frac{du}{dx} &= \\sqrt{1 + u^2} \\\\\n", + "\\int \\frac{du}{\\sqrt{1 + u^2}} &= \\int dx \\\\\n", + "\\sinh^{-1}(u) &= x + c_1, \\quad \\text{where } \\sinh(x) = \\frac{e^x - e^{-x}}{2} \\\\\n", + "u(x) &= \\sinh(x + c_1)\n", + "\\end{align}\n", + "\n", + "Then, we can integrate once again to get $y(x)$:\n", + "\\begin{align}\n", + "\\frac{dy}{dx} &= u(x) = \\sinh(x + c_1) \\\\\n", + "\\int dy &= \\int \\sinh(x + c_1) dx = \\int \\left(\\sinh(x)\\cosh(c_1) + \\cosh(x)\\sinh(c_1)\\right)dx \\\\\n", + "y(x) &= \\cosh(x)\\cosh(c_1) + \\sinh(x)\\sinh(c_1) + c_2 \\\\\n", + "\\rightarrow y(x) &= \\cosh(x + c_1) + c_2\n", + "\\end{align}\n", + "This is the general solution to the catenary problem, and applies to any boundary conditions.\n", + "\n", + "For our specific case, we can apply the boundary conditions and find the particular solution, though it involves some algebra...:\n", + "\\begin{align}\n", + "y(0) &= 0 = \\cosh(c_1) + c_2 \\\\\n", + "\\frac{dy}{dx}\\left(\\frac{L}{2}\\right) &= u(0) = \\sinh \\left(\\frac{L}{2} + c_1\\right) \\\\\n", + "\\rightarrow c_1 &= -\\frac{L}{2} \\\\\n", + "0 &= \\cosh \\left( -\\frac{L}{2} \\right) + c_2 \\\\\n", + "\\rightarrow c_2 &= -\\cosh\\left( -\\frac{L}{2} \\right) = -\\cosh\\left(\\frac{L}{2} \\right)\n", + "\\end{align}\n", + "So, the overall solution for the catenary problem with the given boundary conditions is\n", + "\\begin{equation}\n", + "y(x) = \\cosh \\left(x - \\frac{L}{2}\\right) - \\cosh\\left( \\frac{L}{2} \\right)\n", + "\\end{equation}\n", + "\n", + "Let's see what this looks like:" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "L = 1.0;\n", + "x = linspace(0, 1);\n", + "y = cosh(x - (L/2.)) - cosh(L/2.);\n", + "plot(x, y)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Please note that I've made some simplifications in the above work, and skipped the details of how the ODE is derived. In general, the solution for the shape is\n", + "\\begin{equation}\n", + "y(x) = C \\cosh \\frac{x + c_1}{C} + c_2\n", + "\\end{equation}\n", + "where you would solve for the constants $C$, $c_1$, and $c_2$ using the constraints:\n", + "\\begin{align}\n", + "\\int_{x_a}^{x_b} \\sqrt{1 + (y^{\\prime})^2} dx &= L \\\\\n", + "y(x_a) &= y_a \\\\\n", + "y(x_b) &= y_b \\;,\n", + "\\end{align}\n", + "where $L$ is the length of the rope/chain.\n", + "\n", + "You can read more about the catenary problem here (for example): " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Homogeneous 2nd-order ODEs\n", + "\n", + "An important category of 2nd-order ODEs are those that look like\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + p(x) y^{\\prime} + q(x) y = 0\n", + "\\end{equation}\n", + "\"Homogeneous\" means that the ODE is unforced; that is, the right-hand side is zero.\n", + "\n", + "Depending on what $p(x)$ and $q(x)$ look like, we have a few different solution approaches:\n", + "\n", + "- constant coefficients: $y^{\\prime\\prime} + a y^{\\prime} + by = 0$\n", + "- Euler-Cauchy equations: $x^2 y^{\\prime\\prime} + axy^{\\prime} + by = 0$\n", + "- Series solutions\n", + "\n", + "First, let's talk about the characteristics of linear, homogeneous 2nd-order ODEs:\n", + "\n", + "### Solutions have two parts: \n", + "Solutions have two parts: $y(x) = c_1 y_1 + c_2 y_2$, where $y_1$ and $y_2$ are each a basis of the solution.\n", + "\n", + "### Linearly independent:\n", + "\n", + "The two parts of the solution $y_1$ and $y_2$ are linearly independent.\n", + "\n", + "One way of defining this is that $a_1 y_1 + a_2 y_2 = 0$ only has the trivial solution $a_1=0$ and $a_2=0$.\n", + "\n", + "Another way of thinking about this is that $y_1$ and $y_2$ are linearly *dependent* if one is a multiple of the other, like $y_1 = x$ and $y_2 = 5x$. This *cannot* be solutions to a linear, homogeneous ODE.\n", + "\n", + "### Both parts satisfy the ODE:\n", + "\n", + "$y_1$ and $y_2$ each satisfy the ODE. Meaning, you can plug each of them into the ODE for $y$ and obtain 0.\n", + "\n", + "However, we need both parts together to fully solve the ODE.\n", + "\n", + "### Reduction of order: \n", + "\n", + "If $y_1$ is known, we can get $y_2$ by **reduction of order**. Let $y_2 = u y_1$, where $u$ is some unknown function of $x$. Then, put $y_2$ into the ODE $y^{\\prime}{\\prime} + p(x) y^{\\prime} + q(x) y = 0$:\n", + "\\begin{align}\n", + "y_2 &= u y_1 \\\\\n", + "y_2^{\\prime} &= u y_1^{\\prime} + u^{\\prime} y_1 \\\\\n", + "y_2^{\\prime\\prime} &= 2 u^{\\prime} y_1^{\\prime} + u^{\\prime\\prime} y_1 + u y_1^{\\prime\\prime} \\\\\n", + "\\rightarrow u^{\\prime\\prime} &= - \\left[ p(x) + \\left(\\frac{2 y_1^{\\prime}}{y_1}\\right) \\right] u^{\\prime}\n", + "\\text{or, } u^{\\prime\\prime} &= - \\left( g(x) \\right) u^{\\prime}\n", + "\\end{align}\n", + "Now, we have an ODE with only $u^{\\prime\\prime}$, $u^{\\prime}$, and some function $g(x)$—so we can solve by substitution! Let $u^{\\prime} = v$, and then we have $v^{\\prime} = -g(x) v$:\n", + "\\begin{align}\n", + "\\frac{dv}{dx} &= - \\left( p(x) + \\frac{2 y_1^{\\prime}}{y_1} \\right) v \\\\\n", + "\\int \\frac{dv}{v} &= - \\int \\left(p(x) + \\frac{2 y_1^{\\prime}}{y_1} \\right) dx \\\\\n", + "\\text{Recall } 2 \\frac{d}{dx} \\left( \\ln y_1 \\right) &= 2 \\frac{y_1^{\\prime}}{y_1} \\\\\n", + "\\therefore \\int \\frac{dv}{v} &= - \\int \\left(p(x) + 2 \\frac{d}{dx} \\left( \\ln y_1 \\right) \\right) dx \\\\\n", + "\\ln v &= -\\int p(x) dx - 2 \\ln y_1 \\\\\n", + "\\rightarrow v &= \\frac{\\exp\\left( -\\int p(x)dx \\right)}{y_1^2}\n", + "\\end{align}\n", + "\n", + "So, the actual solution procedure is then:\n", + "\n", + " 1. Solve for $v$: $v = \\frac{\\exp\\left( -\\int p(x)dx \\right)}{y_1^2}$\n", + " 2. Solve for $u$: $u = \\int v dx$\n", + " 3. Get $y_2$: $y_2 = u y_1$\n", + " \n", + "Here's an example, where we know one part of the solution $y_1 = e^{-x}$:\n", + "\\begin{align}\n", + "y^{\\prime\\prime} + 2 y^{\\prime} + y &= 0 \\\\\n", + "\\text{Step 1:} \\quad v = \\frac{\\exp \\left( -\\int 2dx \\right)}{ \\left(e^{-x}\\right)^2} = \\frac{e^{-2x}}{e^{-2x}} &= 1 \\\\\n", + "\\text{Step 2:} \\quad u = \\int v dx = \\int 1 dx &= x \\\\\n", + "\\text{Step 3:} \\quad y_2 &= x e^{-x}\n", + "\\end{align}\n", + "Then, the general solution to the ODE is $y(x) = c_1 e^{-x} + c_2 x e^{-x}$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.11" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/docs/_sources/content/second-order/fourier-series.ipynb b/docs/_sources/content/second-order/fourier-series.ipynb new file mode 100644 index 0000000..bdde538 --- /dev/null +++ b/docs/_sources/content/second-order/fourier-series.ipynb @@ -0,0 +1,480 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Fourier Series\n", + "\n", + "Fourier series are a method we can use to solve inhomogeneous 2nd-order ODEs of the form\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + p(t) y^{\\prime} + q(t) y = r(t) \\;,\n", + "\\end{equation}\n", + "where the forcing function $r(t)$ is periodic. This means looking like one of these examples:" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% periodic square wave\n", + "subplot(3,1,1);\n", + "squareWave = repmat([1,1,1,1,0,0,0,0], [1, 10]);\n", + "t = linspace(0, 1, length(squareWave));\n", + "plot(t, squareWave); ylim([-0.5, 1.5]);\n", + "\n", + "% sin wave\n", + "subplot(3,1,2);\n", + "t = linspace(0, 1, 100);\n", + "plot(t, sin(t*10*pi)); ylim([-1.5, 1.5]);\n", + "\n", + "% sawtooth\n", + "subplot(3,1,3);\n", + "t = linspace(0, 1, 100);\n", + "y = ((mod(t,2*pi/40)/(pi*2/40))*2)-1;\n", + "plot(t, y); ylim([-1.5, 1.5]);" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "(Actually, as we'll see later, we can use a Fourier series to represent generic forcing function!)\n", + "\n", + "Fourier series have been around a while, ever since in 1790 Jean-Baptiste Joseph Fourier found that generic periodic functions could be represented by a sum of series of `sin()` and `cos()` functions, harmonically related by frequency.\n", + "\n", + "In general, a Fourier series represents a function $f(t)$ by\n", + "\\begin{equation}\n", + "f(t) = a_0 + \\sum_{n=1}^{\\infty} a_n \\cos (n \\omega t) + \\sum_{n=1}^{\\infty} b_n \\sin (n \\omega t)\n", + "\\end{equation}\n", + "where $a_0$, $a_n$, and $b_n$ are the Fourier coefficients, $\\omega = \\frac{2\\pi}{T}$ is the frequency of the function $f(t)$, and $T$ is the period. $n$ is an integer used as an index." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Properties of Fourier Series\n", + "\n", + "Considering that $n$ is an integer, and the sine and cosine components of a Fourier series share the same fundamental frequency $\\omega$, Fourier series have some useful properties:\n", + "\n", + "1. The integral of each component trigonometric function over the period is zero:\n", + "\\begin{equation}\n", + "\\int_0^T \\sin (n \\omega t) dt = 0 = \\int_0^T cos (n \\omega t) dt\n", + "\\end{equation}\n", + "\n", + "2. The component trigonometric functions are **orthogonal** over their period:\n", + "\\begin{equation}\n", + "\\int_0^T \\cos(n \\omega t) \\sin (m \\omega t) dt = 0\n", + "\\end{equation}\n", + "for all values of $n, m = 1, 2, \\ldots, \\infty$.\n", + "\n", + "3. The component trigonometric functions are also orthogonal with themselves over their period:\n", + "\\begin{align}\n", + "\\int_0^T \\cos (n \\omega t) \\cos (m \\omega t) dt &= \\begin{cases}0 \\quad m \\neq n \\\\ \\frac{T}{2} \\quad m = n \\end{cases} \\\\\n", + "%\n", + "\\int_0^T \\sin (n \\omega t) \\sin (m \\omega t) dt &= \\begin{cases}0 \\quad m \\neq n \\\\ \\frac{T}{2} \\quad m = n \\end{cases}\n", + "\\end{align}\n", + "for all values of $n, m = 1, 2, \\ldots, \\infty$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Fourier coefficients\n", + "\n", + "We can use the above properties to calculate the Fourier coefficients, given a periodic function $f(t)$.\n", + "First, recall\n", + "\\begin{equation}\n", + "f(t) = a_0 + \\sum_{n=1}^{\\infty} a_n \\cos (n \\omega t) + \\sum_{n=1}^{\\infty} b_n \\sin (n \\omega t)\n", + "\\end{equation}\n", + "\n", + "1. To calculate $a_0$, integrate both sides of the equation over the period:\n", + "\\begin{equation}\n", + "\\int_0^T f(t) dt = \\int_0^T a_0 dt + \\int_0^T \\left( \\sum_{n=1}^{\\infty} a_n \\cos (n \\omega t) \\right)dt + \\int_0^T \\left( \\sum_{n=1}^{\\infty} b_n \\sin (n \\omega t) \\right) dt\n", + "\\end{equation}\n", + "For the integrals of the infinite sums, recall that the integral of the sum of some functions is the same as the sum of the integrals of the functions: $\\int (a + b + c) = \\int a + \\int b + \\int c$. That means that \n", + "\\begin{align}\n", + "\\int_0^T \\left( \\sum_{n=1}^{\\infty} a_n \\cos (n \\omega t) \\right)dt = \\int_0^T a_1 \\cos (\\omega t) dt + \\int_0^T a_2 \\cos (2 \\omega t) dt + \\ldots &= 0 \\;, \\text{ and} \\\\\n", + "\\int_0^T \\left( \\sum_{n=1}^{\\infty} b_n \\sin (n \\omega t) \\right)dt = \\int_0^T b_1 \\sin (\\omega t) dt + \\int_0^T b_2 \\sin (2 \\omega t) dt + \\ldots &= 0 \\;,\n", + "\\end{align}\n", + "since the integrals of the trigonometric functions are all zero over the period. Thus,\n", + "\\begin{equation}\n", + "a_0 = \\frac{1}{T} \\int_0^T f(t) dt\n", + "\\end{equation}\n", + "\n", + "2. To calculate $a_n$, multiply both sides of the equation by $\\cos(m \\omega t)$ and integrate over the period:\n", + "\\begin{equation}\n", + "\\int_0^T f(t) \\cos(m \\omega t) dt = a_0 \\int_0^T \\cos(m \\omega t) dt + \\int_0^T \\left( \\sum_{n=1}^{\\infty} a_n \\cos (n \\omega t) \\cos(m \\omega t) \\right)dt + \\int_0^T \\left( \\sum_{n=1}^{\\infty} b_n \\sin (n \\omega t) \\cos(m \\omega t) \\right) dt\n", + "\\end{equation}\n", + "Let's take a look at each of the three integrals on the right-hand side. First,\n", + "\\begin{equation}\n", + "a_0 \\int_0^T \\cos(m \\omega t) dt = 0\n", + "\\end{equation}\n", + "because it just integrates cosine over the period.\n", + "Skipping to the last term,\n", + "\\begin{equation}\n", + "\\int_0^T \\left( \\sum_{n=1}^{\\infty} b_n \\sin (n \\omega t) \\cos(m \\omega t) \\right) dt = b_1 \\int_0^T sin(\\omega t) \\cos (m \\omega t) dt + b_2 \\int_0^T \\sin (2 \\omega t) \\cos (m \\omega t) dt + \\ldots = 0\n", + "\\end{equation}\n", + "due to orthogonality. We are just left with the middle integral; let's expand a few terms to see what that looks like:\n", + "\\begin{equation}\n", + "\\int_0^T \\left( \\sum_{n=1}^{\\infty} a_n \\cos (n \\omega t) \\cos(m \\omega t) \\right)dt = a_1 \\int_0^T \\cos (\\omega t) \\cos (m \\omega t) dt + a_2 \\int_0^T \\cos (2 \\omega t) \\cos (m \\omega t) dt + \\ldots\n", + "\\end{equation}\n", + "Again, due to orthogonality, all of the terms of this infinite sum of integrals will be zero, *except* for the term where $n = m$. As a result, we are left with\n", + "\\begin{align}\n", + "\\int_0^T f(t) \\cos(m \\omega t) dt &= \\int_0^T a_m \\cos^2 (m \\omega t) dt = a_m \\frac{T}{2} \\\\\n", + "a_n = a_m &= \\frac{2}{T} \\int_0^T f(t) \\cos (n \\omega t) dt\n", + "\\end{align}\n", + "\n", + "3. We can find $b_n$ in the same way, multiplying the equation by $\\sin (m \\omega t)$ and integrating over the period. The work is the same, so let's skip that:\n", + "\\begin{equation}\n", + "b_n = \\frac{2}{T} \\int_0^T f(t) \\sin (n \\omega t) dt\n", + "\\end{equation}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: periodic rectangular wave\n", + "\n", + "Let's find the Fourier series for representing this periodic function $f(t)$:" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x = [0 1 1 2 2 3 3 4]; y = [2 2 1 1 2 2 1 1];\n", + "plot(x,y); ylim([0 2.5]); ylabel('f(t)'); xlabel('t')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "First, we need to identify the fundamental period and frequency: $T = 2$ and then $\\omega = \\frac{2\\pi}{T} = \\pi$. Our work is then to calculate the Fourier coefficients. Since our periodic function $f(t)$ is not easily expressed with a function–hence the need for a Fourier series—we'll use piecewise integration.\n", + "\n", + "First, calculate $a_0$:\n", + "\\begin{align}\n", + "a_0 =& \\frac{1}{T} \\int_0^T f(t) dt \\\\\n", + "&= \\frac{1}{2} \\int_0^2 f(t) dt = \\frac{1}{2}\\left( \\int_0^1 2dt + \\int_1^2 1dt \\right) = \\frac{1}{2} (2\\times 1 + 1 \\times 1) \\\\\n", + "a_0 &= \\frac{3}{2}\n", + "\\end{align}\n", + "\n", + "Then, get $a_n$:\n", + "\\begin{align}\n", + "a_n &= \\frac{2}{T} \\int_0^T f(t) \\cos (n \\omega t) dt \\\\\n", + "&= \\frac{2}{2} \\int_0^2 f(t) \\cos (n \\pi t) dt = \\left( \\int_0^1 2 \\cos (n \\pi t) dt + \\int_1^2 1 \\cos (n \\pi t)dt \\right) \\\\\n", + "&= \\frac{2}{n \\pi} \\sin(n \\pi t)\\Big|_0^1 + \\frac{1}{n\\pi} \\sin(n \\pi t)\\Big|_1^2 \\\\\n", + "&= \\frac{2}{n \\pi}\\left(\\sin(n\\pi) - \\sin(0)\\right) + \\frac{1}{n\\pi}\\left( sin(2n\\pi) - \\sin(n\\pi)\\right) \\\\\n", + "a_n &= 0\n", + "\\end{align}\n", + "\n", + "Finally, we can calculate $b_n$:\n", + "\\begin{align}\n", + "b_n &= \\frac{2}{T} \\int_0^T f(t) \\sin (n \\omega t) dt \\\\\n", + "&= \\frac{2}{2} \\int_0^2 f(t) \\sin (n \\pi t) dt = \\left( \\int_0^1 2 \\sin (n \\pi t) dt + \\int_1^2 1 \\sin (n \\pi t)dt \\right) \\\\\n", + "&= -\\frac{2}{n \\pi} \\cos(n \\pi t)\\Big|_0^1 - \\frac{1}{n\\pi} \\cos(n \\pi t)\\Big|_1^2 \\\\\n", + "&= -\\frac{2}{n \\pi}\\left(\\cos(n\\pi) - \\cos(0)\\right) - \\frac{1}{n\\pi}\\left( cos(2n\\pi) - \\cos(n\\pi)\\right) \\\\\n", + "b_n &= -\\frac{2}{n \\pi}\\left(\\cos(n\\pi) - 1\\right) - \\frac{1}{n\\pi}\\left( 1 - \\cos(n\\pi)\\right) = -\\frac{1}{n\\pi}\\left( \\cos(n\\pi) - 1\\right)\n", + "\\end{align}\n", + "but recall that $n = 1, 2, \\ldots, \\infty$. Then,\n", + "\\begin{align}\n", + "b_n &= -\\frac{1}{n\\pi} \\times \\begin{cases} -2 \\text{ if } n \\text{ odd} \\\\0 \\text{ if } n \\text{ even}\\end{cases} \\\\\n", + "\\rightarrow b_n &= \\frac{2}{n\\pi} \\quad n = \\text{odd}\n", + "\\end{align}\n", + "\n", + "Then, our Fourier series representation for the function shown above is\n", + "\\begin{equation}\n", + "f(t) = \\frac{3}{2} + \\sum_{\\substack{n = 1 \\\\n = \\text{odd}}}^{\\infty} \\frac{2}{n\\pi} \\sin (n \\pi t)\n", + "\\end{equation}\n", + "\n", + "Now, let's see how whether this actually works! Let's start with one term of the infinite sum, then gradually increase." + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "t = linspace(0, 4);\n", + "\n", + "% maximum number of terms\n", + "n_max = [1, 2, 3, 5, 10, 25, 50, 250, 500];\n", + "\n", + "for i = 1 : length(n_max)\n", + " N = n_max(i);\n", + "\n", + " s = 3./2.;\n", + " for n = 1 : 2 : 2*N\n", + " s = s + (2. / (n*pi)) .* sin(n * pi * t);\n", + " end\n", + " subplot(3, 3, i)\n", + " plot(t, s); axis([0 4 0 3]); title(sprintf('%d terms', N));\n", + "end" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As we increase the number of terms, adding higher-frequency sine waves, we are better able to match the original rectangular wave. Notice the discrepancies that remain near the sharp corners even after the rest of the series closely resembles the function: these are known as **Gibbs phenomena**, caused by the Fourier series overshooting or undershooting (or \"ringing\") near discontinuities." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Even and Odd Functions\n", + "\n", + "We can simplify our work generating a Fourier series if we can identify the given periodic function $f(t)$ as an **even function** or an **odd function**.\n", + "\n", + "Even functions are those where $f(-x) = f(x)$.\n", + "\n", + "Odd functions are those where $f(-x) = -f(x)$." + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "y = [-1 0 1 0 -1];\n", + "x = [-2 -1 0 1 2];\n", + "subplot(2,1,1); plot(x,y); title('Even function');\n", + "ax = gca; ax.XAxisLocation = 'origin'; ax.YAxisLocation = 'origin';\n", + "\n", + "y = [0 -1 0 1 0];\n", + "subplot(2,1,2); plot(x,y); title('Odd function');\n", + "ax = gca; ax.XAxisLocation = 'origin'; ax.YAxisLocation = 'origin';" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "An even function's Fourier series simplifies to a Fourier cosine series:\n", + "\\begin{align}\n", + "f(x) &= a_0 + \\sum_{n=1}^{\\infty} a_n \\cos (n \\omega x) dx \\\\\n", + "a_0 &= \\frac{2}{T} \\int_0^{T/2} f(x) dx \\\\\n", + "a_n &= \\frac{4}{T} \\int_0^{T/2} f(x) \\cos(n \\omega x) dx\n", + "\\end{align}\n", + "\n", + "An odd function's Fourier series simplifies to a Fourier sine series:\n", + "\\begin{align}\n", + "f(x) &= \\sum_{n=1}^{\\infty} b_n \\sin (n \\omega x) dx \\\\\n", + "b_n &= \\frac{4}{T} \\int_0^{T/2} f(x) \\sin(n \\omega x) dx\n", + "\\end{align}\n", + "\n", + "Note: not all periodic functions can be considered an even or an odd function." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Application: Inhomogeneous 2nd-order ODE\n", + "\n", + "One way we might use a Fourier series is to solve an inhomogeneous 2nd-order ODE, where the forcing term is given by a periodic function not easily expressed using our typical functions.\n", + "\n", + "### Undamped mass-spring system\n", + "\n", + "For example, let's consider an undamped mass-spring system, where the forcing is given by a periodic rectangular wave:\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + 4y = f(t)\n", + "\\end{equation}\n", + "where the forcing function $f(t)$ is" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "t = [0 0 1 1 2 2 3 3];\n", + "y = [0 1 1 -1 -1 1 1 0];\n", + "plot(t,y); ylim([-1.5 1.5]);\n", + "title('Periodic rectangular forcing function');\n", + "ax = gca; ax.XAxisLocation = 'origin';" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Using recognizing this as an odd function, we could construct a Fourier sine series to represent the forcing function:\n", + "\\begin{equation}\n", + "f(t) = \\sum_{\\substack{n=1\\\\n=\\text{odd}}}^{\\infty} \\frac{4}{n\\pi} \\sin(n \\pi t)\n", + "\\end{equation}\n", + "\n", + "Let's confirm this works:" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "t = linspace(0, 3, 500);\n", + "s = 0;\n", + "for n = 1 : 2 : 1000\n", + " s = s + (4/(n*pi)).*sin(n*pi*t);\n", + "end\n", + "plot(t, s)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Looks good!\n", + "\n", + "To find the exact solution for our displacement $y(t)$, we can follow our usual analytical solution approach: find the homogeneous solution $y_H$, then find the inhomogeneous solution $y_{IH}$; the overall solution is then $y(t) = y_H + y_{IH}$. The homogeneous solution is\n", + "\\begin{equation}\n", + "y_H = c_1 \\sin (2t) + c_2 \\cos (2t)\n", + "\\end{equation}\n", + "We then find the inhomogeneous solution using\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + 4y = \\frac{4}{n\\pi} \\sin (n \\pi t) \\quad n = 1, 3, \\ldots, \\infty\n", + "\\end{equation}\n", + "Solving this will give us a specific $y_{IH, n}$; the complete inhomogeneous solution is then\n", + "\\begin{equation}\n", + "y_{IH} = \\sum_{\\substack{n=1\\\\n=\\text{odd}}}^{\\infty} y_{IH, n} \\;.\n", + "\\end{equation}\n", + "\n", + "Recognizing that our forcing function is sinusoidal, we should use the method of undetermined coefficients:\n", + "\\begin{equation}\n", + "y_{IH, n} = K_1 \\sin (n \\pi t) + K_2 \\cos (n \\pi t)\n", + "\\end{equation}\n", + "Inserting this into the above ODE gives\n", + "\\begin{align}\n", + "K_1 &= \\frac{4}{n \\pi (4 - n^2 \\pi^2)} \\\\\n", + "K_2 &= 0\n", + "\\end{align}\n", + "\n", + "Thus, the overall solution is\n", + "\\begin{equation}\n", + "y(t) = c_1 \\sin(2t) + c_2 \\cos(2t) + \\sum_{\\substack{n=1\\\\n=\\text{odd}}}^{\\infty} \\frac{4}{n \\pi (4 - n^2 \\pi^2)} \\sin (n \\pi t)\n", + "\\end{equation}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Damped mass-spring system\n", + "\n", + "What about a damped mass-spring system? Recall that the homogeneous solution could take one of these three forms:\n", + "\\begin{align}\n", + "y_H &= c_1 e^{-\\lambda_1 t} + c_2 e^{-\\lambda_2 t} \\\\\n", + "y_H &= c_1 e^{-\\lambda_1 t} + c_2 t e^{-\\lambda_2 t} \\text{ or} \\\\\n", + "y_H &= e^{-\\alpha t} (c_1 \\sin(\\beta t) + c_2 \\cos(\\beta t))\n", + "\\end{align}\n", + "while the inhomogeneous solution, given a Fourier series forcing function, will take the form\n", + "\\begin{equation}\n", + "y_{IH} = K_1 \\sin() + K_2 \\cos()\n", + "\\end{equation}\n", + "\n", + "The overall solution combines the homogenenous and inhomogeneous solutions. But, the homogeneous solution in this case is **transient**, because it decays to zero. On the other hand, the inhomogeneous solution remains, and is the **steady-state solution**." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.7" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/docs/_sources/content/second-order/initial-value-problems.ipynb b/docs/_sources/content/second-order/initial-value-problems.ipynb new file mode 100644 index 0000000..5c3e7de --- /dev/null +++ b/docs/_sources/content/second-order/initial-value-problems.ipynb @@ -0,0 +1,308 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Initial-Value Problems\n", + "\n", + "This section focuses on analytical solutions for initial-value problems, meaning problems where we know the values of $y(t)$ and $\\frac{dy}{dt}$ at $t=0$ (or $y(x)$ and $\\frac{dy}{dx}$ at $x=0$): $y(0)$ and $y^{\\prime}(0)$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Equations with constant coefficents\n", + "\n", + "A common category of 2nd-order homogeneous ODEs are equations with constant coefficients, of the form:\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + a y^{\\prime} + by = 0\n", + "\\end{equation}\n", + "Note that these are unforced, and the right-hand side is zero.\n", + "\n", + "Solutions to these equations take the form $y(x) = e^{\\lambda x}$, and inserting this into the ODE gives us the characteristic equation\n", + "\\begin{equation}\n", + "\\lambda^2 + a \\lambda + b = 0\n", + "\\end{equation}\n", + "which we can solve to find the solution for given coefficients $a$ and $b$ and initial conditions. Depending on those coefficients and the solution to the characteristic equation, our solution can fall into one of three cases:\n", + "\n", + "* Real roots: $\\lambda_1$ and $\\lambda_2$. This is an **overdamped** system and the full solution takes the form\n", + "\\begin{equation}\n", + "y(x) = c_1 e^{\\lambda_1 x} + c_2 e^{\\lambda_2 x}\n", + "\\end{equation}\n", + "\n", + "* Repeated roots: $\\lambda_1 = \\lambda_2 = \\lambda$. This is a **critically damped** system and the full solution is\n", + "\\begin{equation}\n", + "y(x) = c_1 e^{\\lambda x} + c_2 x e^{\\lambda x}\n", + "\\end{equation}\n", + "(Where does that second part come from, you might ask? Well, we know that $y_1$ is $e^{\\lambda x}$, but the second part cannot also be $e^{\\lambda x}$ because those are linearly dependent. So, we use *reduction of order* to find $y_2$, which is $x e^{\\lambda x}$.\n", + "\n", + "* Imaginary roots: $\\lambda = \\frac{-a}{2} \\pm \\beta i$, where $\\beta = \\frac{1}{2} \\sqrt{4b - a^2}$. This is an **underdamped** system and the solution takes the form\n", + "\\begin{equation}\n", + "y(x) = e^{-ax/2} \\left( c_1 \\sin \\beta x + c_2 \\cos \\beta x \\right)\n", + "\\end{equation}\n", + "\n", + "Some examples:\n", + "\n", + "1. $y^{\\prime\\prime} + 3 y^{\\prime} + 2y = 0$\n", + "\\begin{align}\n", + "\\rightarrow \\lambda^2 + 3\\lambda + 2 &= 0 \\\\\n", + "(\\lambda + 2)(\\lambda + 1) &= 0 \\\\\n", + "\\lambda &= -2, -1 \\\\\n", + "y(x) &= c_1 e^{-x} + c_2 e^{-2x}\n", + "\\end{align}\n", + "Then, we would use the initial conditions given for $y(0)$ and $y^{\\prime}(0)$ to find $c_1$ and $c_2$.\n", + "\n", + "2. $y^{\\prime\\prime} + 6 y^{\\prime} + 9y = 0$\n", + "\\begin{align}\n", + "\\rightarrow \\lambda^2 + 6\\lambda + 9 &= 0 \\\\\n", + "(\\lambda + 3)(\\lambda + 3) &= 0 \\\\\n", + "\\lambda &= -3 \\\\\n", + "y(x) &= c_1 e^{-3x} + c_2 x e^{-3x}\n", + "\\end{align}\n", + "\n", + "3. $y^{\\prime\\prime} + 6 y^{\\prime} + 25 y = 0$\n", + "\\begin{align}\n", + "\\rightarrow \\lambda^2 + 6\\lambda + 25 &= 0 \\\\\n", + "\\lambda &= -3 \\pm 4i \\\\\n", + "y(x) &= e^{-3x} \\left( c_1 \\sin 4x + c_2 \\cos 4x \\right)\n", + "\\end{align}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Euler-Cauchy equations\n", + "\n", + "Euler-Cauchy equations are of the form\n", + "\\begin{equation}\n", + "x^2 y^{\\prime\\prime} + axy^{\\prime} + by = 0\n", + "\\end{equation}\n", + "\n", + "Solutions take the form $y = x^m$, which when plugged into the ODE leads to a different characterisic equation to find $m$:\n", + "\\begin{align}\n", + "y &= x^m \\\\\n", + "y^{\\prime} &= m x^{m-1} \\\\\n", + "y^{\\prime\\prime} &= m (m-1) x^{m-2} \\\\\n", + "\\rightarrow x^2 m (m-1) x^{m-2} + axmx^{m-1} + bx^m &= 0 \\\\\n", + "m^2 + (a-1)m + b &= 0\n", + "\\end{align}\n", + "This is our new characteristic formula for these problems, and solving for the roots of this equation gives us $m$ and thus our general solution.\n", + "\n", + "Like equations with constant coefficients, we have three solution forms depending on the roots of the characteristic equation:\n", + "\n", + "* Real roots: $y(x) = c_1 x^{m_1} + c_2 x^{m_2}$\n", + "* Repeated roots: $y(x) = c_1 x^m + c_2 x^m \\ln x$\n", + "* Imaginary roots: $m = \\alpha \\pm \\beta i$, and $y(x) = x^{\\alpha} \\left[c_1 \\cos (\\beta \\ln x) + c_2 \\sin (\\beta \\ln x)\\right]$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Inhomogeneous 2nd-order ODEs\n", + "\n", + "Inhomogeneous, or forced, 2nd-order ODEs with constant coefficients take the form\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + a y^{\\prime} + by = F(t)\n", + "\\end{equation}\n", + "with initial conditions $y(0) = y_0$ and $y^{\\prime}(0) = y_0^{\\prime}$. Depending on the form of the forcing function $F(t)$, we can solve with techniques such as\n", + "\n", + " - the method of undetermined coefficients\n", + " - variation of parameters\n", + " - LaPlace transforms\n", + "\n", + "The solution in general to inhomogeneous ODEs includes two parts:\n", + "\\begin{equation}\n", + "y(t) = y_{\\text{H}} + y_{\\text{IH}} = c_1 y_1 + c_2 y_2 + y_{\\text{IH}} \\;,\n", + "\\end{equation}\n", + "where $y_{\\text{H}}$ is the solution from the equivalent homogeneous ODE $y^{\\prime\\prime} + a y^{\\prime} + b y = 0$.\n", + "\n", + "The forcing function $F(t)$ may be \n", + "\n", + " - continuous\n", + " - periodic\n", + " - aperiodic/discontinuous\n", + " \n", + "### Continuous $F(t)$: method of undetermined coefficients\n", + "\n", + "For continuous forcing functions, we have two solution methods: the method of undetermined coefficients, and variation of parameters. \n", + "\n", + "Generally you'll want to use the method of undetermined coefficients when possible, which depends on if $F(t)$ matches one of a set of functions. In that case, the form of the inhomogeneous solution $y_{\\text{IH}}(t)$ follows that of the forcing function $F(t)$, with one or more unknown constants:\n", + "\n", + "| $F(t)$ | $y_{\\text{IH}}(t)$ |\n", + "| ------------- | :-------:|\n", + "| constant | $K$ |\n", + "| $\\cos \\omega t$ | $K_1 \\cos \\omega t + K_2 \\sin \\omega t$ |\n", + "| $\\sin \\omega t$ | $K_1 \\cos \\omega t + K_2 \\sin \\omega t$ |\n", + "| $e^{-at}$ | $K e^{-at}$ |\n", + "| $(A) t$ | $K_0 + K_1 t$ |\n", + "| $t^n$ | $K_0 + K_1 t + K_2 t^2 + \\ldots + K_n t^n$|\n", + "\n", + "For combinations of these functions, we can combine functions; for example, given\n", + "\\begin{align}\n", + "F(t) &= e^{-at} \\cos \\omega t \\quad \\text{or} e^{-at} \\sin \\omega t \\\\\n", + "y_{\\text{IH}} &= K_1 e^{-at} \\cos \\omega t + K_2 e^{-at} \\sin \\omega t\n", + "\\end{align}\n", + "(Note how in all the above cases how the inhomogeneous solution follows the functional form of the forcing function; for example, the exponential decay rate $a$ or the sinusoidal frequency $\\omega$ match.\n", + "\n", + "The method of undetermined coefficients works by plugging the candidate inhomogeneous solutionn $y_{\\text{IH}}$ into the full ODE, and solving for the constants (e.g., $K$)—but **not** from the initial conditions.\n", + "\n", + "For example, let's solve\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + 2y^{\\prime} + y = e^{-x}\n", + "\\end{equation}\n", + "with initial conditions $y(0) = y^{\\prime}(0) = 0$. First, we should find the solution to the homogeneous equation\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + 2y^{\\prime} + y = 0 \\;.\n", + "\\end{equation}\n", + "We can do this by using the associated characteristic formula\n", + "\\begin{align}\n", + "\\lambda^2 + 2 \\lambda + 1 &= 0 \\\\\n", + "(\\lambda + 1)(\\lambda + 1) &= 0 \\\\\n", + "\\rightarrow y_{\\text{H}} &= c_1 e^{-x} + c_2 x e^{-x}\n", + "\\end{align}\n", + "\n", + "To find the inhomogeneous solution, we would look at the table above to find what matches the forcing function $e^{-x}$. Normally, we'd grab $K e^{-x}$, but that would not be linearly independent from the first part of the homogeneous solution $y_{\\text{H}}$. The same is true for $K x e^{-x}$, which is linearly dependent with the second part of $y_{\\text{H}}$, but $K x^2 e^{-x}$ works! Then, we just need to find $K$ by plugging this into the ODE:\n", + "\\begin{align}\n", + "y_{\\text{IH}} &= K x^2 e^{-x} \\\\\n", + "y^{\\prime} &= K e^{-x} (2x - x^2) \\\\\n", + "y^{\\prime\\prime} &= K e^{-x} (x^2 - 4x + 2) \\\\\n", + "2 K &= 1 \\\\\n", + "\\rightarrow K &= \\frac{1}{2} \\\\\n", + "y_{\\text{IH}} &= \\frac{1}{2} x^2 e^{-x}\n", + "\\end{align}\n", + "Thus, the overall general solution is \n", + "\\begin{equation}\n", + "y(x) = c_1 e^{-x} + c_2 x e^{-x} + \\frac{1}{2} x^2 e^{-x}\n", + "\\end{equation}\n", + "and we would solve for the integration constants $c_1$ and $c_2$ using the initial conditions.\n", + "\n", + "Important points to remember:\n", + "\n", + "- The constants of the inhomogeneous solution $y_{\\text{IH}}$ come from the ODE, **not** the initial conditions.\n", + "- Only solve for the integration constants $c_1$ and $c_2$ (part of the homogeneous solution) once you have the full general solution $y = c_1 y_1 + c_2 y_2 + y_{\\text{IH}}$.\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Continuous $F(t)$: variation of parameters\n", + "\n", + "We have the variation of parameters approach to solve for inhomogeneous 2nd-order ODEs that are more general:\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + p(x) y^{\\prime} + q(x) y = r(x)\n", + "\\end{equation}\n", + "In this case, we can assume a solution $y(x) = y_1 u_1 + y_2 u_2$.\n", + "\n", + "The solution procedure is:\n", + "\n", + "1. Obtain $y_1$ and $y_2$ by solving the homogeneous equation: $y^{\\prime\\prime} + p(x) y^{\\prime} + q(x) y = 0$\n", + "\n", + "2. Solve for $u_1$ and $u_2$:\n", + "\\begin{align}\n", + "u_1 &= - \\int \\frac{y_2 r(x)}{W} dx + c_1 \\\\\n", + "u_2 &= \\int \\frac{y_1 r(x)}{W} dx + c_2 \\\\\n", + "W &= \\begin{vmatrix}\n", + "y_1 & y_2\\\\ y_1^{\\prime} & y_2^{\\prime}\\\\\n", + "\\end{vmatrix} = y_1 y_2^{\\prime} - y_2 y_1^{\\prime} \\;,\n", + "\\end{align}\n", + "where $W$ is the Wronksian.\n", + "\n", + "3. Then, we have the general solution:\n", + "\\begin{align}\n", + "y &= u_1 y_1 + u_2 y_2 \\\\\n", + "&= \\left( -\\int \\frac{y_2 r(x)}{W} dx + c_1 \\right) y_1 + \\left( \\int \\frac{y_1 r(x)}{W} dx + c_2 \\right) y_2 \\;,\n", + "\\end{align}\n", + "where we solve for $c_1$ and $c_2$ using the two initial conditions.\n", + "\n", + "#### Example 1: variation of parameters\n", + "\n", + "First, let's try the same example we used for the method of undetermined coefficients above:\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + 2 y^{\\prime} + y = e^{-x}\n", + "\\end{equation}\n", + "We already found the homogeneous solution, so we know that $y_1 = e^{-x}$ and $y_2 = x e^{-x}$.\n", + "Next, let's get the Wronksian, and then $u_1$ and $u_2$.\n", + "\\begin{align}\n", + "W &= \\begin{vmatrix} y_1 & y_2 \\\\ y_1^{\\prime} & y_2^{\\prime} \\end{vmatrix} = e^{-x} e^{-x}(1-x) - x e^{-x} (-e^{-x}) = e^{-2x} \\\\\n", + "%\n", + "u_1 &= -\\int \\frac{x e^{-x} e^{-x}}{e^{-2x}} dx + c_1 = -\\int x dx + c_1 = -\\frac{1}{2} x^2 + c_1 \\\\\n", + "u_2 &= \\int \\frac{e^{-x} e^{-x}}{e^{-2x}} dx + c_2 = \\int dx + c_2 = x + c_2 \\\\\n", + "y(x) &= \\left(-\\frac{1}{2} x^2 + c_1\\right) e^{-x} + (x + c_2) x e^{-x} \\\\\n", + "\\end{align}\n", + "After simplifying, we obtain the same solution as via the method of undetermined coefficients (but with a bit more work):\n", + "\\begin{equation}\n", + "y(x) = x_1 e^{-x} + c_2 x e^{-x} + \\frac{1}{2} x^2 e^{-x}\n", + "\\end{equation}\n", + "\n", + "#### Example 2: variation of parameters\n", + "\n", + "Now let's try an example that we could *not* solve using the method of undetermined coefficients, with a forcing term that involves hyperbolic cosine (cosh); recall that $\\cosh(x) = \\frac{e^x + e^{-x}}{2}$.\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + 4 y^{\\prime} + 4y = \\cosh(x)\n", + "\\end{equation}\n", + "First, we need to find the homogeneous solution:\n", + "\\begin{align}\n", + "y^{\\prime\\prime} + 4 y^{\\prime} + 4y &= 0 \\\\\n", + "\\lambda^2 + 4 \\lambda + 4 &= 0 \\\\\n", + "\\rightarrow \\lambda &= -2\n", + "\\end{align}\n", + "So our homogeneous solution involves repeated roots:\n", + "\\begin{equation}\n", + "y_H = c_1 e^{-2x} + c_2 x e^{-2x}\n", + "\\end{equation}\n", + "where $y_1 = e^{-2x}$ and $y_2 = x e^{-2x}$.\n", + "\n", + "Then, we need to find $u_1$ and $u_2$, so let's get the Wronksian and then solve\n", + "\\begin{align}\n", + "W &= \\begin{vmatrix} y_1 & y_2 \\\\ y_1^{\\prime} & y_2^{\\prime} \\end{vmatrix} = e^{-2x} (e^{-2x}) (1 - 2x) - x e^{-2x}(-2 e^{-2x}) = e^{-4x} \\\\\n", + "%\n", + "u_1 &= - \\int \\frac{x e^{-2x} \\cosh x}{e^{-4x}} dx + c_1 = -\\int \\frac{x \\frac{1}{2}(e^x + e^{-x})}{e^{-2x}} dx + c_1 \\\\\n", + " &= -\\frac{1}{2} \\int x (e^{3x} + e^x) dx + c_1 = -\\frac{1}{2} \\left[ \\frac{1}{9} e^{3x}(3x-1) + e^x(x-1) \\right] + c_1 \\\\\n", + "u_1 &= -\\frac{1}{18} e^{3x}(3x-1) - \\frac{1}{2} e^x (x-1) + c_1 \\\\\n", + "%\n", + "u_2 &= \\int \\frac{e^{-2x} \\cosh x}{e^{-4x}} dx + c_2 = \\frac{1}{2} \\int e^{2x}(e^x + e^{-x}) dx + c_2 = \\frac{1}{2} \\int (e^{3x} + e^x) dx + c_2 \\\\ \n", + "u_2 &= \\frac{1}{6} e^{3x} + \\frac{1}{2} e^x + c_2\n", + "\\end{align}\n", + "\n", + "Then, when we put these all together, we get the full (complicated) solution:\n", + "\\begin{equation}\n", + "y(x) = \\left[ -\\frac{1}{18} e^{3x} (3x-1) - \\frac{1}{2} e^x (x-1) + c_1 \\right] e^{-2x} + \\left( \\frac{1}{6} e^{3x} + \\frac{1}{2} e^x + c_2 \\right) x e^{-2x}\n", + "\\end{equation}" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.11" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/docs/_sources/content/second-order/numerical-methods.ipynb b/docs/_sources/content/second-order/numerical-methods.ipynb new file mode 100644 index 0000000..7c18b38 --- /dev/null +++ b/docs/_sources/content/second-order/numerical-methods.ipynb @@ -0,0 +1,545 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Numerical methods for 2nd-order ODEs\n", + "\n", + "We've gone over how to solve 1st-order ODEs using numerical methods, but what about 2nd-order or any higher-order ODEs? We can use the same methods we've already discussed by transforming our higher-order ODEs into a **system of first-order ODEs**.\n", + "\n", + "Meaning, if we are given a 2nd-order ODE\n", + "\\begin{equation}\n", + "\\frac{d^2 y}{dx^2} = y^{\\prime\\prime} = f(x, y, y^{\\prime})\n", + "\\end{equation}\n", + "we can transform this into a system of **two 1st-order ODEs** that are coupled:\n", + "\\begin{align}\n", + "\\frac{dy}{dx} &= y^{\\prime} = u \\\\\n", + "\\frac{du}{dx} &= u^{\\prime} = y^{\\prime\\prime} = f(x, y, u)\n", + "\\end{align}\n", + "where $f(x, y, u)$ is the same as that given above for $\\frac{d^2 y}{dx^2}$.\n", + "\n", + "Thus, instead of a 2nd-order ODE to solve, we have two 1st-order ODEs:\n", + "\\begin{align}\n", + "y^{\\prime} &= u \\\\\n", + "u^{\\prime} &= f(x, y, u)\n", + "\\end{align}\n", + "\n", + "So, we can use all of the methods we have talked about so far to solve 2nd-order ODEs by transforming the one equation into a system of two 1st-order equations.\n", + "\n", + "## Higher-order ODEs\n", + "\n", + "This works for higher-order ODEs too! For example, if we have a 3rd-order ODE, we can transform it into a system of three 1st-order ODEs:\n", + "\\begin{align}\n", + "\\frac{d^3 y}{dx^3} &= f(x, y, y^{\\prime}, y^{\\prime\\prime}) \\\\\n", + "\\rightarrow y^{\\prime} &= u \\\\\n", + "u^{\\prime} &= y^{\\prime\\prime} = w \\\\\n", + "w^{\\prime} &= y^{\\prime\\prime\\prime} = f(x, y, u, w)\n", + "\\end{align}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: mass-spring problem\n", + "\n", + "For example, let's solve a forced damped mass-spring problem given by a 2nd-order ODE:\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + 5y^{\\prime} + 6y = 10 \\sin \\omega t\n", + "\\end{equation}\n", + "with the initial conditions $y(0) = 0$ and $y^{\\prime}(0) = 5$.\n", + "\n", + "We start by transforming the equation into two 1st-order ODEs. Let's use the variables $z_1 = y$ and $z_2 = y^{\\prime}$:\n", + "\\begin{align}\n", + "\\frac{dz_1}{dt} &= z_1^{\\prime} = z_2 \\\\\n", + "\\frac{dz_2}{dt} &= z_2^{\\prime} = y^{\\prime\\prime} = 10 \\sin \\omega t - 5z_2 - 6z_1\n", + "\\end{align}\n", + "\n", + "### Forward Euler\n", + "\n", + "Then, let's solve numerically using the forward Euler method. Recall that the recursion formula for forward Euler is:\n", + "\\begin{equation}\n", + "y_{i+1} = y_i + \\Delta x f(x_i, y_i)\n", + "\\end{equation}\n", + "where $f(x,y) = \\frac{dy}{dx}$.\n", + "\n", + "Let's solve using $\\omega = 1$ and with a step size of $\\Delta t = 0.1$, over $0 \\leq t \\leq 3$.\n", + "\n", + "We can compare this against the exact solution, obtainable using the method of undetermined coefficients:\n", + "\\begin{equation}\n", + "y(t) = -6 e^{-3t} + 7 e^{-2t} + \\sin t - \\cos t\n", + "\\end{equation}" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% plot exact solution first\n", + "t = linspace(0, 3);\n", + "y_exact = -6*exp(-3*t) + 7*exp(-2*t) + sin(t) - cos(t);\n", + "plot(t, y_exact); hold on\n", + "\n", + "omega = 1;\n", + "\n", + "dt = 0.1;\n", + "t = [0 : dt : 3];\n", + "\n", + "f = @(t,z1,z2) 10*sin(omega*t) - 5*z2 - 6*z1;\n", + "\n", + "z1 = zeros(length(t), 1);\n", + "z2 = zeros(length(t), 1);\n", + "z1(1) = 0;\n", + "z2(1) = 5;\n", + "for i = 1 : length(t)-1\n", + " z1(i+1) = z1(i) + dt * z2(i);\n", + " z2(i+1) = z2(i) + dt * f(t(i), z1(i), z2(i));\n", + "end\n", + "\n", + "plot(t, z1, 'o--')\n", + "xlabel('time'); ylabel('displacement')\n", + "legend('Exact', 'Forward Euler', 'Location','southeast')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Heun's Method\n", + "\n", + "For schemes that involve more than one stage, like Heun's method, we'll need to implement both stages for each 1st-order ODE. For example:" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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1TA3BkqlAg6QwSVB6SdaZasMKDhyCQAIwMWrq6xw/N8KL1VyAc6gtzwO9HDn3MAlddgCmZJj6ysV/nwI/eDrZJIVx8mESAgnAZOQKjSTlJKa+AuMCvRy4+DAJgQRgGnKFJib7fEtqMNII2GCOnzvntqhAIAGYQNaZ6pjs85lRA5kuBOABTycb2YLBcoVGrtAwXcuTQiABdNaDhRgSR6JtBGyTFCaJyT7PdBVPCqPsAJ5Cq/VSCSEf3J9oWIgBgG0MD5M4sd0JAgngKVBLLBsCafamQ3NPRgVllzJbFUA75vi5x2RXZJ2pZrqQx0MgATwp4/VSCSFB6SWeruIBrwRhvVRgM08nm38PuJz3xQdWXZ+pld6lDrLzbyyeIQE8KcOuaCq1Pii9JNDLITNqAA92RQPeq5Wm/j4u7m/dZlObodgHRipj/Zku6iEQSABPilov1bAQA7XxK9ZLBZajWvax0eFz/NypUeCs3e8cgQTwpKhd0V7/eLtswWBqIQaslwrsZ2jZv9TwX+pfUYSt+53jGRLAk/rpd7ug7mv2uciwXipwiPF+5wbsbNkjkACeCLUQg2zBkECvYKZrAXgKVO+cYb9z8kfLntpAklUQSACPR0195fRenCBYhv3OvVSnqVF2rG3ZI5AAHiM5XylXaDD1FTjNOTJBcTNtQmQ804W0B4EE0J6Y7ArjPcgBwHwwyg7gkYLSSwghSCMAeiCQAB7CeOor07UACAW67ABaL5mqUus/uD8RDSMAmiGQAP6yZKpcoZm96dB/Gt8l5BTTdQEIC7rsQOiMl0ylJht9/V4oOxdWAeA3BBIInWFhFcOur4FeDuxcWAWA39BlB0JHLayy1ftvxpON2LmwCgC/oYUEQifyCag4IWsoLzSMYsCSqQCMQAsJhC50X43na9kbLA9iyVQAZiGQQLhUan1QRnH0MPekMAkhmG8EwDAEEgiUXKEJSi+WLRgS6OXAdC0AQAieIYEwUQPqkEYArIIWEghOcr4yq6gaq3cDsA0CCYSFWi8VaQTAQggk4KdWy9MRQn4fFxeTXeHpZIP1UgHYCYEE/GS8PB0h5EhhmdUC/zkJP0b7uTNbGAA8CgY1AA8ZL09HCJErNO8U/C7yCZii3MtsYQDQDgQS8JBheTpCSHK+Mib7vDJx5ND5iVieDoDNEEjAQ9TydNQme4YV6rA8HQDLIZCAh0Q+AbrywmUb9wR6OVAr1GF5OgD2w6AG4KGffrcL6r5mX3dZ4O2cWikhWJ4OgAsQSMA31LxX2YIhgV7BTNcCAE+B24F0/vz5o0ePLly4kOlCgBVUan1MdgXBvFcAbuLwM6SampqVK1fm5+czXQiwglyhkaScNDw0AgDO4WoLKTw8/MKFC83Nzd7e3kzXAsyjuulaUtFHB8BhXA2kVatWNTY25uXlFRUVMV0LMAnddAC8wdVA8vHxIYScPXsWgSQorVaoU6n1EbVBf+ywBwDcxtVAehI5OTk5OTnU1wUFBcwWAyZhvEJdcr7ycEVZwW+LJYmnmK4LgL2Cgx90ZVdVVcXHxzNbTPv4HEjh4eEs/+nDUzGsUGfopvt6YajN3oJaaSpmvAI8iuGf42lpacxW8lgcHmUHQkOtUGc8ms7TycY5MgEr1AHwA59bSMAzzpEJZ7emxFi+Zbz1OFaoA+ANzreQLCwsmC4B6KBS6ycet9GVF55/zcKQRlihDoBPuN1Cmjdv3rx585iuAsyOmmaUNNEvKKm0VpraUH6SOo4V6gD4hNuBBLzXdpoR2kMAfIVAAvZ60DAKlWDfcQAhQCABSwWllxCsvwAgJAgkYFirxRcIIWk9wrOKqjOjBhoGLwCAECCQgGHGiy/IFZrlXx9fVfFaUnYp03UBAN04P+wbOM148YXkfGVM9vnVs8cMeCWoVprKdGkAQDcEEjCJWnwhOV8pSTmZFCZRJo4M9HLA4gsAwkRHIBUXF587d874iFar3b9/f2NjIw1XBzb7PSTus6VLVHV6462MsPgCgDDR8Qxp586d9vb2L774ouGISqVKSEgoLCx0cnKioQBgIZVan3xIefmiw26R0s1XYzhOLb7QJ3kfg7UBACPMG0gRERENDQ2//fablZXV2bNnDcfr6uqsra3t7OzMenVgJ5Vav/NMNTXBKDNqKiFTsfgCABBzB1JAQEBDQ8Px48e7devm7+//51W7dBk+fLi1tbVZrw5sY4ii6GHuxhOMsPgCABBzB1JCQgIhZOXKlXZ2dosXLzbrtYBV2m7tul/yRtsoAgAwoOMZ0ieffFJVVfXDDz80NDQYH588eXK3bt1oKADoZzy7iBAyLTlvlRyziwCgPXQEkkwmi4uLa25ubnU8KCgIgcRL1OyiG+6Dd+YrqVbRiaSpv6Vha1cAaA8dgfT111+LxeLPPvvM29vb+HjXrl1puDrQTyuTHp2SlpJRbNxB5xyZUJkUgUACgEehI5A0Gs2ECRN8fHxouBYwK+tMdfIh5XSHcP/DW5Qbs4zfwuwiAGgfHRNjX3755bKyMhouBEyhFv6xSCg4ptDIYod8uviN4foKXXmh4QRs7QoAj0VHCyksLCwuLu7dd98NDAy0tbU1Po5eO66TKzQ7z1TLFXXRw9yNVlsQSzJOYXYRADwVOgJpz549Wq1WJpPJZDLj46NGjcJKDRxiPJKbmlG02SE80Mtxjp9bZtSAtuejPQQAT4WOQFq3bt3q1avbHn/mmWdouDqYCjWSe12lW1ZRNSEkcYb1le/jJQtOMV0XAPAEHYHUrVs3CwuL/Pz8EydO9OvXb/Dgwb169RKLxTRcGkxCpdbXSlOP6CRf5rZEDyOy2CGeTjaEkN+uBGAkNwCYCh2BpNVqo6OjKyoqunTpEhQURAjZtGnTsmXLZs+eTcPVoWNUar1cUXdVrV9xSOnpZLOjTBqevG/Z838ZuI+R3ABgQnQE0rZt26qqqvLy8nbt2nX79u23335bq9Vu27YNgcRC1MMhuUKjqtMFejl6OtpQQxW0ssSGw1vI838ZmICR3ABgQnQEUlFRUWRkpGFWrJWV1axZs7Zt23bp0qX+/fvTUAC0QyuTnsvZKW2QUC89nGzIuDjZgsGtThP5BNRKU3XlhYbVgLBPBACYFh2BZGtrW19fb3ykpqaGYFAD05LzlXKF5pPjKV96vzdwcnBSmIQQ0nTzWmVSBAlrPVTB2hUjuQHAvOgIpJCQkNWrV/v6+up0uubm5l9++WX58uV9+/Z99tlnabg6GKNCSK6o83SyiR7mvsHyYO9Xgr6Nn2s4wdpVTDWGHvpkCI+LAMB86AikqKioy5cvJyYm3r9/nxBy5MgRDw+PtLQ0Gi4NhJDkfCU1UNvTURTo5ZAUJpF5PeiRU8bmObfpc8NQBQBgBB2BRAj5+OOPZ82aVVpaqtFo+vXrFxAQ0KULTZcWoLyvtmrl0tPdBng42UT7uccTMic2jhqo3YpzZAI1u8j4IIYqAAAj6EuFvn379u3bl7bLCU1yvpIQ8scQ7XTrN9ZsmjqReqvp5jVt4liS8ZAZrBiqAADsQUcg1dfXp6WlXbx4sdWWSGlpafb29jQUwFfGQ7Sjh7kTQlpSg2ulqU09gtz+SCPS7mMhDFUAAPagI5A2bNiQnZ09ZMgQFxcX4+PW1tY0XJ1/qByiGkPRw9yTwiSBXg6Gd7Uyadv2TfuPhfC4CADYgI5AKi0tjYqKWrFiBQ3X4jfD8IS/Lq39F3gsBAAcRUcg9ezZs7GxkYYL8RXVJCrN2bXJrSyeethzm9RKH96ywWMhAOAoOgLp9ddfX7ZsmZ+fX3BwsHE3nUgksrCwoKEATpOknCSERA9zT7U86Bz5uSFmmm5eU8b6S9oMVcBjIQDgKDoCyc3NzcrK6sMPP2x1vLCwEPshPYpcoYnJPh/o5UgtrV0rTW3yCTCkEcEMVgDgHToCaevWrRYWFkuWLGk1qMF491gwUKn1MdkVhPy5ywPp0FAFAABuoSOQKisrZ82aNW/ePBquxWlUFKnqdJlRA40HzhEMVQAAAbCk4RovvfTS+fPnabgQp8kVGknKyUAvB2XiyFZpRAgR+QToygt15YWGI9RQBTSPAIA36GghjR8/fv78+e+///6YMWOsrKwMx8PCwrp27UpDAewXlF6iqtMpE0c+dIEfgqEKACAAdATS3r17dTrdgQMHDhw4YHx81KhRGNQgV2iC0otXhEqSwlpvQdQW2kMAwGN0BNK6detWr17d9jj2QzqXs6uL7Nua0cHkdkmtlBBEDgAIGB2B1K1bNwsLi/z8/BMnTvTr12/w4MG9evUSi8U0XJrNYrIr3spNHfHxl4+dWgQAIAR0DGrQarVRUVFLly49ePBgWVlZSUnJxIkTv/76axouzVpB6SV2h7cMeCXooVOLGCwMAIApdATStm3bqqqq8vLyXnvtNULI22+/HR0dvW3bNhouzU5B6SWeTjaLu55p20HnHJmglUkZqQoAgFl0BFJRUVFkZKS3tzf10srKatasWTdu3Lh06RINV2cbKo0yowZQU4tavYupRQAgWHQEkq2tbX19vfGRmpoaIshBDYY0IphaBADwV3QMaggJCVm9erWvr69Op2tubv7ll1+WL1/et2/fZ599loars4dxGhFMLQIA+Cs6AikqKury5cuJiYn3798nhBw5csTDwyMtLY2GS7NHqzQyQHsIAIBCRyARQmJiYtzc3FxcXDQajU6nW7BgAT3XZYnkfCUhpG0aAQCAAR2BdOnSpTfeeMPKyurUqVOEkEmTJn311VeffPIJNeiO9+QKTVZRtTJxJNOFAACwGk3Dvu3t7Q0Tj7755ptXX331q6++ouHSjFOp9UHpxZlRA5kuBACA7egIpF9//XXChAmGYd/29vbvvvvu5cuXVSoVDVdnVkx2hWzBkLardwMAQCt0BJKzs/ONGzeMj+j1ekII7/cvD0ovCfRyQBoBADwJOgJpzJgx+/fv37Jly/Xr17VabWFh4dKlS3v37v3cc8/RcHWmyBUaQkhSmITpQgAAuIGOQQ1vvvnmhQsXNm/evGnTJupIz549N2/ezO8WUkz2eTw6AgB4cnQEkqWl5Zo1a+bOnVtWVnb79m2xWDxy5EiRSETDpRmhlUnztm/d90qw59nC2rOEYLIRAMAToGkeEiGkd+/ehqUZbt26tWDBgm3btvFysYZaaWrZkL/NfvcN6iU2lQAAeBJ0BNLFixfj4uKuXbvW6nhTUxMNV6dZrTRVqpNMmDbRcMSwqQTaSQAA7aBjUMOWLVv0ev3SpUt79OgRERGRmJjo4uISFxfn4eFBw9Vpdi5nl3NkQquRddhUAgDgsegIJKVSOXPmzLlz54aFhTk4OLz11ltr167NzMxsbGyk4eo0+17yxuQre1odxKYSAACPRUcg2djY1NXVEUK8vb0VCgUh5OWXX25oaKC+5pOg9JKJUydiUwkAgA6g4xnSyy+/nJeX9/zzz/v6+m7cuLGkpKS6upoQYmlJRxzShpp4NC7AlwRgUwkAgKdGRyDNnz//7NmzW7duPXz4sEQiiYqKIoT069fPsJgQPyTnKw3TYNEeAgB4WnQEkouLS05OTl1dnYWFxc6dOw8dOtTc3BwaGsqnFlLWmWpPJxusEgQA0GH0zUNydHQkhNjZ2c2YMYO2i9Im+ZBSFjuE6SoAADjMjIFUVFRUUVHRzgn/8z//Y2NjY74CaJN1pjrQy9HTiQ/3AgDAFDMGUkFBQVZWVjsnTJo0iR+BhOYRAEDnmTGQli5dunTpUsNLrVZ74cKF+vp6sVjMp+EMaB4BAJgETc+Q0tLS/vnPf1LbIBFCXnzxxdTUVH5sP4HmEQCASdAxzu1///d/N2/eHBISsmvXrn379iUlJVVVVb333ns0XNrc0DwCADAVOlpIhw4dGjNmzIYNG6iXvr6+np6eMTEx169f5/pq3zvP/IYt+AAATIKOFlJNTY2vr6/xkf79+xNC1Go1DVc3H8w9AgAwIToCydfX99ixY8abTRw6dEgkEvXt25eGq5vPzjO/zfFzZ7oKAACeoKPLLiAgQCqVzpgxY+rUqba2tiUlJfv37x8+fPj3339PCOnatev06dNpKMO0VGq9qk6H5hEAgKnQEUg//PBDc3PzxYsX169fbzhYWFhYWFhICLG2tuZiICUfUiaF4ukRAIDJ0BFIn332WWpqKg0X4sS5HgAAGUdJREFUolPWmerMqAFMVwEAwB90BFKrRVSbmpqsra1puK75JOcrV6B5BABgUnQMamhpadm7d+/WrVsJIWvXrn355ZdHjRpVUFBAw6XNJKuoGsMZAABMi45A2rVrV3Jyck1NzcWLFzMzM0eMGOHt7f3RRx8Zj7vjEEyGBQAwBzoC6fDhw5MmTUpMTDx27Jitre3mzZs//fRTjUZz5coVGq5ucscUmrEYXAcAYGp0BJJGoxk4cCAh5PTp00OHDu3evbutrS0hRKfT0XB1k8s6Ux2N/joAAFOjI5DEYrFcLpfL5YWFhaNGjSKE5ObmWlpa9unTh4armxbSCADATOgIpDlz5pSUlMyfP9/BwWHatGmZmZmrVq2aOnWqi4sLDVc3LazOAABgJnQM+x4xYsT+/fsvXbo0YsSIHj16+Pr6btiwYeLEiTRc2lS0MqlWLhX5BPieVfq6SGrPEufIBKaLAgDgFbNvYR4eHn727NmrV68SQvLy8gzv7t27l0NbmNdKU93iP19Q5jD2XQdnP/emm9eUsf6SjFNM1wUAwB9mDCSZTJaZmTl27Njc3Nwff/yx7Qlc2cK8Vpoq8gkQ+QTIc09SywVZu4pFPgG10lS0kwAATMWipaWF6RrMJS0tLT4+vvPfRxnr3yd5356rXY4pNIblgppuXqtMikAjCQC4wlS/Es2HjkENXOccmVArTW01/ejGlsX2QZEMVgUAwDNm7LLbsmXLv/71r3ZOyM3NdXDgwAxTqnfuvKV3ZlQcdURXXth081qf5H3MFgYAwCdmDKQXX3yRmvra0NCwd+9eR0fH0NDQZ5555r///W9RUdHIkSO7detmvqubkLWr+NjbuZHS1FrpgzXLdeWFbvGfM1sVAADPmDGQRo8ePXr0aELI+vXrn3vuuX379tnb21NvbdmyZdeuXRxa8/uYQjM2MsEZM5AAAMyGjmdIJ0+enDBhgiGNCCGRkZEajebXX3+l4eomgQUaAADMjY5Asre3v3HjhvGRa9euEUK6d+9Ow9U7D2kEAEADOgJp1KhRubm56enp169f12q1crl8yZIlHh4ezz33HA1X7zws7w0AQAM6lg6aO3dueXn5F1988cUXX1BH+vTpk5aWRsOlTUKuqEvC/rAAAGZGRyBZWVkZoohzsB0fAAA9MDH2MdBfBwBADwTSY8gVdYFejkxXAQDAfwikx1Cp9eivAwCgAQKpPRjwDQBAGwRSe7A/LAAAbRBI7ZEr6gIxogEAgBYIpEdCfx0AAJ0QSI9kvB0fAACYGx0TYzugubk5Nze3oqLCyclpypQpYrG41QnFxcWlpaWGl8HBwR4eHqatQa6oM+03BACAdrA0kOLj43/++eeQkJDTp09v27YtJyenb9++xifs3bv33LlzEsmDFX0GDRpk2kCSKzSejiITfkMAAGgfGwOpvLy8oKAgIyMjODi4sbFx8uTJ27dvT0lJMT7nwoUL8+fPnzFjhplqUKl1mH4EAEAnNj5DKi0tFYlEgYGBhJCuXbsGBwcb984RQpqamq5cueLg4HDw4MHCwsKmpiaT14AVgwAAaMbGFpJarXZ1dbW0fBCWbm5uarXa+ASFQtHc3Lxo0aLevXtXVlaKxeLdu3e7uLi0+j45OTk5OTnU1wUFBU9VQ9aZaoxoAAAeCA4Opr6oqqqKj49ntpj2sTGQ7t+/b/zS0tKyubnZ+Mi9e/emTZu2cOFCsVh89erV119/PS0tbcWKFa2+T3h4eMd++hjwDQC8YfjnOPs3/WFjIHXr1q2hocHwsr6+3sbmL49zBg0atG7dOuprDw+PyZMnFxcXm7CAq2q9pyMeIAEA0IqNz5DEYvGtW7dqamqolxUVFa2GfUul0vXr1xte1tfXW1tbm7AAuUIzth9W+AYAoBUbA2nkyJHdu3dfuXJlVVXV/v37jxw5EhISQghJTEzcs2cPIcTe3n7Hjh05OTl37twpKCg4cOBAaGioCQvAikEAAPRjY5edg4PDunXrli9fHhwcbGlpOWnSpFmzZhFC8vLyGhoaZs2aNX78+PPnzyclJX300UfdunWbNWtWdHS0qa6OB0gAAIxgYyARQkJCQsaNG1dTU2NnZ2d4gFRWVmY44f3333/vvffUarWzs7OVlZUJL40B3wAAjGBjlx3FwsKiZ8+erYYzGOvSpYurq6tp04hgi1gAAIawN5CYgi1iAQAYgUD6CzxAAgBgCgLpL/AACQCAKQikv8ADJAAApiCQ/gIPkAAAmIJA+hMeIAEAMAiB9CcsYQcAwCAE0p9UdXoP9NcBADAEgfQndNkBADAIgfSAXKHB+DoAAAYhkB5QqXUYXwcAwCAE0gOYEgsAwCwE0gOYEgsAwCwE0gOYEgsAwCwEEiEYXwcAwAIIJEIwJRYAgAUQSIRgSiwAAAsgkAjBiAYAABbownQBTNLKpFq5VOQTMOWK0u5IYS0hzpEJTBcFACBQgg6kWmmqW/zn3zZ4/h6icY4c0HTzmjLWX5Jxium6AACESLhddrXSVJFPgMgnwDAl1tpVLPIJqJWmMl0aAIAQCTeQtDIp1UFn/ADJOTJBK5MyWhcAgEAJN5CcIxOoxpDxlNgbWxbbB0UyWhcAgEAJN5BEPgG68kJdeaFhSqyuvLDp5jWMawAAYIRwBzVYu4olGafOfpkySFFXa+lOCNGVF7rFf850XQAAAiXcQKKkOYSPjXRwxrpBAABME26XHQVTYgEAWELogYRFvgEAWELQgYRtywEA2EPQgYRtywEA2EPQgYRtywEA2EPQgYQRDQAA7CHoQMKIBgAA9hBuICGNAABYRbiBhP46AABWEW4gXVXrPR3RQgIAYAvhBpKqTu+BLjsAANYQbiChyw4AgFWEG0gY1AAAwCoCDaSsM9XRWOEbAIBNBBpIAADANgINJCwaBADANgINJIxoAABgG4EGEkY0AACwjRADCdsgAQCwkBADCdsgAQCwkBADCSMaAABYSIiBhBENAAAsJMRAIoSgyw4AgG2EGEgqtZ7pEgAAoDXBBRKG2AEAsJPgAglD7AAA2ElwgYQhdgAA7CS4QAIAAHYSXCBhzDcAADsJLpCwih0AADsJK5AwxA4AgLWEFUgYYgcAwFrCCqSrar2nIwIJAICNhBVIqjq9B1pIAACs1IXpAmglV9QlhUqYrgIATCktLY3pEtgiPj6e6RI6RViBhCF2ALzE9V/EppKWlsbpH4WAuuyQRgAAbCagQMKUWAAANhNQIAEAAJsJKJCwrCoAAJsJKJDQZQcAwGYCCiQAAHqoVCrLv3J1de3MN0xNTVWpVCaqjr0ENOwbo+wAgB73799vaWn56aefJJIHEx8tLTv1r/8NGzYMHTrU09PTBMWxmFBaSFlnqqP93JmuAgAEpGfPnu5/6NWr14oVKwIDAxsbGwkhc+fOnTdvHnXamjVr3N3dbWxsfH19y8rKCCF5eXm+vr62trbDhw8vLi5+6aWXbt269eabb6anpzN5P+YnoBYSAAhBTHYFI9fNjBrQ6kh2drahp27gwIHx8fEZGRlr1qwZNGjQt99+W1paSgg5ffr03//+9+3btw8ZMuRvf/vbxo0bExMTw8PDExMTQ0NDP/3006ioqPT09BkzZsTFxYWGhtJ9V/QSSiBhiB2AQLQNBqZ8++23NjYPHhOEh4ePHj06IyPjjTfeeOaZZ9avX+/l5UUIcXBw2L179+zZs2tqapycnGpqar755pvBgwcnJycTQrZu3bpjx45Ro0Z17959xIgR/fr1Y/J+zE8ogaRS6+f4iZiuAgAE5LvvvvP29jY+Eh4eLhaLq6qqZs+eTR3p06fPF1988dFHH9XX19va2g4ZMkSlUvn6+lLvuru7JyYm0l03c4TyDElVp8PGEwDArO3bt9fV1fXp02fJkiXUkc8++yw/P//w4cN1dXWxsbGEEEdHx5s3b1Lv3rp1a9myZXfu3GGsYnoJqIWEIXYAQKcbN250796d+trS0rK5ufn999/fvHmzl5fXmDFjIiIixo0bV1lZKZFI+vfvX11d/c0334jF4ldffXXTpk0nT54cPHjw6tWrf/nlF1tbW0tLy5s3bzY1NVlbWzN7U2YliBYSdi4HAPqNHj26zx/69+8/f/58f3//t956a9SoUfPnz587d+6dO3dmz55dVFTk4OAwaNCgUaNG5efn3717d/HixUFBQY6OjmfPnv3HP/5BCHn11Vdnzpz56aefMn1P5iWIFhJ2LgcAOvXt27elpaWdEwwDuF955ZW6ujrD8a1btxJCpk2btnbtWuPzd+3atWvXLjNUyi6CaCFh53IAAPYTRCBh53IAAPYTRiCp9Z5OGPMNAMBqwggkjPkGAGA9YQQSxnwDALAe/wMJaQQANKO2n/j1118NR9auXevn58dgSZzA/2Hf2JcPANqnlUm1cqnIJ8BwxDkyoTPfkNp+wnjkd3Nzc/sDwYEIoYUEANC+Wmmqc2SC4T/7wEhlrL/5Lnf48OEXX3zR0dHxtddeq62tlclkhmW8L1++TC1kJ5PJpk+fvnDhQhcXF7FY/MMPP2i12unTpz/zzDOOjo5xcXH37983X4VM4X8gYZ1vAGhHrTRV5BNg3DyydhWLfAJqpamd/M7Z2dlb//Dzzz9TB69evTpt2rR58+bl5+f//vvvc+bMqa+vVyqV1LsNDQ2XLl0ihNTX1+fm5t65c+fAgQP+/v4ffPDBl19+WVZWJpfLv/766127duXn53eyPBbif5cdAEA7tDJpn+R9rQ46RyZUJkV0suPOePuJ6upqd3d3QsiePXuGDBkSERFBCFm5cuXo0aNnzpz50D/evXv39PR0Gxub999/f9y4cdbW1rdv366srAwLCysoKOjVq1dnamMn/reQ8AwJANrhHJnQtjF0Y8ti+6DITn7n7777rugPCxYsoA5euXLlp59+6t27d+/evV955ZWWlpaampqH/nFqG1lCiK2tbXNz89y5cydNmjRr1iwHB4eUlBRerrLK/0DCKDsAaIfIJ0BXXqgrLzQc0ZUXNt281snm0aO4ubmFh4c3NDQ0NDRotdpjx455eHg0NzdT79bW1hrOtLCwMP6DarV67dq1tbW13333XUlJSVJSkjnKYxb/AwkAoB3WrmJJxqmG8pO10lTDf27xn5vpchMmTPjhhx/OnTvXrVu3L7/8cuHCha6uriqVqrS09O7duxs3bnzUH0xNTaW29Rs/fvxLL7107949M1XIIJ4/Q8LGEwDwJMzUHjJmaWlJCBk1alRiYmJQUFBLS4u3t3daWtqIESNCQkIGDx5sa2sbHh5OndyqeUQISUhImDJliqOjY5cuXdzc3HJzc81dMP0seDw0Pi0t7Rn/GccUmsyoAUzXAgDmkpaWFh8fz3QVrND+j4L9Pyied9lh4wkAAK7geSBh4wkAAK7geSBhzDcAAFfwfFADAPDeqVOnmC4BTIPngYRJSAC8t3v3bqZLANPgc5ed/Ff01wEAcAafA0lVh+YRAABn8DqQMOYbAIA7+BxITSInjPkGAOAKPgfSve7Onk4ipqsAAIAnwtVRds3Nzbm5uRUVFU5OTlOmTBGLxW3PaRI5o8sOAIAruNpCio+PX716tVar/fHHH6dMmXLlypW25zR1d+bloIbg4GCmSzALvt5XWloa0yWYC18/Mr7eF/txsoVUXl5eUFCQkZERHBzc2Ng4efLk7du3p6SkUO9qZVKtXCryCVioUdZKSwgt6/gCAEAncTKQSktLRSJRYGAgIaRr167BwcEnTpwwvEvtZSLyCfj9foVz5ICmm9eUsf6SDMzlBgBgNU522anValdXV2pzEUKIm5ubWq2mvq6Vpop8AkQ+AYaTrV3FIp+AtlsUAwAAq3ByP6RNmzbt37//0KFD1Mtdu3Zt2bKFWs9KGevfJ3mftauYGPXdu1rff/bAuv+GfshUwQAAbMDy/ZA42WXXrVu3hoYGw8v6+nobmweDF5wjEwzbDxt+9JVJEaLZi1+JZPUnAQAgcJzsshOLxbdu3aqpqaFeVlRUGIZ9i3wCdOWFuvJCw8m68sKmm9cwrgEAgOU42UIaOXJk9+7dV65cuWzZspKSkiNHjnz44YPuOGtXsSTjVK00taH8JHVEV15INZgAAIDNOPkMiRBy+PDh5cuXazQaS0vLSZMmrV27tksXToYrAABQuBpIhJCWlpaamho7OzvDAyQAAOAuDgcSAADwCScHNQAAAP/w8LnLk6y7ygmPvZHi4uLS0lLDy+DgYA8PD3prNKXz588fPXp04cKFTBfSWe3cCD8+ssbGxtzc3EuXLrm4uLz22mtubm5MV9Rxj70XfnxkOp1u3759SqWyV69e06dPd3V1Zbqih+Nhl11sbOzPP/8cEhJy4cKF//u//8vJyenbty/TRXXEY2/kgw8+OHfunEQioV7OmzfPz8+PiUpNoKamJj4+vr6+fv/+/UzX0int3wgPPrLm5uaZM2dWVlYGBgaWlJTcuHEjJyfH09OT6bo64knuhR8fWUREhFqt9vf3P336tF6v//7773v27Ml0XQ/Twi+//PJL//79jx492tLScvfu3ZCQkL///e9MF9URT3IjkyZN2rdvHxPVmdj06dMHDBjQv3//SZMmMV1Lpzz2RnjwkR06dOiFF15QKBQtLS0NDQ1jxoxZt24d00V10JPcCw8+sqNHj/r4+Fy/fr2lpaWysvKFF1749ttvmS7q4fjWZdf+uqsc8tgbaWpqunLlioODw8GDBx0dHYcNG2Ztbc1MrZ22atWqxsbGvLy8oqIipmvplPZvhB8fWW1t7bBhw6jGukgkcnd3r6urY7qoDnrsvfDjI7O1tX3nnXfc3d0JISKRyNLS0s7OjumiHo5vgdTOuqvc8tgbUSgUzc3NixYt6t27d2VlpVgs3r17t4uLCxPFdpaPjw8h5OzZs1wPpPZvhB8fWVRUVFRUFPV1UVHRuXPnZs6cyWxJHfbYe+HHR+bv7+/v769Wq7ds2SKTyUaOHMnaDZ/4Nsru/v37xi8tLS2bm5uZKqYzHnsj9+7dmzZt2sGDB3/88ccDBw7U1dXxeCM4fuDTR9bS0rJ37965c+dOmDBh6tSpTJfTKe3cC58+svv373ft2tXd3b2iouLcuXNMl/NwfGshtbPuKrc89kYGDRq0bt066msPD4/JkycXFxfTWiI8Jd58ZDdv3kxISKioqPjwww+joqIsLCyYrqjj2r8XfnxkOp2OEOLi4rJs2TJCyNtvv719+3Z2Ds3gWwupnXVXueWxNyKVStevX294WV9fz8XebUHhx0em0+lmz55NCPnhhx9mzpzJ6TR67L3w4yNLSUkx9EwSQry8vKqqqhispx18CyTDuqtVVVX79+8/cuRISEgI00V1xKNuJDExcc+ePYQQe3v7HTt25OTk3Llzp6Cg4MCBA6GhoUxXDQ/Bs4/s22+/vXXr1kcffVRfX69UKpVKZW1tLdNFddCj7oVnH9nw4cMvXLjw1Vdf1dTU/PTTT7m5uQEBAY//Y4xgepif6R06dGj48OH9+/d/4YUXEhISmpqamK6ogx56I4MGDVq0aBF1Qmpq6qBBg/r37+/r67tmzZp79+4xWm9n/fOf/5wyZQrTVZhAqxvh2UcWHx/f/69Wr17NdFEd9Kh74dlH1tLSsm7dOh8fH8Mvk4aGBqYrejgeTowlPFp39bE3cu/ePbVa7ezsbGVlRXNt0DH4yDiHHx9Zc3NzTU2No6Nj165dma7lkfgZSAAAwDl8e4YEAAAchUACAABWQCABAAArIJAAAIAVEEgAAMAKCCQAAGAFBBJAxzU3N9+7d48QotfrlyxZ8t///pfpigA4DIEE0HFLliyJjo4mhNy9ezcvL+/69etMVwTAYXxb7RuAToZdQuzt7YuLi7m+MggAs9BCAuigefPmnTp16sKFC2+++WZNTc2777575syZ33777c033zx+/HhsbOzYsWPnzJlz7dq1rKysadOmBQYGLl261LCryN27dzds2DB58uTx48evWLFCq9UyezsAjEMgAXTQ4MGDHR0dRSLR0KFDe/Tocfr06dra2rt3754+fTouLq5Pnz5z5869ePHi9OnTs7KygoODJ06c+O9//3vHjh3UH1+wYMG//vWvqVOnRkREHD58ODo6mqObSQKYCrrsADooLi7u8uXLNTU1ixYtun37tvFbb7/99uLFiwkhNTU1W7du3bJli7+/PyHk+PHjSqWSEFJYWHjixIndu3cPHz6cEDJixIgZM2YcO3aMtXtLA9AAgQRgeobtOHv06NG1a1cqdaiX1GOnsrIyQkhmZmZWVpbhT5WUlCCQQMgQSACmZ7z3qKWlZdutSPV6vYWFhZubm+HIG2+88fzzz9NUHwArIZAAGODt7d3S0vLWW29JJBJCyJ07dz7//PPevXszXRcAkzCoAaDjunbtWlVVdfz48R49ejzVHwwODn722WcXLVr0888/l5SULF++fP/+/f369TNTnQCcgEAC6LhXX321vr7+nXfeuXDhAjHqnbO0/PP/rFbdd9Rb3bp127Ztm5WVVXR0dFRUVFVVVUpKip2dHe13AMAi2DEWgEn19fWNjY1OTk5MFwLAPAQSAACwArrsAACAFRBIAADACggkAABgBQQSAACwAgIJAABYAYEEAACsgEACAABW+P9FNt2INsFqIAAAAABJRU5ErkJggg==\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "% plot exact solution first\n", + "t = linspace(0, 3);\n", + "y_exact = -6*exp(-3*t) + 7*exp(-2*t) + sin(t) - cos(t);\n", + "plot(t, y_exact); hold on\n", + "\n", + "omega = 1;\n", + "\n", + "dt = 0.1;\n", + "t = [0 : dt : 3];\n", + "\n", + "f = @(t,z1,z2) 10*sin(omega*t) - 5*z2 - 6*z1;\n", + "\n", + "z1 = zeros(length(t), 1);\n", + "z2 = zeros(length(t), 1);\n", + "z1(1) = 0;\n", + "z2(1) = 5;\n", + "for i = 1 : length(t)-1\n", + " % predictor\n", + " z1p = z1(i) + z2(i)*dt;\n", + " z2p = z2(i) + f(t(i), z1(i), z2(i))*dt;\n", + "\n", + " % corrector\n", + " z1(i+1) = z1(i) + 0.5*dt*(z2(i) + z2p);\n", + " z2(i+1) = z2(i) + 0.5*dt*(f(t(i), z1(i), z2(i)) + f(t(i+1), z1p, z2p));\n", + "end\n", + "plot(t, z1, 'o')\n", + "xlabel('time'); ylabel('displacement')\n", + "legend('Exact', 'Heuns', 'Location','southeast')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Runge-Kutta: `ode45`\n", + "\n", + "We can also solve using `ode45`, by providing a separate function file that defines the system of 1st-order ODEs. In this case, we'll need to use a single **array** variable, `Z`, to store $z_1$ and $z_2$. The first column of `Z` will store $z_1$ (`Z(:,1)`) and the second column will store $z_2$ (`Z(:,2)`)." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Created file '/Users/kyle/projects/ME373/docs/mass_spring.m'.\n" + ] + } + ], + "source": [ + "%%file mass_spring.m\n", + "function dzdt = mass_spring(t, z)\n", + " omega = 1;\n", + " dzdt = zeros(2,1);\n", + " \n", + " dzdt(1) = z(2);\n", + " dzdt(2) = 10*sin(omega*t) - 6*z(1) - 5*z(2);\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% plot exact solution first\n", + "t = linspace(0, 3);\n", + "y_exact = -6*exp(-3*t) + 7*exp(-2*t) + sin(t) - cos(t);\n", + "plot(t, y_exact); hold on\n", + "\n", + "% solution via ode45:\n", + "[T, Z] = ode45('mass_spring', [0 3], [0 5]);\n", + "\n", + "plot(T, Z(:,1), 'o')\n", + "xlabel('time'); ylabel('displacement')\n", + "legend('Exact', 'ode45', 'Location','southeast')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Backward Euler for 2nd-order ODEs\n", + "\n", + "We saw how to implement the Backward Euler method for a 1st-order ODE, but what about a 2nd-order ODE? (Or in general a system of 1st-order ODEs?)\n", + "\n", + "The recursion formula is the same, except now our dependent variable is an array/vector:\n", + "\\begin{equation}\n", + "\\mathbf{y}_{i+1} = \\mathbf{y}_i + \\Delta t \\, \\mathbf{f} \\left( t_{i+1} , \\mathbf{y}_{i+1} \\right)\n", + "\\end{equation}\n", + "where the bolded $\\mathbf{y}$ and $\\mathbf{f}$ indicate array quantities (in other words, they hold more than one value).\n", + "\n", + "In general, we can use Backward Euler to solve 2nd-order ODEs in a similar fashion as our other numerical methods:\n", + "\n", + "1. Convert the 2nd-order ODE into a system of two 1st-order ODEs\n", + "2. Insert the ODEs into the Backward Euler recursion formula and solve for $\\mathbf{y}_{i+1}$\n", + "\n", + "The main difference is that we will now have a system of two equations and two unknowns: $y_{1, i+1}$ and $y_{2, i+1}$.\n", + "\n", + "Let's demonstrate with an example:\n", + "\\begin{equation}\n", + "y^{\\prime\\prime} + 6 y^{\\prime} + 5y = 10 \\quad y(0) = 0 \\quad y^{\\prime}(0) = 5\n", + "\\end{equation}\n", + "where the exact solution is\n", + "\\begin{equation}\n", + "y(t) = -\\frac{3}{4} e^{-5t} - \\frac{5}{4} e^{-t} + 2\n", + "\\end{equation}\n", + "\n", + "To solve numerically,\n", + "\n", + "1. Convert the 2nd-order ODE into a system of two 1st-order ODEs:\n", + "\\begin{gather}\n", + "y_1 = y \\quad y_1(t=0) = 0 \\\\\n", + "y_2 = y^{\\prime} \\quad y_2 (t=0) = 5\n", + "\\end{gather}\n", + "Then, for the derivatives of these variables:\n", + "\\begin{align}\n", + "y_1^{\\prime} &= y_2 \\\\\n", + "y_2^{\\prime} &= 10 - 6 y_2 - 5 y_1\n", + "\\end{align}\n", + "\n", + "2. Then plug these derivatives into the Backward Euler recursion formulas and solve for $y_{1,i+1}$ and $y_{2,i+1}$:\n", + "\\begin{align}\n", + "y_{1, i+1} &= y_{1, i} + \\Delta t \\, y_{2,i+1} \\\\\n", + "y_{2, i+1} &= y_{2, i} + \\Delta t \\left( 10 - 6 y_{2, i+1} - 5 y_{1,i+1} \\right) \\\\\n", + "\\\\\n", + "y_{1, i+1} - \\Delta t \\, y_{2, i+1} &= y_{1,i} \\\\\n", + "5 \\Delta t \\, y_{1, i+1} + (1 + 6 \\Delta t) y_{2, i+1} &= y_{2,i} + 10 \\Delta t \\\\\n", + "\\text{or} \\quad \n", + "\\begin{bmatrix} 1 & -\\Delta t \\\\ 5 \\Delta t & (1+6\\Delta t)\\end{bmatrix} \n", + "\\begin{bmatrix} y_{1, i+1} \\\\ y_{2, i+1} \\end{bmatrix} &= \n", + "\\begin{bmatrix} y_{1,i} \\\\ y_{2,i} + 10 \\Delta t \\\\ \\end{bmatrix} \\\\\n", + "\\mathbf{A} \\mathbf{y}_{i+1} &= \\mathbf{b}\n", + "\\end{align}\n", + "To isolate $\\mathbf{y}_{i+1}$ and get a usable recursion formula, we need to solve this system of two equations. We could solve this by hand using the substitution method, or we can use Cramer's rule:\n", + "\\begin{align}\n", + "y_{1, i+1} &= \\frac{ y_{1,i} (1 + 6 \\Delta t) + \\Delta t \\left( y_{2,i} + 10 \\Delta t \\right)}{1 + 6 \\Delta t + 5 \\Delta t^2} \\\\\n", + "y_{2, i+1} &= \\frac{ y_{2,i} + 10 \\Delta t - 5 \\Delta t y_{1,i}}{1 + 6 \\Delta t + 5 \\Delta t^2}\n", + "\\end{align}\n", + "\n", + "Let's confirm that this gives us a good, well-behaved numerical solution and compare with the Forward Euler method:" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Maximum error of Forward Euler: 0.099\n", + "Maximum error of Backward Euler: 0.068" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "\n", + "% Exact solution\n", + "t = linspace(0, 5);\n", + "y_exact = @(t) -(3/4)*exp(-5*t) - (5/4)*exp(-t) + 2;\n", + "plot(t, y_exact(t)); hold on\n", + "\n", + "dt = 0.1;\n", + "t = 0 : dt : 5;\n", + "\n", + "% Forward Euler\n", + "f = @(t, y1, y2) 10 - 6*y2 - 5*y1;\n", + "y1 = zeros(length(t), 1); y2 = zeros(length(t), 1);\n", + "y1(1) = 0; y2(1) = 5;\n", + "for i = 1 : length(t) - 1\n", + " y1(i+1) = y1(i) + dt*y2(i);\n", + " y2(i+1) = y2(i) + dt*f(t(i), y1(i), y2(i));\n", + "end\n", + "plot(t, y1, '+')\n", + "\n", + "Y = zeros(length(t), 2);\n", + "Y(1,1) = 0;\n", + "Y(1,2) = 5;\n", + "for i = 1 : length(t) - 1\n", + " D = 1 + 6*dt + 5*dt^2;\n", + " Y(i+1, 1) = (Y(i,1)*(1 + 6*dt) + dt*(Y(i,2) + 10*dt)) / D;\n", + " Y(i+1, 2) = (Y(i,2) + 10*dt - Y(i,1)*5*dt) / D;\n", + "end\n", + "plot(t, Y(:,1), 'o')\n", + "legend('Exact', 'Forward Euler', 'Backward Euler', 'location', 'southeast')\n", + "\n", + "fprintf('Maximum error of Forward Euler: %5.3f\\n', max(abs(y1(:) - y_exact(t)')));\n", + "fprintf('Maximum error of Backward Euler: %5.3f', max(abs(Y(:,1) - y_exact(t)')));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "So, for $\\Delta t = 0.1$, we see that the Forward and Backward Euler methods give an error $\\mathcal{O}(\\Delta t)$, as expected since both methods are *first-order* accurate.\n", + "\n", + "Let's see how they compare for a larger step size:" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Maximum error of Forward Euler: 43.242\n", + "Maximum error of Backward Euler: 0.228" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "\n", + "% Exact solution\n", + "t = linspace(0, 5);\n", + "y_exact = @(t) -(3/4)*exp(-5*t) - (5/4)*exp(-t) + 2;\n", + "plot(t, y_exact(t)); hold on\n", + "\n", + "dt = 0.5;\n", + "t = 0 : dt : 5;\n", + "\n", + "% Forward Euler\n", + "f = @(t, y1, y2) 10 - 6*y2 - 5*y1;\n", + "y1 = zeros(length(t), 1); y2 = zeros(length(t), 1);\n", + "y1(1) = 0; y2(1) = 5;\n", + "for i = 1 : length(t) - 1\n", + " y1(i+1) = y1(i) + dt*y2(i);\n", + " y2(i+1) = y2(i) + dt*f(t(i), y1(i), y2(i));\n", + "end\n", + "plot(t, y1, 'o')\n", + "\n", + "% Backward Euler\n", + "\n", + "Y = zeros(length(t), 2);\n", + "Y(1,1) = 0;\n", + "Y(1,2) = 5;\n", + "for i = 1 : length(t) - 1\n", + " D = 1 + 6*dt + 5*dt^2;\n", + " Y(i+1, 1) = (Y(i,1)*(1 + 6*dt) + dt*(Y(i,2) + 10*dt)) / D;\n", + " Y(i+1, 2) = (Y(i,2) + 10*dt - Y(i,1)*5*dt) / D;\n", + "end\n", + "plot(t, Y(:,1), 'o')\n", + "legend('Exact', 'Backward Euler', 'location', 'southeast')\n", + "\n", + "%fprintf('Maximum error of Forward Euler: %5.3f\\n', max(abs(y1(:) - y_exact(t)')));\n", + "%fprintf('Maximum error of Backward Euler: %5.3f', max(abs(Y(:,1) - y_exact(t)')));\n", + "\n", + "fprintf('Maximum error of Forward Euler: %5.3f\\n', max(abs(y1(:) - y_exact(t)')));\n", + "fprintf('Maximum error of Backward Euler: %5.3f', max(abs(Y(:,1) - y_exact(t)')));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Backward Euler, since it is unconditionally stable, remains well-behaved at this larger step size, while the Forward Euler method blows up.\n", + "\n", + "One other thing: instead of using Cramer's rule to get expressions for $y_{1,i+1}$ and $y_{2,i+1}$, we could instead use Matlab to solve the linear system of equations at each time step. To do that, we could replace\n", + "```OCTAVE\n", + "A = [1 -dt; 5*dt (1+6*dt)];\n", + "b = [Y(i,1); Y(i,2)+10*dt];\n", + "Y(i+1,:) = (A\\b)';\n", + "```\n", + "where `A\\b` is equivalent to `inv(A)*b`, but faster. Let's confirm that this gives the same answer:" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "clear\n", + "\n", + "% Exact solution\n", + "t = linspace(0, 5);\n", + "y_exact = @(t) -(3/4)*exp(-5*t) - (5/4)*exp(-t) + 2;\n", + "plot(t, y_exact(t)); hold on\n", + "\n", + "dt = 0.1;\n", + "t = 0 : dt : 5;\n", + "\n", + "Y = zeros(length(t), 2);\n", + "Y(1,1) = 0;\n", + "Y(1,2) = 5;\n", + "for i = 1 : length(t) - 1\n", + " A = [1 -dt; 5*dt (1+6*dt)];\n", + " b = [Y(i,1); Y(i,2)+10*dt];\n", + " Y(i+1,:) = (A\\b)';\n", + "end\n", + "plot(t, Y(:,1), 'o')\n", + "legend('Exact', 'Backward Euler', 'location', 'southeast')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Cramer's Rule\n", + "\n", + "Cramer's Rule provides a solution method for a system of linear equations, where the number of equations equals the number of unknowns. It works for any number of equations/unknowns, but isn't really practical for more than two or three. We'll focus on using it for a system of two equations, with two unknowns $x_1$ and $x_2$:\n", + "\\begin{gather}\n", + "a_{11} + x_1 + a_{12} x_2 = b_1 \\\\\n", + "a_{21} + x_1 + a_{22} x_2 = b_2 \\\\\n", + "\\text{or } \\mathbf{A} \\mathbf{x} = \\mathbf{b}\n", + "\\end{gather}\n", + "where\n", + "\\begin{gather}\n", + "\\mathbf{A} = \\begin{bmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{bmatrix} \\\\\n", + "\\mathbf{x} = \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix} \\\\\n", + "\\mathbf{b} = \\begin{bmatrix} b_1 \\\\ b_2 \\end{bmatrix}\n", + "\\end{gather}\n", + "\n", + "The solutions for the unknowns are then\n", + "\\begin{align}\n", + "x_1 &= \\frac{ \\begin{vmatrix} b_1 & a_{12} \\\\ b_2 & a_{22} \\end{vmatrix} }{D} = \\frac{b_1 a_{22} - a_{12} b_2}{D} \\\\\n", + "x_2 &= \\frac{ \\begin{vmatrix} a_{11} & b_1 \\\\ a_{21} & b_2 \\end{vmatrix} }{D} = \\frac{a_{11} b_2 - b_1 a_{21}}{D}\n", + "\\end{align}\n", + "where $D$ is the determinant of $\\mathbf{A}$:\n", + "\\begin{equation}\n", + "D = \\det(\\mathbf{A}) = \\begin{vmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{vmatrix} = a_{11} a_{22} - a_{12} a_{21}\n", + "\\end{equation}\n", + "This works as long as the determinant does not equal zero." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Matlab", + "language": "matlab", + "name": "matlab" + }, + "language_info": { + "codemirror_mode": "octave", + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-octave", + "name": "matlab", + "version": "0.16.7" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/docs/_sources/content/second-order/power-series.md b/docs/_sources/content/second-order/power-series.md new file mode 100644 index 0000000..5f5bc58 --- /dev/null +++ b/docs/_sources/content/second-order/power-series.md @@ -0,0 +1,198 @@ +# Power Series Solutions + +**Power series solutions** are another technique we can use to solve 2nd-order homogeneous ODEs of the form +\begin{equation} +y^{\prime\prime} + p(x) y^{\prime} + q(x) y = 0 +\end{equation} +This is useful for more-general cases where our other techniques fail. + +For example, how would you find the solution to this ODE? +\begin{equation} +(1 + x^2) y^{\prime\prime} - 4 x y^{\prime} + 6y = 0 +\end{equation} +None of the methods we've discussed so far would allow us to find an analytical solution to this problem—but we can using a power series solution. + +Power series solutions will be of the form +\begin{equation} +y = \sum_{n=0}^{\infty} a_n x^n +\end{equation} +where the coefficients $a_n$ are what we need to find. + +1. First, for power series to be a valid solution, we need to check whether $x=0$ is an **ordinary point** of the ODE: is the ODE *continuous* and *bounded* at $x=0$? + +*Continuous* means that there should be no discontinuity at $x=0$. + +*Bounded* means that the solution should be finite at $x=0$. + +For example, consider the ODE +\begin{equation} +y^{\prime\prime} - 4xy^{\prime} + (4x^2 - 2)y = 0 +\end{equation} +Both $p(x) = -4x$ and $q(x) = (4x^2 - 2)$ are continuous and bounded at $x=0$, so $x=0$ **is** an ordinary point. + +On the other hand, what about +\begin{equation} +y^{\prime\prime} + x^3 y^{\prime} + \frac{1}{x} y = 0 \text{ ?} +\end{equation} +In this case, the solution is unbounded at $x=0$, and so it is **not** an ordinary point. + +2. If $x=0$ is an ordinary point, then we can find a solution in the form of a power series: +\begin{equation} +y = \sum_{n=0}^{\infty} a_n x^n +\end{equation} + +We then solve for the coefficients $a_n$ by plugging this in to the ODE. To do that, we'll need to take advantage of certain properties of power series. + +## Properties of power series + +- **Dummy index rule**. We can replace the index variable used in the power series with another index variable arbitrarily: +\begin{equation} +\sum_{n=0}^{\infty} a_n x^n = \sum_{m=0}^{\infty} a_m x^m +\end{equation} +This is because the index variable is just a "dummy" that only has meaning inside the sum. + +- **Product rule**. We can bring variables, including $x$, multiplying an entire power series into the power series: +\begin{equation} +x \sum_{n=0}^{\infty} a_n x^n = \sum_{n=0}^{\infty} a_n x^{n+1} +\end{equation} + +- **Derivatives**. We can take derivatives of our power series: +\begin{align} +y^{\prime} &= \sum_{n=0}^{\infty} a_n (n) x^{n-1} = \sum_{n=1}^{\infty} a_n (n) x^{n-1} \\ +y^{\prime\prime} &= \sum_{n=0}^{\infty} a_n (n)(n-1) x^{n-2} = \sum_{n=2}^{\infty} a_n (n)(n-1) x^{n-2} +\end{align} +Notice that we can change where the sums in the power series start, because for $y^{\prime}$ the term corresponding to $n=1$ would just be zero, and similar for the first two terms of $y^{\prime\prime}$. + +- **Index shift**. We can redefine the index used within a sum to shift where it starts. For example, if we let $m=n-1$, or $n=m+1$, then: +\begin{equation} +\sum_{n=1}^{\infty} a_n (n) x^{n-1} = \sum_{m=0}^{\infty} a_{m+1} (m+1) x^m +\end{equation} +Or, in other case, if we let $m=n-2$, or $n=m+2$, then: +\begin{equation} +\sum_{n=2}^{\infty} a_n (n)(n-1) x^{n-2} = \sum_{m=0}^{\infty} a_{m+2} (m+2)(m+1) x^m +\end{equation} + +Now, let's apply these properties to solve ODEs. + +## Power series example 1 + +Let's try to apply the power series approach to solve +\begin{equation} +y^{\prime\prime} + y = 0 \;, +\end{equation} +where we know the solution will be $y(x) = c_1 \sin x + c_2 \cos x$. + +1. Is $x=0$ an ordinary point? Yes, the ODE is continuous and bounded at $x=0$. +So, we can find a solution of the form $y(x) = \sum_{n=0}^{\infty} a_n x^n$. + +2. Now, we solve for the coefficents by plugging the power series into the ODE: +\begin{equation} +\sum_{n=2}^{\infty} a_n (n)(n-1) x^{n-2} + \sum_{n=0}^{\infty} a_n x^n = 0 +\end{equation} +Let's use the index shift rule on the first part of that: +\begin{equation} +\sum_{n=2}^{\infty} a_n (n)(n-1) x^{n-2} \rightarrow \sum_{m=0}^{\infty} a_{m+2} (m+2)(m+1) x^m +\end{equation} +Then, we can use the dummy index rule to change $m$ back to $n$: +\begin{equation} +\sum_{m=0}^{\infty} a_m (m+2)(m+1) x^m \rightarrow \sum_{n=0}^{\infty} a_n (n+2)(n+1) x^n +\end{equation} +Now, let's replace the first term in the ODE with that, merge both terms into a single sum, and simplify: +\begin{align} +\sum_{n=0}^{\infty} a_n (n+2)(n+1) x^n + \sum_{n=0}^{\infty} a_n x^n &= 0 \\ +\sum_{n=0}^{\infty} x^n \left[ a_{n+2}(n+2)(n+1) + a_n \right] &= 0 +\end{align} + +There are infinite terms in this sum, involving the continuous variable $x$; the only way that equation can be satisfied is if + +- $x=0$ always, which cannot be true, or +- $a_{n+2}(n+2)(n+1) + a_n = 0$ for all values of $n$. This is what we can use to find the coefficients of our power series solution. + +Use that expression to define a recursive formula for the coefficients: +\begin{equation} +a_{n+2} = \frac{-a_n}{(n+1)(n+2)} +\end{equation} +We can see that the even coefficients will be related to each other, and the odd coefficients will be related. Let's try to identify a pattern with each, starting with the even terms: +\begin{align} +n=0: \quad a_2 &= \frac{-a_0}{1 \cdot 2} = \frac{-a_0}{2!} \\ +n=2: \quad a_4 &= \frac{-a_2}{3 \cdot 4} = \frac{a_0}{4!} \\ +n=4: \quad a_6 &= \frac{-a_4}{5 \cdot 6} = \frac{-a_0}{6!} +\end{align} +and the odd terms: +\begin{align} +n=1: \quad a_3 &= \frac{-a_1}{2 \cdot 3} = \frac{-a_1}{3!} \\ +n=3: \quad a_5 &= \frac{-a_3}{4 \cdot 5} = \frac{a_1}{5!} \\ +n=5: \quad a_7 &= \frac{-a_5}{6 \cdot 7} = \frac{-a_1}{7!} +\end{align} + +Now, let's put that all together: +\begin{align} +y(x) &= a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \dots \\ +y &= a_0 \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \dots \right) + a_1 \left( x - \frac{x^3}{3!} + \frac{x^5}{5!} + \dots \right) +\end{align} +which you might recognize as being the Taylor series expansion of sine and cosine: +\begin{equation} +y(x) = a_0 \cos x + a_1 \sin x +\end{equation} +So, our unknown coefficients end up being our integration constants, which we can use our two constraints to find. + +## Power series example 2 + +Find the solution to the ODE +\begin{equation} +(1 + x^2) y^{\prime\prime} - 4x y^{\prime} + 6y = 0 +\end{equation} + +First, rearrange into standard form: +\begin{equation} +y^{\prime\prime} - \frac{4x}{1+x^2} y^{\prime} + \frac{6}{1+x^2} y = 0 +\end{equation} +Then, check whether $x=0$ is an ordinary point: yes, it is. + +Now, let's insert the power series into the ODE: +\begin{align} +y^{\prime\prime} + x^2 y^{\prime\prime} - 4 x y^{\prime} + 6 y &= 0 \\ +\sum_{n=2}^{\infty} a_n (n)(n-1)x^{n-2} + x^2 \sum_{n=0}^{\infty} a_n (n)(n-1)x^{n-2} - 4 x \sum_{n=1} a_n (n) x^{n-1} + 6 \sum_{n=0}^{\infty} a_n x^n &= 0 +\end{align} +First, we'll use the power rule: +\begin{equation} +\sum_{n=2}^{\infty} a_n (n)(n-1)x^{n-2} + \sum_{n=2}^{\infty} a_n (n)(n-1)x^{n} - 4 \sum_{n=1} a_n (n) x^{n} + 6 \sum_{n=0}^{\infty} a_n x^n = 0 +\end{equation} +and then the index shift and dummy index rules on the first term: +\begin{equation} +\sum_{n=2}^{\infty} a_n (n)(n-1)x^{n-2} \rightarrow \sum_{m=0}^{\infty} a_{m+2} (m+2)(m+1) x^m \rightarrow \sum_{n=0}^{\infty} a_{n+2} (n+2)(n+1) x^n +\end{equation} + +Then, put that back into the full equation and combine the sums: +\begin{align} +\sum_{n=0}^{\infty} a_{n+2} (n+2)(n+1) x^n + \sum_{n=0}^{\infty} a_n (n)(n-1)x^{n} - 4 \sum_{n=1} a_n (n) x^{n} + 6 \sum_{n=0}^{\infty} a_n x^n &= 0 \\ +\sum_{n=0}^{\infty} x^n \left[ a_{n+2} (n+2)(n+1) + a_n (n)(n-1) - 4a_n (n) + 6a_n \right] &= 0 \\ +a_{n+2} (n+2)(n+1) + a_n (n^2 -5n + 6) &= 0 \\ +a_{n+2} (n+2)(n+1) + a_n (n-3)(n-2) &= 0 \\ +\end{align} + +Thus, our recursion formula for the coefficients $a_n$ is +\begin{equation} +a_{n+2} = -a_n \frac{(n-3)(n-2)}{(n+1)(n+2)} +\end{equation} +Again, we can see that the even terms will be related and the odd terms will be related: +\begin{align} +n=0: \quad a_2 &= -a_0 \frac{6}{2} = -3 a_0 \\ +n=2: \quad a_4 &= 0 \\ +n=4: \quad a_6 &= -a_4 \frac{2}{30} = 0 \\ +&\ldots +\end{align} +and the odd terms: +\begin{align} +n=1: \quad a_3 &= -a_1 \frac{2}{6} = \frac{-a_1}{3} \\ +n=3: \quad a_5 &= 0 \\ +n=5: \quad a_7 &= -a_5 \frac{6}{42} = 0 \\ +&\ldots +\end{align} + +The solution is then +\begin{align} +y(x) &= a_0 + a_1 x - 3 a_0 x^2 - \frac{a_1}{3} x^3 \\ +y &= a_0 \left(1 - 3x^2 \right) + a_1 \left( x - \frac{x^3}{3} \right) +\end{align} +where we find $a_0$ and $a_1$ using our initial or boundary conditions. diff --git a/docs/_sources/content/second-order/second-order.md b/docs/_sources/content/second-order/second-order.md new file mode 100644 index 0000000..d3a025a --- /dev/null +++ b/docs/_sources/content/second-order/second-order.md @@ -0,0 +1,3 @@ +# Second-order Ordinary Differential Equations + +This chapter focuses on analytical and numerical methods for solving 2nd-order ordinary differential equations (ODEs), including initial-value problems (IVPs) and boundary-value problems (BVPs). diff --git a/docs/_static/__init__.py b/docs/_static/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/docs/_static/__pycache__/__init__.cpython-38.pyc b/docs/_static/__pycache__/__init__.cpython-38.pyc new file mode 100644 index 0000000..e202972 Binary files /dev/null and b/docs/_static/__pycache__/__init__.cpython-38.pyc differ diff --git a/docs/_static/basic.css b/docs/_static/basic.css new file mode 100644 index 0000000..5d8ae08 --- /dev/null +++ b/docs/_static/basic.css @@ -0,0 +1,861 @@ +/* + * basic.css + * ~~~~~~~~~ + * + * Sphinx stylesheet -- basic theme. + * + * :copyright: Copyright 2007-2021 by the Sphinx team, see AUTHORS. + * :license: BSD, see LICENSE for details. + * + */ + +/* -- main layout ----------------------------------------------------------- */ + +div.clearer { + clear: both; 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+ display: flex; + top: .3em; + right: .5em; + width: 1.7em; + height: 1.7em; + opacity: 0; + transition: opacity 0.3s, border .3s, background-color .3s; + user-select: none; + padding: 0; + border: none; + outline: none; + border-radius: 0.4em; + border: #e1e1e1 1px solid; + background-color: rgb(245, 245, 245); +} + +button.copybtn.success { + border-color: #22863a; +} + +button.copybtn img { + width: 100%; + padding: .2em; +} + +div.highlight { + position: relative; +} + +.highlight:hover button.copybtn { + opacity: 1; +} + +.highlight button.copybtn:hover { + background-color: rgb(235, 235, 235); +} + +.highlight button.copybtn:active { + background-color: rgb(187, 187, 187); +} + +/** + * A minimal CSS-only tooltip copied from: + * https://codepen.io/mildrenben/pen/rVBrpK + * + * To use, write HTML like the following: + * + *

Short

+ */ + .o-tooltip--left { + position: relative; + } + + .o-tooltip--left:after { + opacity: 0; + visibility: hidden; + position: absolute; + content: attr(data-tooltip); + padding: .2em; + font-size: .8em; + left: -.2em; + background: grey; + color: white; + white-space: nowrap; + z-index: 2; + border-radius: 2px; + transform: translateX(-102%) translateY(0); + transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1); +} + +.o-tooltip--left:hover:after { + display: block; + opacity: 1; + visibility: visible; + transform: translateX(-100%) translateY(0); + transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1); + transition-delay: .5s; +} diff --git a/docs/_static/copybutton.js b/docs/_static/copybutton.js new file mode 100644 index 0000000..482bda0 --- /dev/null +++ b/docs/_static/copybutton.js @@ -0,0 +1,197 @@ +// Localization support +const messages = { + 'en': { + 'copy': 'Copy', + 'copy_to_clipboard': 'Copy to clipboard', + 'copy_success': 'Copied!', + 'copy_failure': 'Failed to copy', + }, + 'es' : { + 'copy': 'Copiar', + 'copy_to_clipboard': 'Copiar al portapapeles', + 'copy_success': '¡Copiado!', + 'copy_failure': 'Error al copiar', + }, + 'de' : { + 'copy': 'Kopieren', + 'copy_to_clipboard': 'In die Zwischenablage kopieren', + 'copy_success': 'Kopiert!', + 'copy_failure': 'Fehler beim Kopieren', + }, + 'fr' : { + 'copy': 'Copier', + 'copy_to_clipboard': 'Copié dans le presse-papier', + 'copy_success': 'Copié !', + 'copy_failure': 'Échec de la copie', + }, + 'ru': { + 'copy': 'Скопировать', + 'copy_to_clipboard': 'Скопировать в буфер', + 'copy_success': 'Скопировано!', + 'copy_failure': 'Не удалось скопировать', + }, + 'zh-CN': { + 'copy': '复制', + 'copy_to_clipboard': '复制到剪贴板', + 'copy_success': '复制成功!', + 'copy_failure': '复制失败', + } +} + +let locale = 'en' +if( document.documentElement.lang !== undefined + && messages[document.documentElement.lang] !== undefined ) { + locale = document.documentElement.lang +} + +let doc_url_root = DOCUMENTATION_OPTIONS.URL_ROOT; +if (doc_url_root == '#') { + doc_url_root = ''; +} + +const path_static = `${doc_url_root}_static/`; + +/** + * Set up copy/paste for code blocks + */ + +const runWhenDOMLoaded = cb => { + if (document.readyState != 'loading') { + cb() + } else if (document.addEventListener) { + document.addEventListener('DOMContentLoaded', cb) + } else { + document.attachEvent('onreadystatechange', function() { + if (document.readyState == 'complete') cb() + }) + } +} + +const codeCellId = index => `codecell${index}` + +// Clears selected text since ClipboardJS will select the text when copying +const clearSelection = () => { + if (window.getSelection) { + window.getSelection().removeAllRanges() + } else if (document.selection) { + document.selection.empty() + } +} + +// Changes tooltip text for two seconds, then changes it back +const temporarilyChangeTooltip = (el, oldText, newText) => { + el.setAttribute('data-tooltip', newText) + el.classList.add('success') + setTimeout(() => el.setAttribute('data-tooltip', oldText), 2000) + setTimeout(() => el.classList.remove('success'), 2000) +} + +// Changes the copy button icon for two seconds, then changes it back +const temporarilyChangeIcon = (el) => { + img = el.querySelector("img"); + img.setAttribute('src', `${path_static}check-solid.svg`) + setTimeout(() => img.setAttribute('src', `${path_static}copy-button.svg`), 2000) +} + +const addCopyButtonToCodeCells = () => { + // If ClipboardJS hasn't loaded, wait a bit and try again. This + // happens because we load ClipboardJS asynchronously. + if (window.ClipboardJS === undefined) { + setTimeout(addCopyButtonToCodeCells, 250) + return + } + + // Add copybuttons to all of our code cells + const codeCells = document.querySelectorAll('div.highlight pre') + codeCells.forEach((codeCell, index) => { + const id = codeCellId(index) + codeCell.setAttribute('id', id) + + const clipboardButton = id => + `` + codeCell.insertAdjacentHTML('afterend', clipboardButton(id)) + }) + +function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string +} + +// Callback when a copy button is clicked. Will be passed the node that was clicked +// should then grab the text and replace pieces of text that shouldn't be used in output +function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") { + + var regexp; + var match; + + // Do we check for line continuation characters and "HERE-documents"? + var useLineCont = !!lineContinuationChar + var useHereDoc = !!hereDocDelim + + // create regexp to capture prompt and remaining line + if (isRegexp) { + regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)') + } else { + regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)') + } + + const outputLines = []; + var promptFound = false; + var gotLineCont = false; + var gotHereDoc = false; + const lineGotPrompt = []; + for (const line of textContent.split('\n')) { + match = line.match(regexp) + if (match || gotLineCont || gotHereDoc) { + promptFound = regexp.test(line) + lineGotPrompt.push(promptFound) + if (removePrompts && promptFound) { + outputLines.push(match[2]) + } else { + outputLines.push(line) + } + gotLineCont = line.endsWith(lineContinuationChar) & useLineCont + if (line.includes(hereDocDelim) & useHereDoc) + gotHereDoc = !gotHereDoc + } else if (!onlyCopyPromptLines) { + outputLines.push(line) + } else if (copyEmptyLines && line.trim() === '') { + outputLines.push(line) + } + } + + // If no lines with the prompt were found then just use original lines + if (lineGotPrompt.some(v => v === true)) { + textContent = outputLines.join('\n'); + } + + // Remove a trailing newline to avoid auto-running when pasting + if (textContent.endsWith("\n")) { + textContent = textContent.slice(0, -1) + } + return textContent +} + + +var copyTargetText = (trigger) => { + var target = document.querySelector(trigger.attributes['data-clipboard-target'].value); + return formatCopyText(target.innerText, '', false, true, true, true, '', '') +} + + // Initialize with a callback so we can modify the text before copy + const clipboard = new ClipboardJS('.copybtn', {text: copyTargetText}) + + // Update UI with error/success messages + clipboard.on('success', event => { + clearSelection() + temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_success']) + temporarilyChangeIcon(event.trigger) + }) + + clipboard.on('error', event => { + temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_failure']) + }) +} + +runWhenDOMLoaded(addCopyButtonToCodeCells) \ No newline at end of file diff --git a/docs/_static/copybutton_funcs.js b/docs/_static/copybutton_funcs.js new file mode 100644 index 0000000..b9168c5 --- /dev/null +++ b/docs/_static/copybutton_funcs.js @@ -0,0 +1,58 @@ +function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string +} + +// Callback when a copy button is clicked. Will be passed the node that was clicked +// should then grab the text and replace pieces of text that shouldn't be used in output +export function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") { + + var regexp; + var match; + + // Do we check for line continuation characters and "HERE-documents"? + var useLineCont = !!lineContinuationChar + var useHereDoc = !!hereDocDelim + + // create regexp to capture prompt and remaining line + if (isRegexp) { + regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)') + } else { + regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)') + } + + const outputLines = []; + var promptFound = false; + var gotLineCont = false; + var gotHereDoc = false; + const lineGotPrompt = []; + for (const line of textContent.split('\n')) { + match = line.match(regexp) + if (match || gotLineCont || gotHereDoc) { + promptFound = regexp.test(line) + lineGotPrompt.push(promptFound) + if (removePrompts && promptFound) { + outputLines.push(match[2]) + } else { + outputLines.push(line) + } + gotLineCont = line.endsWith(lineContinuationChar) & useLineCont + if (line.includes(hereDocDelim) & useHereDoc) + gotHereDoc = !gotHereDoc + } else if (!onlyCopyPromptLines) { + outputLines.push(line) + } else if (copyEmptyLines && line.trim() === '') { + outputLines.push(line) + } + } + + // If no lines with the prompt were found then just use original lines + if (lineGotPrompt.some(v => v === true)) { + textContent = outputLines.join('\n'); + } + + // Remove a trailing newline to avoid auto-running when pasting + if (textContent.endsWith("\n")) { + textContent = textContent.slice(0, -1) + } + return textContent +} diff --git a/docs/_static/css/index.c5995385ac14fb8791e8eb36b4908be2.css b/docs/_static/css/index.c5995385ac14fb8791e8eb36b4908be2.css new file mode 100644 index 0000000..655656d --- /dev/null +++ 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(min-width:1200px){.toc-entry a{padding-right:0}}.toc-entry a:hover{color:rgba(var(--pst-color-toc-link-hover),1);text-decoration:none}.bd-sidebar{padding-top:1em}@media (min-width:720px){.bd-sidebar{border-right:1px solid rgba(0,0,0,.1)}@supports (position:-webkit-sticky) or (position:sticky){.bd-sidebar{position:-webkit-sticky;position:sticky;top:calc(var(--pst-header-height) + 20px);z-index:1000;height:calc(100vh - var(--pst-header-height) - 20px)}}}.bd-sidebar.no-sidebar{border-right:0}.bd-links{padding-top:1rem;padding-bottom:1rem;margin-right:-15px;margin-left:-15px}@media (min-width:720px){.bd-links{display:block!important}@supports (position:-webkit-sticky) or (position:sticky){.bd-links{max-height:calc(100vh - 11rem);overflow-y:auto}}}.bd-sidenav{display:none}.bd-content{padding-top:20px}.bd-content .section{max-width:100%}.bd-content .section table{display:block;overflow:auto}.bd-toc-link{display:block;padding:.25rem 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li>a:hover{color:rgba(var(--pst-color-sidebar-link-hover),1);text-decoration:none;background-color:transparent}.bd-sidebar .nav li>a.reference.external:after{font-family:Font Awesome\ 5 Free;font-weight:900;content:"\f35d";font-size:.75em;margin-left:.3em}.bd-sidebar .nav .active:hover>a,.bd-sidebar .nav .active>a{font-weight:600;color:rgba(var(--pst-color-sidebar-link-active),1)}.toc-h2{font-size:.85rem}.toc-h3{font-size:.75rem}.toc-h4{font-size:.65rem}.toc-entry>.nav-link.active{font-weight:600;color:#130654;color:rgba(var(--pst-color-toc-link-active),1);background-color:transparent;border-left:2px solid rgba(var(--pst-color-toc-link-active),1)}.nav-link:hover{border-style:none}#navbar-main-elements li.nav-item i{font-size:.7rem;padding-left:2px;vertical-align:middle}.bd-toc .nav .nav{display:none}.bd-toc .nav .nav.visible,.bd-toc .nav>.active>ul{display:block}.prev-next-bottom{margin:20px 0}.prev-next-bottom a.left-prev,.prev-next-bottom a.right-next{padding:10px;border:1px solid rgba(0,0,0,.2);max-width:45%;overflow-x:hidden;color:rgba(0,0,0,.65)}.prev-next-bottom a.left-prev{float:left}.prev-next-bottom a.left-prev:before{content:"<< "}.prev-next-bottom a.right-next{float:right}.prev-next-bottom a.right-next:after{content:" >>"}.alert{padding-bottom:0}.alert-info a{color:#e83e8c}#navbar-icon-links i.fa,#navbar-icon-links i.fab,#navbar-icon-links i.far,#navbar-icon-links i.fas{vertical-align:middle;font-style:normal;font-size:1.5rem;line-height:1.25}#navbar-icon-links i.fa-github-square:before{color:#333}#navbar-icon-links i.fa-twitter-square:before{color:#55acee}#navbar-icon-links i.fa-gitlab:before{color:#548}#navbar-icon-links i.fa-bitbucket:before{color:#0052cc}.tocsection{border-left:1px solid #eee;padding:.3rem 1.5rem}.tocsection i{padding-right:.5rem}.editthispage{padding-top:2rem}.editthispage a{color:#130754}.xr-wrap[hidden]{display:block!important}.toctree-checkbox{position:absolute;display:none}.toctree-checkbox~ul{display:none}.toctree-checkbox~label i{transform:rotate(0deg)}.toctree-checkbox:checked~ul{display:block}.toctree-checkbox:checked~label i{transform:rotate(180deg)}.bd-sidebar li{position:relative}.bd-sidebar label{position:absolute;top:0;right:0;height:30px;width:30px;cursor:pointer;display:flex;justify-content:center;align-items:center}.bd-sidebar label:hover{background:rgba(var(--pst-color-sidebar-expander-background-hover),1)}.bd-sidebar label i{display:inline-block;font-size:.75rem;text-align:center}.bd-sidebar label i:hover{color:rgba(var(--pst-color-sidebar-link-hover),1)}.bd-sidebar li.has-children>.reference{padding-right:30px}div.doctest>div.highlight span.gp,span.linenos,table.highlighttable td.linenos{user-select:none!important;-webkit-user-select:text!important;-webkit-user-select:none!important;-moz-user-select:none!important;-ms-user-select:none!important} \ No newline at end of file diff --git a/docs/_static/css/theme.css b/docs/_static/css/theme.css new file mode 100644 index 0000000..3f6e79d --- /dev/null +++ b/docs/_static/css/theme.css @@ -0,0 +1,117 @@ +:root { + /***************************************************************************** + * Theme config + **/ + --pst-header-height: 60px; + + /***************************************************************************** + * Font size + **/ + --pst-font-size-base: 15px; /* base font size - applied at body / html level */ + + /* heading font sizes */ + --pst-font-size-h1: 36px; + --pst-font-size-h2: 32px; + --pst-font-size-h3: 26px; + --pst-font-size-h4: 21px; + --pst-font-size-h5: 18px; + --pst-font-size-h6: 16px; + + /* smaller then heading font sizes*/ + --pst-font-size-milli: 12px; + + --pst-sidebar-font-size: .9em; + --pst-sidebar-caption-font-size: .9em; + + /***************************************************************************** + * Font family + **/ + /* These are adapted from https://systemfontstack.com/ */ + --pst-font-family-base-system: -apple-system, BlinkMacSystemFont, Segoe UI, "Helvetica Neue", + Arial, sans-serif, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol; + --pst-font-family-monospace-system: "SFMono-Regular", Menlo, Consolas, Monaco, + Liberation Mono, Lucida Console, monospace; + + --pst-font-family-base: var(--pst-font-family-base-system); + --pst-font-family-heading: var(--pst-font-family-base); + --pst-font-family-monospace: var(--pst-font-family-monospace-system); + + /***************************************************************************** + * Color + * + * Colors are defined in rgb string way, "red, green, blue" + **/ + --pst-color-primary: 19, 6, 84; + --pst-color-success: 40, 167, 69; + --pst-color-info: 0, 123, 255; /*23, 162, 184;*/ + --pst-color-warning: 255, 193, 7; + --pst-color-danger: 220, 53, 69; + --pst-color-text-base: 51, 51, 51; + + --pst-color-h1: var(--pst-color-primary); + --pst-color-h2: var(--pst-color-primary); + --pst-color-h3: var(--pst-color-text-base); + --pst-color-h4: var(--pst-color-text-base); + --pst-color-h5: var(--pst-color-text-base); + --pst-color-h6: var(--pst-color-text-base); + --pst-color-paragraph: var(--pst-color-text-base); + --pst-color-link: 0, 91, 129; + --pst-color-link-hover: 227, 46, 0; + --pst-color-headerlink: 198, 15, 15; + --pst-color-headerlink-hover: 255, 255, 255; + --pst-color-preformatted-text: 34, 34, 34; + --pst-color-preformatted-background: 250, 250, 250; + --pst-color-inline-code: 232, 62, 140; + + --pst-color-active-navigation: 19, 6, 84; + --pst-color-navbar-link: 77, 77, 77; + --pst-color-navbar-link-hover: var(--pst-color-active-navigation); + --pst-color-navbar-link-active: var(--pst-color-active-navigation); + --pst-color-sidebar-link: 77, 77, 77; + --pst-color-sidebar-link-hover: var(--pst-color-active-navigation); + --pst-color-sidebar-link-active: var(--pst-color-active-navigation); + --pst-color-sidebar-expander-background-hover: 244, 244, 244; + --pst-color-sidebar-caption: 77, 77, 77; + --pst-color-toc-link: 119, 117, 122; + --pst-color-toc-link-hover: var(--pst-color-active-navigation); + --pst-color-toc-link-active: var(--pst-color-active-navigation); + + /***************************************************************************** + * Icon + **/ + + /* font awesome icons*/ + --pst-icon-check-circle: '\f058'; + --pst-icon-info-circle: '\f05a'; + --pst-icon-exclamation-triangle: '\f071'; + --pst-icon-exclamation-circle: '\f06a'; + --pst-icon-times-circle: '\f057'; + --pst-icon-lightbulb: '\f0eb'; + + /***************************************************************************** + * Admonitions + **/ + + --pst-color-admonition-default: var(--pst-color-info); + --pst-color-admonition-note: var(--pst-color-info); + --pst-color-admonition-attention: var(--pst-color-warning); + --pst-color-admonition-caution: var(--pst-color-warning); + --pst-color-admonition-warning: var(--pst-color-warning); + --pst-color-admonition-danger: var(--pst-color-danger); + --pst-color-admonition-error: var(--pst-color-danger); + --pst-color-admonition-hint: var(--pst-color-success); + --pst-color-admonition-tip: var(--pst-color-success); + --pst-color-admonition-important: var(--pst-color-success); + + --pst-icon-admonition-default: var(--pst-icon-info-circle); + --pst-icon-admonition-note: var(--pst-icon-info-circle); + --pst-icon-admonition-attention: var(--pst-icon-exclamation-circle); + --pst-icon-admonition-caution: var(--pst-icon-exclamation-triangle); + --pst-icon-admonition-warning: var(--pst-icon-exclamation-triangle); + --pst-icon-admonition-danger: var(--pst-icon-exclamation-triangle); + --pst-icon-admonition-error: var(--pst-icon-times-circle); + --pst-icon-admonition-hint: var(--pst-icon-lightbulb); + --pst-icon-admonition-tip: var(--pst-icon-lightbulb); + --pst-icon-admonition-important: var(--pst-icon-exclamation-circle); + +} diff --git a/docs/_static/doctools.js b/docs/_static/doctools.js new file mode 100644 index 0000000..61ac9d2 --- /dev/null +++ b/docs/_static/doctools.js @@ -0,0 +1,321 @@ +/* + * doctools.js + * ~~~~~~~~~~~ + * + * Sphinx JavaScript utilities for all documentation. + * + * :copyright: Copyright 2007-2021 by the Sphinx team, see AUTHORS. + * :license: BSD, see LICENSE for details. + * + */ + +/** + * select a different prefix for underscore + */ +$u = _.noConflict(); + +/** + * make the code below compatible with browsers without + * an installed firebug like debugger +if (!window.console || !console.firebug) { + var names = ["log", "debug", "info", "warn", "error", "assert", "dir", + "dirxml", "group", "groupEnd", "time", "timeEnd", "count", "trace", + "profile", "profileEnd"]; + window.console = {}; + for (var i = 0; i < names.length; ++i) + window.console[names[i]] = function() {}; +} + */ + +/** + * small helper function to urldecode strings + * + * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/decodeURIComponent#Decoding_query_parameters_from_a_URL + */ +jQuery.urldecode = function(x) { + if (!x) { + return x + } + return decodeURIComponent(x.replace(/\+/g, ' ')); +}; + +/** + * small helper function to urlencode strings + */ +jQuery.urlencode = encodeURIComponent; + +/** + * This function returns the parsed url parameters of the + * current request. Multiple values per key are supported, + * it will always return arrays of strings for the value parts. + */ +jQuery.getQueryParameters = function(s) { + if (typeof s === 'undefined') + s = document.location.search; + var parts = s.substr(s.indexOf('?') + 1).split('&'); + var result = {}; + for (var i = 0; i < parts.length; i++) { + var tmp = parts[i].split('=', 2); + var key = jQuery.urldecode(tmp[0]); + var value = jQuery.urldecode(tmp[1]); + if (key in result) + result[key].push(value); + else + result[key] = [value]; + } + return result; +}; + +/** + * highlight a given string on a jquery object by wrapping it in + * span elements with the given class name. + */ +jQuery.fn.highlightText = function(text, className) { + function highlight(node, addItems) { + if (node.nodeType === 3) { + var val = node.nodeValue; + var pos = val.toLowerCase().indexOf(text); + if (pos >= 0 && + !jQuery(node.parentNode).hasClass(className) && + !jQuery(node.parentNode).hasClass("nohighlight")) { + var span; + var isInSVG = jQuery(node).closest("body, svg, foreignObject").is("svg"); + if (isInSVG) { + span = document.createElementNS("http://www.w3.org/2000/svg", "tspan"); + } else { + span = document.createElement("span"); + span.className = className; + } + span.appendChild(document.createTextNode(val.substr(pos, text.length))); + node.parentNode.insertBefore(span, node.parentNode.insertBefore( + document.createTextNode(val.substr(pos + text.length)), + node.nextSibling)); + node.nodeValue = val.substr(0, pos); + if (isInSVG) { + var rect = document.createElementNS("http://www.w3.org/2000/svg", "rect"); + var bbox = node.parentElement.getBBox(); + rect.x.baseVal.value = bbox.x; + rect.y.baseVal.value = bbox.y; + rect.width.baseVal.value = bbox.width; + rect.height.baseVal.value = bbox.height; + rect.setAttribute('class', className); + addItems.push({ + "parent": node.parentNode, + "target": rect}); + } + } + } + else if (!jQuery(node).is("button, select, textarea")) { + jQuery.each(node.childNodes, function() { + highlight(this, addItems); + }); + } + } + var addItems = []; + var result = this.each(function() { + highlight(this, addItems); + }); + for (var i = 0; i < addItems.length; ++i) { + jQuery(addItems[i].parent).before(addItems[i].target); + } + return result; +}; + +/* + * backward compatibility for jQuery.browser + * This will be supported until firefox bug is fixed. + */ +if (!jQuery.browser) { + jQuery.uaMatch = function(ua) { + ua = ua.toLowerCase(); + + var match = /(chrome)[ \/]([\w.]+)/.exec(ua) || + /(webkit)[ \/]([\w.]+)/.exec(ua) || + /(opera)(?:.*version|)[ \/]([\w.]+)/.exec(ua) || + /(msie) ([\w.]+)/.exec(ua) || + ua.indexOf("compatible") < 0 && /(mozilla)(?:.*? rv:([\w.]+)|)/.exec(ua) || + []; + + return { + browser: match[ 1 ] || "", + version: match[ 2 ] || "0" + }; + }; + jQuery.browser = {}; + jQuery.browser[jQuery.uaMatch(navigator.userAgent).browser] = true; +} + +/** + * Small JavaScript module for the documentation. + */ +var Documentation = { + + init : function() { + this.fixFirefoxAnchorBug(); + this.highlightSearchWords(); + this.initIndexTable(); + if (DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) { + this.initOnKeyListeners(); + } + }, + + /** + * i18n support + */ + TRANSLATIONS : {}, + PLURAL_EXPR : function(n) { return n === 1 ? 0 : 1; }, + LOCALE : 'unknown', + + // gettext and ngettext don't access this so that the functions + // can safely bound to a different name (_ = Documentation.gettext) + gettext : function(string) { + var translated = Documentation.TRANSLATIONS[string]; + if (typeof translated === 'undefined') + return string; + return (typeof translated === 'string') ? translated : translated[0]; + }, + + ngettext : function(singular, plural, n) { + var translated = Documentation.TRANSLATIONS[singular]; + if (typeof translated === 'undefined') + return (n == 1) ? singular : plural; + return translated[Documentation.PLURALEXPR(n)]; + }, + + addTranslations : function(catalog) { + for (var key in catalog.messages) + this.TRANSLATIONS[key] = catalog.messages[key]; + this.PLURAL_EXPR = new Function('n', 'return +(' + catalog.plural_expr + ')'); + this.LOCALE = catalog.locale; + }, + + /** + * add context elements like header anchor links + */ + addContextElements : function() { + $('div[id] > :header:first').each(function() { + $('\u00B6'). + attr('href', '#' + this.id). + attr('title', _('Permalink to this headline')). + appendTo(this); + }); + $('dt[id]').each(function() { + $('\u00B6'). + attr('href', '#' + this.id). + attr('title', _('Permalink to this definition')). + appendTo(this); + }); + }, + + /** + * workaround a firefox stupidity + * see: https://bugzilla.mozilla.org/show_bug.cgi?id=645075 + */ + fixFirefoxAnchorBug : function() { + if (document.location.hash && $.browser.mozilla) + window.setTimeout(function() { + document.location.href += ''; + }, 10); + }, + + /** + * highlight the search words provided in the url in the text + */ + highlightSearchWords : function() { + var params = $.getQueryParameters(); + var terms = (params.highlight) ? params.highlight[0].split(/\s+/) : []; + if (terms.length) { + var body = $('div.body'); + if (!body.length) { + body = $('body'); + } + window.setTimeout(function() { + $.each(terms, function() { + body.highlightText(this.toLowerCase(), 'highlighted'); + }); + }, 10); + $('') + .appendTo($('#searchbox')); + } + }, + + /** + * init the domain index toggle buttons + */ + initIndexTable : function() { + var togglers = $('img.toggler').click(function() { + var src = $(this).attr('src'); + var idnum = $(this).attr('id').substr(7); + $('tr.cg-' + idnum).toggle(); + if (src.substr(-9) === 'minus.png') + $(this).attr('src', src.substr(0, src.length-9) + 'plus.png'); + else + $(this).attr('src', src.substr(0, src.length-8) + 'minus.png'); + }).css('display', ''); + if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) { + togglers.click(); + } + }, + + /** + * helper function to hide the search marks again + */ + hideSearchWords : function() { + $('#searchbox .highlight-link').fadeOut(300); + $('span.highlighted').removeClass('highlighted'); + }, + + /** + * make the url absolute + */ + makeURL : function(relativeURL) { + return DOCUMENTATION_OPTIONS.URL_ROOT + '/' + relativeURL; + }, + + /** + * get the current relative url + */ + getCurrentURL : function() { + var path = document.location.pathname; + var parts = path.split(/\//); + $.each(DOCUMENTATION_OPTIONS.URL_ROOT.split(/\//), function() { + if (this === '..') + parts.pop(); + }); + var url = parts.join('/'); + return path.substring(url.lastIndexOf('/') + 1, path.length - 1); + }, + + initOnKeyListeners: function() { + $(document).keydown(function(event) { + var activeElementType = document.activeElement.tagName; + // don't navigate when in search box, textarea, dropdown or button + if (activeElementType !== 'TEXTAREA' && activeElementType !== 'INPUT' && activeElementType !== 'SELECT' + && activeElementType !== 'BUTTON' && !event.altKey && !event.ctrlKey && !event.metaKey + && !event.shiftKey) { + switch (event.keyCode) { + case 37: // left + var prevHref = $('link[rel="prev"]').prop('href'); + if (prevHref) { + window.location.href = prevHref; + return false; + } + case 39: // right + var nextHref = $('link[rel="next"]').prop('href'); + if (nextHref) { + window.location.href = nextHref; + return false; + } + } + } + }); + } +}; + +// quick alias for translations +_ = Documentation.gettext; + +$(document).ready(function() { + Documentation.init(); +}); diff --git a/docs/_static/documentation_options.js b/docs/_static/documentation_options.js new file mode 100644 index 0000000..93b7c24 --- /dev/null +++ b/docs/_static/documentation_options.js @@ -0,0 +1,12 @@ +var DOCUMENTATION_OPTIONS = { + URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'), + VERSION: '', + LANGUAGE: 'None', + COLLAPSE_INDEX: false, + BUILDER: 'html', + FILE_SUFFIX: '.html', + LINK_SUFFIX: '.html', + HAS_SOURCE: true, + SOURCELINK_SUFFIX: '', + NAVIGATION_WITH_KEYS: true +}; \ No newline at end of file diff --git a/docs/_static/file.png b/docs/_static/file.png new file mode 100644 index 0000000..a858a41 Binary files /dev/null and b/docs/_static/file.png differ diff --git a/docs/_static/images/logo_binder.svg b/docs/_static/images/logo_binder.svg new file mode 100644 index 0000000..45fecf7 --- /dev/null +++ b/docs/_static/images/logo_binder.svg @@ -0,0 +1,19 @@ + + + + +logo + + + + + + + + diff --git a/docs/_static/images/logo_colab.png b/docs/_static/images/logo_colab.png new file mode 100644 index 0000000..b7560ec Binary files /dev/null and b/docs/_static/images/logo_colab.png differ diff --git a/docs/_static/images/logo_jupyterhub.svg b/docs/_static/images/logo_jupyterhub.svg new file mode 100644 index 0000000..60cfe9f --- /dev/null +++ b/docs/_static/images/logo_jupyterhub.svg @@ -0,0 +1 @@ +logo_jupyterhubHub diff --git a/docs/_static/jquery-3.5.1.js b/docs/_static/jquery-3.5.1.js new file mode 100644 index 0000000..5093733 --- /dev/null +++ b/docs/_static/jquery-3.5.1.js @@ -0,0 +1,10872 @@ +/*! + * jQuery JavaScript Library v3.5.1 + * https://jquery.com/ + * + * Includes Sizzle.js + * https://sizzlejs.com/ + * + * Copyright JS Foundation and other contributors + * Released under the MIT license + * https://jquery.org/license + * + * Date: 2020-05-04T22:49Z + */ +( function( global, factory ) { + + "use strict"; + + if ( typeof module === "object" && typeof module.exports === "object" ) { + + // For CommonJS and CommonJS-like environments where a proper `window` + // is present, execute the factory and get jQuery. + // For environments that do not have a `window` with a `document` + // (such as Node.js), expose a factory as module.exports. + // This accentuates the need for the creation of a real `window`. + // e.g. var jQuery = require("jquery")(window); + // See ticket #14549 for more info. + module.exports = global.document ? + factory( global, true ) : + function( w ) { + if ( !w.document ) { + throw new Error( "jQuery requires a window with a document" ); + } + return factory( w ); + }; + } else { + factory( global ); + } + +// Pass this if window is not defined yet +} )( typeof window !== "undefined" ? window : this, function( window, noGlobal ) { + +// Edge <= 12 - 13+, Firefox <=18 - 45+, IE 10 - 11, Safari 5.1 - 9+, iOS 6 - 9.1 +// throw exceptions when non-strict code (e.g., ASP.NET 4.5) accesses strict mode +// arguments.callee.caller (trac-13335). But as of jQuery 3.0 (2016), strict mode should be common +// enough that all such attempts are guarded in a try block. +"use strict"; + +var arr = []; + +var getProto = Object.getPrototypeOf; + +var slice = arr.slice; + +var flat = arr.flat ? function( array ) { + return arr.flat.call( array ); +} : function( array ) { + return arr.concat.apply( [], array ); +}; + + +var push = arr.push; + +var indexOf = arr.indexOf; + +var class2type = {}; + +var toString = class2type.toString; + +var hasOwn = class2type.hasOwnProperty; + +var fnToString = hasOwn.toString; + +var ObjectFunctionString = fnToString.call( Object ); + +var support = {}; + +var isFunction = function isFunction( obj ) { + + // Support: Chrome <=57, Firefox <=52 + // In some browsers, typeof returns "function" for HTML elements + // (i.e., `typeof document.createElement( "object" ) === "function"`). + // We don't want to classify *any* DOM node as a function. + return typeof obj === "function" && typeof obj.nodeType !== "number"; + }; + + +var isWindow = function isWindow( obj ) { + return obj != null && obj === obj.window; + }; + + +var document = window.document; + + + + var preservedScriptAttributes = { + type: true, + src: true, + nonce: true, + noModule: true + }; + + function DOMEval( code, node, doc ) { + doc = doc || document; + + var i, val, + script = doc.createElement( "script" ); + + script.text = code; + if ( node ) { + for ( i in preservedScriptAttributes ) { + + // Support: Firefox 64+, Edge 18+ + // Some browsers don't support the "nonce" property on scripts. + // On the other hand, just using `getAttribute` is not enough as + // the `nonce` attribute is reset to an empty string whenever it + // becomes browsing-context connected. + // See https://github.com/whatwg/html/issues/2369 + // See https://html.spec.whatwg.org/#nonce-attributes + // The `node.getAttribute` check was added for the sake of + // `jQuery.globalEval` so that it can fake a nonce-containing node + // via an object. + val = node[ i ] || node.getAttribute && node.getAttribute( i ); + if ( val ) { + script.setAttribute( i, val ); + } + } + } + doc.head.appendChild( script ).parentNode.removeChild( script ); + } + + +function toType( obj ) { + if ( obj == null ) { + return obj + ""; + } + + // Support: Android <=2.3 only (functionish RegExp) + return typeof obj === "object" || typeof obj === "function" ? + class2type[ toString.call( obj ) ] || "object" : + typeof obj; +} +/* global Symbol */ +// Defining this global in .eslintrc.json would create a danger of using the global +// unguarded in another place, it seems safer to define global only for this module + + + +var + version = "3.5.1", + + // Define a local copy of jQuery + jQuery = function( selector, context ) { + + // The jQuery object is actually just the init constructor 'enhanced' + // Need init if jQuery is called (just allow error to be thrown if not included) + return new jQuery.fn.init( selector, context ); + }; + +jQuery.fn = jQuery.prototype = { + + // The current version of jQuery being used + jquery: version, + + constructor: jQuery, + + // The default length of a jQuery object is 0 + length: 0, + + toArray: function() { + return slice.call( this ); + }, + + // Get the Nth element in the matched element set OR + // Get the whole matched element set as a clean array + get: function( num ) { + + // Return all the elements in a clean array + if ( num == null ) { + return slice.call( this ); + } + + // Return just the one element from the set + return num < 0 ? this[ num + this.length ] : this[ num ]; + }, + + // Take an array of elements and push it onto the stack + // (returning the new matched element set) + pushStack: function( elems ) { + + // Build a new jQuery matched element set + var ret = jQuery.merge( this.constructor(), elems ); + + // Add the old object onto the stack (as a reference) + ret.prevObject = this; + + // Return the newly-formed element set + return ret; + }, + + // Execute a callback for every element in the matched set. + each: function( callback ) { + return jQuery.each( this, callback ); + }, + + map: function( callback ) { + return this.pushStack( jQuery.map( this, function( elem, i ) { + return callback.call( elem, i, elem ); + } ) ); + }, + + slice: function() { + return this.pushStack( slice.apply( this, arguments ) ); + }, + + first: function() { + return this.eq( 0 ); + }, + + last: function() { + return this.eq( -1 ); + }, + + even: function() { + return this.pushStack( jQuery.grep( this, function( _elem, i ) { + return ( i + 1 ) % 2; + } ) ); + }, + + odd: function() { + return this.pushStack( jQuery.grep( this, function( _elem, i ) { + return i % 2; + } ) ); + }, + + eq: function( i ) { + var len = this.length, + j = +i + ( i < 0 ? len : 0 ); + return this.pushStack( j >= 0 && j < len ? [ this[ j ] ] : [] ); + }, + + end: function() { + return this.prevObject || this.constructor(); + }, + + // For internal use only. + // Behaves like an Array's method, not like a jQuery method. + push: push, + sort: arr.sort, + splice: arr.splice +}; + +jQuery.extend = jQuery.fn.extend = function() { + var options, name, src, copy, copyIsArray, clone, + target = arguments[ 0 ] || {}, + i = 1, + length = arguments.length, + deep = false; + + // Handle a deep copy situation + if ( typeof target === "boolean" ) { + deep = target; + + // Skip the boolean and the target + target = arguments[ i ] || {}; + i++; + } + + // Handle case when target is a string or something (possible in deep copy) + if ( typeof target !== "object" && !isFunction( target ) ) { + target = {}; + } + + // Extend jQuery itself if only one argument is passed + if ( i === length ) { + target = this; + i--; + } + + for ( ; i < length; i++ ) { + + // Only deal with non-null/undefined values + if ( ( options = arguments[ i ] ) != null ) { + + // Extend the base object + for ( name in options ) { + copy = options[ name ]; + + // Prevent Object.prototype pollution + // Prevent never-ending loop + if ( name === "__proto__" || target === copy ) { + continue; + } + + // Recurse if we're merging plain objects or arrays + if ( deep && copy && ( jQuery.isPlainObject( copy ) || + ( copyIsArray = Array.isArray( copy ) ) ) ) { + src = target[ name ]; + + // Ensure proper type for the source value + if ( copyIsArray && !Array.isArray( src ) ) { + clone = []; + } else if ( !copyIsArray && !jQuery.isPlainObject( src ) ) { + clone = {}; + } else { + clone = src; + } + copyIsArray = false; + + // Never move original objects, clone them + target[ name ] = jQuery.extend( deep, clone, copy ); + + // Don't bring in undefined values + } else if ( copy !== undefined ) { + target[ name ] = copy; + } + } + } + } + + // Return the modified object + return target; +}; + +jQuery.extend( { + + // Unique for each copy of jQuery on the page + expando: "jQuery" + ( version + Math.random() ).replace( /\D/g, "" ), + + // Assume jQuery is ready without the ready module + isReady: true, + + error: function( msg ) { + throw new Error( msg ); + }, + + noop: function() {}, + + isPlainObject: function( obj ) { + var proto, Ctor; + + // Detect obvious negatives + // Use toString instead of jQuery.type to catch host objects + if ( !obj || toString.call( obj ) !== "[object Object]" ) { + return false; + } + + proto = getProto( obj ); + + // Objects with no prototype (e.g., `Object.create( null )`) are plain + if ( !proto ) { + return true; + } + + // Objects with prototype are plain iff they were constructed by a global Object function + Ctor = hasOwn.call( proto, "constructor" ) && proto.constructor; + return typeof Ctor === "function" && fnToString.call( Ctor ) === ObjectFunctionString; + }, + + isEmptyObject: function( obj ) { + var name; + + for ( name in obj ) { + return false; + } + return true; + }, + + // Evaluates a script in a provided context; falls back to the global one + // if not specified. + globalEval: function( code, options, doc ) { + DOMEval( code, { nonce: options && options.nonce }, doc ); + }, + + each: function( obj, callback ) { + var length, i = 0; + + if ( isArrayLike( obj ) ) { + length = obj.length; + for ( ; i < length; i++ ) { + if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { + break; + } + } + } else { + for ( i in obj ) { + if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { + break; + } + } + } + + return obj; + }, + + // results is for internal usage only + makeArray: function( arr, results ) { + var ret = results || []; + + if ( arr != null ) { + if ( isArrayLike( Object( arr ) ) ) { + jQuery.merge( ret, + typeof arr === "string" ? + [ arr ] : arr + ); + } else { + push.call( ret, arr ); + } + } + + return ret; + }, + + inArray: function( elem, arr, i ) { + return arr == null ? -1 : indexOf.call( arr, elem, i ); + }, + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + merge: function( first, second ) { + var len = +second.length, + j = 0, + i = first.length; + + for ( ; j < len; j++ ) { + first[ i++ ] = second[ j ]; + } + + first.length = i; + + return first; + }, + + grep: function( elems, callback, invert ) { + var callbackInverse, + matches = [], + i = 0, + length = elems.length, + callbackExpect = !invert; + + // Go through the array, only saving the items + // that pass the validator function + for ( ; i < length; i++ ) { + callbackInverse = !callback( elems[ i ], i ); + if ( callbackInverse !== callbackExpect ) { + matches.push( elems[ i ] ); + } + } + + return matches; + }, + + // arg is for internal usage only + map: function( elems, callback, arg ) { + var length, value, + i = 0, + ret = []; + + // Go through the array, translating each of the items to their new values + if ( isArrayLike( elems ) ) { + length = elems.length; + for ( ; i < length; i++ ) { + value = callback( elems[ i ], i, arg ); + + if ( value != null ) { + ret.push( value ); + } + } + + // Go through every key on the object, + } else { + for ( i in elems ) { + value = callback( elems[ i ], i, arg ); + + if ( value != null ) { + ret.push( value ); + } + } + } + + // Flatten any nested arrays + return flat( ret ); + }, + + // A global GUID counter for objects + guid: 1, + + // jQuery.support is not used in Core but other projects attach their + // properties to it so it needs to exist. + support: support +} ); + +if ( typeof Symbol === "function" ) { + jQuery.fn[ Symbol.iterator ] = arr[ Symbol.iterator ]; +} + +// Populate the class2type map +jQuery.each( "Boolean Number String Function Array Date RegExp Object Error Symbol".split( " " ), +function( _i, name ) { + class2type[ "[object " + name + "]" ] = name.toLowerCase(); +} ); + +function isArrayLike( obj ) { + + // Support: real iOS 8.2 only (not reproducible in simulator) + // `in` check used to prevent JIT error (gh-2145) + // hasOwn isn't used here due to false negatives + // regarding Nodelist length in IE + var length = !!obj && "length" in obj && obj.length, + type = toType( obj ); + + if ( isFunction( obj ) || isWindow( obj ) ) { + return false; + } + + return type === "array" || length === 0 || + typeof length === "number" && length > 0 && ( length - 1 ) in obj; +} +var Sizzle = +/*! + * Sizzle CSS Selector Engine v2.3.5 + * https://sizzlejs.com/ + * + * Copyright JS Foundation and other contributors + * Released under the MIT license + * https://js.foundation/ + * + * Date: 2020-03-14 + */ +( function( window ) { +var i, + support, + Expr, + getText, + isXML, + tokenize, + compile, + select, + outermostContext, + sortInput, + hasDuplicate, + + // Local document vars + setDocument, + document, + docElem, + documentIsHTML, + rbuggyQSA, + rbuggyMatches, + matches, + contains, + + // Instance-specific data + expando = "sizzle" + 1 * new Date(), + preferredDoc = window.document, + dirruns = 0, + done = 0, + classCache = createCache(), + tokenCache = createCache(), + compilerCache = createCache(), + nonnativeSelectorCache = createCache(), + sortOrder = function( a, b ) { + if ( a === b ) { + hasDuplicate = true; + } + return 0; + }, + + // Instance methods + hasOwn = ( {} ).hasOwnProperty, + arr = [], + pop = arr.pop, + pushNative = arr.push, + push = arr.push, + slice = arr.slice, + + // Use a stripped-down indexOf as it's faster than native + // https://jsperf.com/thor-indexof-vs-for/5 + indexOf = function( list, elem ) { + var i = 0, + len = list.length; + for ( ; i < len; i++ ) { + if ( list[ i ] === elem ) { + return i; + } + } + return -1; + }, + + booleans = "checked|selected|async|autofocus|autoplay|controls|defer|disabled|hidden|" + + "ismap|loop|multiple|open|readonly|required|scoped", + + // Regular expressions + + // http://www.w3.org/TR/css3-selectors/#whitespace + whitespace = "[\\x20\\t\\r\\n\\f]", + + // https://www.w3.org/TR/css-syntax-3/#ident-token-diagram + identifier = "(?:\\\\[\\da-fA-F]{1,6}" + whitespace + + "?|\\\\[^\\r\\n\\f]|[\\w-]|[^\0-\\x7f])+", + + // Attribute selectors: http://www.w3.org/TR/selectors/#attribute-selectors + attributes = "\\[" + whitespace + "*(" + identifier + ")(?:" + whitespace + + + // Operator (capture 2) + "*([*^$|!~]?=)" + whitespace + + + // "Attribute values must be CSS identifiers [capture 5] + // or strings [capture 3 or capture 4]" + "*(?:'((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\"|(" + identifier + "))|)" + + whitespace + "*\\]", + + pseudos = ":(" + identifier + ")(?:\\((" + + + // To reduce the number of selectors needing tokenize in the preFilter, prefer arguments: + // 1. quoted (capture 3; capture 4 or capture 5) + "('((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\")|" + + + // 2. simple (capture 6) + "((?:\\\\.|[^\\\\()[\\]]|" + attributes + ")*)|" + + + // 3. anything else (capture 2) + ".*" + + ")\\)|)", + + // Leading and non-escaped trailing whitespace, capturing some non-whitespace characters preceding the latter + rwhitespace = new RegExp( whitespace + "+", "g" ), + rtrim = new RegExp( "^" + whitespace + "+|((?:^|[^\\\\])(?:\\\\.)*)" + + whitespace + "+$", "g" ), + + rcomma = new RegExp( "^" + whitespace + "*," + whitespace + "*" ), + rcombinators = new RegExp( "^" + whitespace + "*([>+~]|" + whitespace + ")" + whitespace + + "*" ), + rdescend = new RegExp( whitespace + "|>" ), + + rpseudo = new RegExp( pseudos ), + ridentifier = new RegExp( "^" + identifier + "$" ), + + matchExpr = { + "ID": new RegExp( "^#(" + identifier + ")" ), + "CLASS": new RegExp( "^\\.(" + identifier + ")" ), + "TAG": new RegExp( "^(" + identifier + "|[*])" ), + "ATTR": new RegExp( "^" + attributes ), + "PSEUDO": new RegExp( "^" + pseudos ), + "CHILD": new RegExp( "^:(only|first|last|nth|nth-last)-(child|of-type)(?:\\(" + + whitespace + "*(even|odd|(([+-]|)(\\d*)n|)" + whitespace + "*(?:([+-]|)" + + whitespace + "*(\\d+)|))" + whitespace + "*\\)|)", "i" ), + "bool": new RegExp( "^(?:" + booleans + ")$", "i" ), + + // For use in libraries implementing .is() + // We use this for POS matching in `select` + "needsContext": new RegExp( "^" + whitespace + + "*[>+~]|:(even|odd|eq|gt|lt|nth|first|last)(?:\\(" + whitespace + + "*((?:-\\d)?\\d*)" + whitespace + "*\\)|)(?=[^-]|$)", "i" ) + }, + + rhtml = /HTML$/i, + rinputs = /^(?:input|select|textarea|button)$/i, + rheader = /^h\d$/i, + + rnative = /^[^{]+\{\s*\[native \w/, + + // Easily-parseable/retrievable ID or TAG or CLASS selectors + rquickExpr = /^(?:#([\w-]+)|(\w+)|\.([\w-]+))$/, + + rsibling = /[+~]/, + + // CSS escapes + // http://www.w3.org/TR/CSS21/syndata.html#escaped-characters + runescape = new RegExp( "\\\\[\\da-fA-F]{1,6}" + whitespace + "?|\\\\([^\\r\\n\\f])", "g" ), + funescape = function( escape, nonHex ) { + var high = "0x" + escape.slice( 1 ) - 0x10000; + + return nonHex ? + + // Strip the backslash prefix from a non-hex escape sequence + nonHex : + + // Replace a hexadecimal escape sequence with the encoded Unicode code point + // Support: IE <=11+ + // For values outside the Basic Multilingual Plane (BMP), manually construct a + // surrogate pair + high < 0 ? + String.fromCharCode( high + 0x10000 ) : + String.fromCharCode( high >> 10 | 0xD800, high & 0x3FF | 0xDC00 ); + }, + + // CSS string/identifier serialization + // https://drafts.csswg.org/cssom/#common-serializing-idioms + rcssescape = /([\0-\x1f\x7f]|^-?\d)|^-$|[^\0-\x1f\x7f-\uFFFF\w-]/g, + fcssescape = function( ch, asCodePoint ) { + if ( asCodePoint ) { + + // U+0000 NULL becomes U+FFFD REPLACEMENT CHARACTER + if ( ch === "\0" ) { + return "\uFFFD"; + } + + // Control characters and (dependent upon position) numbers get escaped as code points + return ch.slice( 0, -1 ) + "\\" + + ch.charCodeAt( ch.length - 1 ).toString( 16 ) + " "; + } + + // Other potentially-special ASCII characters get backslash-escaped + return "\\" + ch; + }, + + // Used for iframes + // See setDocument() + // Removing the function wrapper causes a "Permission Denied" + // error in IE + unloadHandler = function() { + setDocument(); + }, + + inDisabledFieldset = addCombinator( + function( elem ) { + return elem.disabled === true && elem.nodeName.toLowerCase() === "fieldset"; + }, + { dir: "parentNode", next: "legend" } + ); + +// Optimize for push.apply( _, NodeList ) +try { + push.apply( + ( arr = slice.call( preferredDoc.childNodes ) ), + preferredDoc.childNodes + ); + + // Support: Android<4.0 + // Detect silently failing push.apply + // eslint-disable-next-line no-unused-expressions + arr[ preferredDoc.childNodes.length ].nodeType; +} catch ( e ) { + push = { apply: arr.length ? + + // Leverage slice if possible + function( target, els ) { + pushNative.apply( target, slice.call( els ) ); + } : + + // Support: IE<9 + // Otherwise append directly + function( target, els ) { + var j = target.length, + i = 0; + + // Can't trust NodeList.length + while ( ( target[ j++ ] = els[ i++ ] ) ) {} + target.length = j - 1; + } + }; +} + +function Sizzle( selector, context, results, seed ) { + var m, i, elem, nid, match, groups, newSelector, + newContext = context && context.ownerDocument, + + // nodeType defaults to 9, since context defaults to document + nodeType = context ? context.nodeType : 9; + + results = results || []; + + // Return early from calls with invalid selector or context + if ( typeof selector !== "string" || !selector || + nodeType !== 1 && nodeType !== 9 && nodeType !== 11 ) { + + return results; + } + + // Try to shortcut find operations (as opposed to filters) in HTML documents + if ( !seed ) { + setDocument( context ); + context = context || document; + + if ( documentIsHTML ) { + + // If the selector is sufficiently simple, try using a "get*By*" DOM method + // (excepting DocumentFragment context, where the methods don't exist) + if ( nodeType !== 11 && ( match = rquickExpr.exec( selector ) ) ) { + + // ID selector + if ( ( m = match[ 1 ] ) ) { + + // Document context + if ( nodeType === 9 ) { + if ( ( elem = context.getElementById( m ) ) ) { + + // Support: IE, Opera, Webkit + // TODO: identify versions + // getElementById can match elements by name instead of ID + if ( elem.id === m ) { + results.push( elem ); + return results; + } + } else { + return results; + } + + // Element context + } else { + + // Support: IE, Opera, Webkit + // TODO: identify versions + // getElementById can match elements by name instead of ID + if ( newContext && ( elem = newContext.getElementById( m ) ) && + contains( context, elem ) && + elem.id === m ) { + + results.push( elem ); + return results; + } + } + + // Type selector + } else if ( match[ 2 ] ) { + push.apply( results, context.getElementsByTagName( selector ) ); + return results; + + // Class selector + } else if ( ( m = match[ 3 ] ) && support.getElementsByClassName && + context.getElementsByClassName ) { + + push.apply( results, context.getElementsByClassName( m ) ); + return results; + } + } + + // Take advantage of querySelectorAll + if ( support.qsa && + !nonnativeSelectorCache[ selector + " " ] && + ( !rbuggyQSA || !rbuggyQSA.test( selector ) ) && + + // Support: IE 8 only + // Exclude object elements + ( nodeType !== 1 || context.nodeName.toLowerCase() !== "object" ) ) { + + newSelector = selector; + newContext = context; + + // qSA considers elements outside a scoping root when evaluating child or + // descendant combinators, which is not what we want. + // In such cases, we work around the behavior by prefixing every selector in the + // list with an ID selector referencing the scope context. + // The technique has to be used as well when a leading combinator is used + // as such selectors are not recognized by querySelectorAll. + // Thanks to Andrew Dupont for this technique. + if ( nodeType === 1 && + ( rdescend.test( selector ) || rcombinators.test( selector ) ) ) { + + // Expand context for sibling selectors + newContext = rsibling.test( selector ) && testContext( context.parentNode ) || + context; + + // We can use :scope instead of the ID hack if the browser + // supports it & if we're not changing the context. + if ( newContext !== context || !support.scope ) { + + // Capture the context ID, setting it first if necessary + if ( ( nid = context.getAttribute( "id" ) ) ) { + nid = nid.replace( rcssescape, fcssescape ); + } else { + context.setAttribute( "id", ( nid = expando ) ); + } + } + + // Prefix every selector in the list + groups = tokenize( selector ); + i = groups.length; + while ( i-- ) { + groups[ i ] = ( nid ? "#" + nid : ":scope" ) + " " + + toSelector( groups[ i ] ); + } + newSelector = groups.join( "," ); + } + + try { + push.apply( results, + newContext.querySelectorAll( newSelector ) + ); + return results; + } catch ( qsaError ) { + nonnativeSelectorCache( selector, true ); + } finally { + if ( nid === expando ) { + context.removeAttribute( "id" ); + } + } + } + } + } + + // All others + return select( selector.replace( rtrim, "$1" ), context, results, seed ); +} + +/** + * Create key-value caches of limited size + * @returns {function(string, object)} Returns the Object data after storing it on itself with + * property name the (space-suffixed) string and (if the cache is larger than Expr.cacheLength) + * deleting the oldest entry + */ +function createCache() { + var keys = []; + + function cache( key, value ) { + + // Use (key + " ") to avoid collision with native prototype properties (see Issue #157) + if ( keys.push( key + " " ) > Expr.cacheLength ) { + + // Only keep the most recent entries + delete cache[ keys.shift() ]; + } + return ( cache[ key + " " ] = value ); + } + return cache; +} + +/** + * Mark a function for special use by Sizzle + * @param {Function} fn The function to mark + */ +function markFunction( fn ) { + fn[ expando ] = true; + return fn; +} + +/** + * Support testing using an element + * @param {Function} fn Passed the created element and returns a boolean result + */ +function assert( fn ) { + var el = document.createElement( "fieldset" ); + + try { + return !!fn( el ); + } catch ( e ) { + return false; + } finally { + + // Remove from its parent by default + if ( el.parentNode ) { + el.parentNode.removeChild( el ); + } + + // release memory in IE + el = null; + } +} + +/** + * Adds the same handler for all of the specified attrs + * @param {String} attrs Pipe-separated list of attributes + * @param {Function} handler The method that will be applied + */ +function addHandle( attrs, handler ) { + var arr = attrs.split( "|" ), + i = arr.length; + + while ( i-- ) { + Expr.attrHandle[ arr[ i ] ] = handler; + } +} + +/** + * Checks document order of two siblings + * @param {Element} a + * @param {Element} b + * @returns {Number} Returns less than 0 if a precedes b, greater than 0 if a follows b + */ +function siblingCheck( a, b ) { + var cur = b && a, + diff = cur && a.nodeType === 1 && b.nodeType === 1 && + a.sourceIndex - b.sourceIndex; + + // Use IE sourceIndex if available on both nodes + if ( diff ) { + return diff; + } + + // Check if b follows a + if ( cur ) { + while ( ( cur = cur.nextSibling ) ) { + if ( cur === b ) { + return -1; + } + } + } + + return a ? 1 : -1; +} + +/** + * Returns a function to use in pseudos for input types + * @param {String} type + */ +function createInputPseudo( type ) { + return function( elem ) { + var name = elem.nodeName.toLowerCase(); + return name === "input" && elem.type === type; + }; +} + +/** + * Returns a function to use in pseudos for buttons + * @param {String} type + */ +function createButtonPseudo( type ) { + return function( elem ) { + var name = elem.nodeName.toLowerCase(); + return ( name === "input" || name === "button" ) && elem.type === type; + }; +} + +/** + * Returns a function to use in pseudos for :enabled/:disabled + * @param {Boolean} disabled true for :disabled; false for :enabled + */ +function createDisabledPseudo( disabled ) { + + // Known :disabled false positives: fieldset[disabled] > legend:nth-of-type(n+2) :can-disable + return function( elem ) { + + // Only certain elements can match :enabled or :disabled + // https://html.spec.whatwg.org/multipage/scripting.html#selector-enabled + // https://html.spec.whatwg.org/multipage/scripting.html#selector-disabled + if ( "form" in elem ) { + + // Check for inherited disabledness on relevant non-disabled elements: + // * listed form-associated elements in a disabled fieldset + // https://html.spec.whatwg.org/multipage/forms.html#category-listed + // https://html.spec.whatwg.org/multipage/forms.html#concept-fe-disabled + // * option elements in a disabled optgroup + // https://html.spec.whatwg.org/multipage/forms.html#concept-option-disabled + // All such elements have a "form" property. + if ( elem.parentNode && elem.disabled === false ) { + + // Option elements defer to a parent optgroup if present + if ( "label" in elem ) { + if ( "label" in elem.parentNode ) { + return elem.parentNode.disabled === disabled; + } else { + return elem.disabled === disabled; + } + } + + // Support: IE 6 - 11 + // Use the isDisabled shortcut property to check for disabled fieldset ancestors + return elem.isDisabled === disabled || + + // Where there is no isDisabled, check manually + /* jshint -W018 */ + elem.isDisabled !== !disabled && + inDisabledFieldset( elem ) === disabled; + } + + return elem.disabled === disabled; + + // Try to winnow out elements that can't be disabled before trusting the disabled property. + // Some victims get caught in our net (label, legend, menu, track), but it shouldn't + // even exist on them, let alone have a boolean value. + } else if ( "label" in elem ) { + return elem.disabled === disabled; + } + + // Remaining elements are neither :enabled nor :disabled + return false; + }; +} + +/** + * Returns a function to use in pseudos for positionals + * @param {Function} fn + */ +function createPositionalPseudo( fn ) { + return markFunction( function( argument ) { + argument = +argument; + return markFunction( function( seed, matches ) { + var j, + matchIndexes = fn( [], seed.length, argument ), + i = matchIndexes.length; + + // Match elements found at the specified indexes + while ( i-- ) { + if ( seed[ ( j = matchIndexes[ i ] ) ] ) { + seed[ j ] = !( matches[ j ] = seed[ j ] ); + } + } + } ); + } ); +} + +/** + * Checks a node for validity as a Sizzle context + * @param {Element|Object=} context + * @returns {Element|Object|Boolean} The input node if acceptable, otherwise a falsy value + */ +function testContext( context ) { + return context && typeof context.getElementsByTagName !== "undefined" && context; +} + +// Expose support vars for convenience +support = Sizzle.support = {}; + +/** + * Detects XML nodes + * @param {Element|Object} elem An element or a document + * @returns {Boolean} True iff elem is a non-HTML XML node + */ +isXML = Sizzle.isXML = function( elem ) { + var namespace = elem.namespaceURI, + docElem = ( elem.ownerDocument || elem ).documentElement; + + // Support: IE <=8 + // Assume HTML when documentElement doesn't yet exist, such as inside loading iframes + // https://bugs.jquery.com/ticket/4833 + return !rhtml.test( namespace || docElem && docElem.nodeName || "HTML" ); +}; + +/** + * Sets document-related variables once based on the current document + * @param {Element|Object} [doc] An element or document object to use to set the document + * @returns {Object} Returns the current document + */ +setDocument = Sizzle.setDocument = function( node ) { + var hasCompare, subWindow, + doc = node ? node.ownerDocument || node : preferredDoc; + + // Return early if doc is invalid or already selected + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( doc == document || doc.nodeType !== 9 || !doc.documentElement ) { + return document; + } + + // Update global variables + document = doc; + docElem = document.documentElement; + documentIsHTML = !isXML( document ); + + // Support: IE 9 - 11+, Edge 12 - 18+ + // Accessing iframe documents after unload throws "permission denied" errors (jQuery #13936) + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( preferredDoc != document && + ( subWindow = document.defaultView ) && subWindow.top !== subWindow ) { + + // Support: IE 11, Edge + if ( subWindow.addEventListener ) { + subWindow.addEventListener( "unload", unloadHandler, false ); + + // Support: IE 9 - 10 only + } else if ( subWindow.attachEvent ) { + subWindow.attachEvent( "onunload", unloadHandler ); + } + } + + // Support: IE 8 - 11+, Edge 12 - 18+, Chrome <=16 - 25 only, Firefox <=3.6 - 31 only, + // Safari 4 - 5 only, Opera <=11.6 - 12.x only + // IE/Edge & older browsers don't support the :scope pseudo-class. + // Support: Safari 6.0 only + // Safari 6.0 supports :scope but it's an alias of :root there. + support.scope = assert( function( el ) { + docElem.appendChild( el ).appendChild( document.createElement( "div" ) ); + return typeof el.querySelectorAll !== "undefined" && + !el.querySelectorAll( ":scope fieldset div" ).length; + } ); + + /* Attributes + ---------------------------------------------------------------------- */ + + // Support: IE<8 + // Verify that getAttribute really returns attributes and not properties + // (excepting IE8 booleans) + support.attributes = assert( function( el ) { + el.className = "i"; + return !el.getAttribute( "className" ); + } ); + + /* getElement(s)By* + ---------------------------------------------------------------------- */ + + // Check if getElementsByTagName("*") returns only elements + support.getElementsByTagName = assert( function( el ) { + el.appendChild( document.createComment( "" ) ); + return !el.getElementsByTagName( "*" ).length; + } ); + + // Support: IE<9 + support.getElementsByClassName = rnative.test( document.getElementsByClassName ); + + // Support: IE<10 + // Check if getElementById returns elements by name + // The broken getElementById methods don't pick up programmatically-set names, + // so use a roundabout getElementsByName test + support.getById = assert( function( el ) { + docElem.appendChild( el ).id = expando; + return !document.getElementsByName || !document.getElementsByName( expando ).length; + } ); + + // ID filter and find + if ( support.getById ) { + Expr.filter[ "ID" ] = function( id ) { + var attrId = id.replace( runescape, funescape ); + return function( elem ) { + return elem.getAttribute( "id" ) === attrId; + }; + }; + Expr.find[ "ID" ] = function( id, context ) { + if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { + var elem = context.getElementById( id ); + return elem ? [ elem ] : []; + } + }; + } else { + Expr.filter[ "ID" ] = function( id ) { + var attrId = id.replace( runescape, funescape ); + return function( elem ) { + var node = typeof elem.getAttributeNode !== "undefined" && + elem.getAttributeNode( "id" ); + return node && node.value === attrId; + }; + }; + + // Support: IE 6 - 7 only + // getElementById is not reliable as a find shortcut + Expr.find[ "ID" ] = function( id, context ) { + if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { + var node, i, elems, + elem = context.getElementById( id ); + + if ( elem ) { + + // Verify the id attribute + node = elem.getAttributeNode( "id" ); + if ( node && node.value === id ) { + return [ elem ]; + } + + // Fall back on getElementsByName + elems = context.getElementsByName( id ); + i = 0; + while ( ( elem = elems[ i++ ] ) ) { + node = elem.getAttributeNode( "id" ); + if ( node && node.value === id ) { + return [ elem ]; + } + } + } + + return []; + } + }; + } + + // Tag + Expr.find[ "TAG" ] = support.getElementsByTagName ? + function( tag, context ) { + if ( typeof context.getElementsByTagName !== "undefined" ) { + return context.getElementsByTagName( tag ); + + // DocumentFragment nodes don't have gEBTN + } else if ( support.qsa ) { + return context.querySelectorAll( tag ); + } + } : + + function( tag, context ) { + var elem, + tmp = [], + i = 0, + + // By happy coincidence, a (broken) gEBTN appears on DocumentFragment nodes too + results = context.getElementsByTagName( tag ); + + // Filter out possible comments + if ( tag === "*" ) { + while ( ( elem = results[ i++ ] ) ) { + if ( elem.nodeType === 1 ) { + tmp.push( elem ); + } + } + + return tmp; + } + return results; + }; + + // Class + Expr.find[ "CLASS" ] = support.getElementsByClassName && function( className, context ) { + if ( typeof context.getElementsByClassName !== "undefined" && documentIsHTML ) { + return context.getElementsByClassName( className ); + } + }; + + /* QSA/matchesSelector + ---------------------------------------------------------------------- */ + + // QSA and matchesSelector support + + // matchesSelector(:active) reports false when true (IE9/Opera 11.5) + rbuggyMatches = []; + + // qSa(:focus) reports false when true (Chrome 21) + // We allow this because of a bug in IE8/9 that throws an error + // whenever `document.activeElement` is accessed on an iframe + // So, we allow :focus to pass through QSA all the time to avoid the IE error + // See https://bugs.jquery.com/ticket/13378 + rbuggyQSA = []; + + if ( ( support.qsa = rnative.test( document.querySelectorAll ) ) ) { + + // Build QSA regex + // Regex strategy adopted from Diego Perini + assert( function( el ) { + + var input; + + // Select is set to empty string on purpose + // This is to test IE's treatment of not explicitly + // setting a boolean content attribute, + // since its presence should be enough + // https://bugs.jquery.com/ticket/12359 + docElem.appendChild( el ).innerHTML = "" + + ""; + + // Support: IE8, Opera 11-12.16 + // Nothing should be selected when empty strings follow ^= or $= or *= + // The test attribute must be unknown in Opera but "safe" for WinRT + // https://msdn.microsoft.com/en-us/library/ie/hh465388.aspx#attribute_section + if ( el.querySelectorAll( "[msallowcapture^='']" ).length ) { + rbuggyQSA.push( "[*^$]=" + whitespace + "*(?:''|\"\")" ); + } + + // Support: IE8 + // Boolean attributes and "value" are not treated correctly + if ( !el.querySelectorAll( "[selected]" ).length ) { + rbuggyQSA.push( "\\[" + whitespace + "*(?:value|" + booleans + ")" ); + } + + // Support: Chrome<29, Android<4.4, Safari<7.0+, iOS<7.0+, PhantomJS<1.9.8+ + if ( !el.querySelectorAll( "[id~=" + expando + "-]" ).length ) { + rbuggyQSA.push( "~=" ); + } + + // Support: IE 11+, Edge 15 - 18+ + // IE 11/Edge don't find elements on a `[name='']` query in some cases. + // Adding a temporary attribute to the document before the selection works + // around the issue. + // Interestingly, IE 10 & older don't seem to have the issue. + input = document.createElement( "input" ); + input.setAttribute( "name", "" ); + el.appendChild( input ); + if ( !el.querySelectorAll( "[name='']" ).length ) { + rbuggyQSA.push( "\\[" + whitespace + "*name" + whitespace + "*=" + + whitespace + "*(?:''|\"\")" ); + } + + // Webkit/Opera - :checked should return selected option elements + // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked + // IE8 throws error here and will not see later tests + if ( !el.querySelectorAll( ":checked" ).length ) { + rbuggyQSA.push( ":checked" ); + } + + // Support: Safari 8+, iOS 8+ + // https://bugs.webkit.org/show_bug.cgi?id=136851 + // In-page `selector#id sibling-combinator selector` fails + if ( !el.querySelectorAll( "a#" + expando + "+*" ).length ) { + rbuggyQSA.push( ".#.+[+~]" ); + } + + // Support: Firefox <=3.6 - 5 only + // Old Firefox doesn't throw on a badly-escaped identifier. + el.querySelectorAll( "\\\f" ); + rbuggyQSA.push( "[\\r\\n\\f]" ); + } ); + + assert( function( el ) { + el.innerHTML = "" + + ""; + + // Support: Windows 8 Native Apps + // The type and name attributes are restricted during .innerHTML assignment + var input = document.createElement( "input" ); + input.setAttribute( "type", "hidden" ); + el.appendChild( input ).setAttribute( "name", "D" ); + + // Support: IE8 + // Enforce case-sensitivity of name attribute + if ( el.querySelectorAll( "[name=d]" ).length ) { + rbuggyQSA.push( "name" + whitespace + "*[*^$|!~]?=" ); + } + + // FF 3.5 - :enabled/:disabled and hidden elements (hidden elements are still enabled) + // IE8 throws error here and will not see later tests + if ( el.querySelectorAll( ":enabled" ).length !== 2 ) { + rbuggyQSA.push( ":enabled", ":disabled" ); + } + + // Support: IE9-11+ + // IE's :disabled selector does not pick up the children of disabled fieldsets + docElem.appendChild( el ).disabled = true; + if ( el.querySelectorAll( ":disabled" ).length !== 2 ) { + rbuggyQSA.push( ":enabled", ":disabled" ); + } + + // Support: Opera 10 - 11 only + // Opera 10-11 does not throw on post-comma invalid pseudos + el.querySelectorAll( "*,:x" ); + rbuggyQSA.push( ",.*:" ); + } ); + } + + if ( ( support.matchesSelector = rnative.test( ( matches = docElem.matches || + docElem.webkitMatchesSelector || + docElem.mozMatchesSelector || + docElem.oMatchesSelector || + docElem.msMatchesSelector ) ) ) ) { + + assert( function( el ) { + + // Check to see if it's possible to do matchesSelector + // on a disconnected node (IE 9) + support.disconnectedMatch = matches.call( el, "*" ); + + // This should fail with an exception + // Gecko does not error, returns false instead + matches.call( el, "[s!='']:x" ); + rbuggyMatches.push( "!=", pseudos ); + } ); + } + + rbuggyQSA = rbuggyQSA.length && new RegExp( rbuggyQSA.join( "|" ) ); + rbuggyMatches = rbuggyMatches.length && new RegExp( rbuggyMatches.join( "|" ) ); + + /* Contains + ---------------------------------------------------------------------- */ + hasCompare = rnative.test( docElem.compareDocumentPosition ); + + // Element contains another + // Purposefully self-exclusive + // As in, an element does not contain itself + contains = hasCompare || rnative.test( docElem.contains ) ? + function( a, b ) { + var adown = a.nodeType === 9 ? a.documentElement : a, + bup = b && b.parentNode; + return a === bup || !!( bup && bup.nodeType === 1 && ( + adown.contains ? + adown.contains( bup ) : + a.compareDocumentPosition && a.compareDocumentPosition( bup ) & 16 + ) ); + } : + function( a, b ) { + if ( b ) { + while ( ( b = b.parentNode ) ) { + if ( b === a ) { + return true; + } + } + } + return false; + }; + + /* Sorting + ---------------------------------------------------------------------- */ + + // Document order sorting + sortOrder = hasCompare ? + function( a, b ) { + + // Flag for duplicate removal + if ( a === b ) { + hasDuplicate = true; + return 0; + } + + // Sort on method existence if only one input has compareDocumentPosition + var compare = !a.compareDocumentPosition - !b.compareDocumentPosition; + if ( compare ) { + return compare; + } + + // Calculate position if both inputs belong to the same document + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + compare = ( a.ownerDocument || a ) == ( b.ownerDocument || b ) ? + a.compareDocumentPosition( b ) : + + // Otherwise we know they are disconnected + 1; + + // Disconnected nodes + if ( compare & 1 || + ( !support.sortDetached && b.compareDocumentPosition( a ) === compare ) ) { + + // Choose the first element that is related to our preferred document + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( a == document || a.ownerDocument == preferredDoc && + contains( preferredDoc, a ) ) { + return -1; + } + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( b == document || b.ownerDocument == preferredDoc && + contains( preferredDoc, b ) ) { + return 1; + } + + // Maintain original order + return sortInput ? + ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : + 0; + } + + return compare & 4 ? -1 : 1; + } : + function( a, b ) { + + // Exit early if the nodes are identical + if ( a === b ) { + hasDuplicate = true; + return 0; + } + + var cur, + i = 0, + aup = a.parentNode, + bup = b.parentNode, + ap = [ a ], + bp = [ b ]; + + // Parentless nodes are either documents or disconnected + if ( !aup || !bup ) { + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + /* eslint-disable eqeqeq */ + return a == document ? -1 : + b == document ? 1 : + /* eslint-enable eqeqeq */ + aup ? -1 : + bup ? 1 : + sortInput ? + ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : + 0; + + // If the nodes are siblings, we can do a quick check + } else if ( aup === bup ) { + return siblingCheck( a, b ); + } + + // Otherwise we need full lists of their ancestors for comparison + cur = a; + while ( ( cur = cur.parentNode ) ) { + ap.unshift( cur ); + } + cur = b; + while ( ( cur = cur.parentNode ) ) { + bp.unshift( cur ); + } + + // Walk down the tree looking for a discrepancy + while ( ap[ i ] === bp[ i ] ) { + i++; + } + + return i ? + + // Do a sibling check if the nodes have a common ancestor + siblingCheck( ap[ i ], bp[ i ] ) : + + // Otherwise nodes in our document sort first + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + /* eslint-disable eqeqeq */ + ap[ i ] == preferredDoc ? -1 : + bp[ i ] == preferredDoc ? 1 : + /* eslint-enable eqeqeq */ + 0; + }; + + return document; +}; + +Sizzle.matches = function( expr, elements ) { + return Sizzle( expr, null, null, elements ); +}; + +Sizzle.matchesSelector = function( elem, expr ) { + setDocument( elem ); + + if ( support.matchesSelector && documentIsHTML && + !nonnativeSelectorCache[ expr + " " ] && + ( !rbuggyMatches || !rbuggyMatches.test( expr ) ) && + ( !rbuggyQSA || !rbuggyQSA.test( expr ) ) ) { + + try { + var ret = matches.call( elem, expr ); + + // IE 9's matchesSelector returns false on disconnected nodes + if ( ret || support.disconnectedMatch || + + // As well, disconnected nodes are said to be in a document + // fragment in IE 9 + elem.document && elem.document.nodeType !== 11 ) { + return ret; + } + } catch ( e ) { + nonnativeSelectorCache( expr, true ); + } + } + + return Sizzle( expr, document, null, [ elem ] ).length > 0; +}; + +Sizzle.contains = function( context, elem ) { + + // Set document vars if needed + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( ( context.ownerDocument || context ) != document ) { + setDocument( context ); + } + return contains( context, elem ); +}; + +Sizzle.attr = function( elem, name ) { + + // Set document vars if needed + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( ( elem.ownerDocument || elem ) != document ) { + setDocument( elem ); + } + + var fn = Expr.attrHandle[ name.toLowerCase() ], + + // Don't get fooled by Object.prototype properties (jQuery #13807) + val = fn && hasOwn.call( Expr.attrHandle, name.toLowerCase() ) ? + fn( elem, name, !documentIsHTML ) : + undefined; + + return val !== undefined ? + val : + support.attributes || !documentIsHTML ? + elem.getAttribute( name ) : + ( val = elem.getAttributeNode( name ) ) && val.specified ? + val.value : + null; +}; + +Sizzle.escape = function( sel ) { + return ( sel + "" ).replace( rcssescape, fcssescape ); +}; + +Sizzle.error = function( msg ) { + throw new Error( "Syntax error, unrecognized expression: " + msg ); +}; + +/** + * Document sorting and removing duplicates + * @param {ArrayLike} results + */ +Sizzle.uniqueSort = function( results ) { + var elem, + duplicates = [], + j = 0, + i = 0; + + // Unless we *know* we can detect duplicates, assume their presence + hasDuplicate = !support.detectDuplicates; + sortInput = !support.sortStable && results.slice( 0 ); + results.sort( sortOrder ); + + if ( hasDuplicate ) { + while ( ( elem = results[ i++ ] ) ) { + if ( elem === results[ i ] ) { + j = duplicates.push( i ); + } + } + while ( j-- ) { + results.splice( duplicates[ j ], 1 ); + } + } + + // Clear input after sorting to release objects + // See https://github.com/jquery/sizzle/pull/225 + sortInput = null; + + return results; +}; + +/** + * Utility function for retrieving the text value of an array of DOM nodes + * @param {Array|Element} elem + */ +getText = Sizzle.getText = function( elem ) { + var node, + ret = "", + i = 0, + nodeType = elem.nodeType; + + if ( !nodeType ) { + + // If no nodeType, this is expected to be an array + while ( ( node = elem[ i++ ] ) ) { + + // Do not traverse comment nodes + ret += getText( node ); + } + } else if ( nodeType === 1 || nodeType === 9 || nodeType === 11 ) { + + // Use textContent for elements + // innerText usage removed for consistency of new lines (jQuery #11153) + if ( typeof elem.textContent === "string" ) { + return elem.textContent; + } else { + + // Traverse its children + for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { + ret += getText( elem ); + } + } + } else if ( nodeType === 3 || nodeType === 4 ) { + return elem.nodeValue; + } + + // Do not include comment or processing instruction nodes + + return ret; +}; + +Expr = Sizzle.selectors = { + + // Can be adjusted by the user + cacheLength: 50, + + createPseudo: markFunction, + + match: matchExpr, + + attrHandle: {}, + + find: {}, + + relative: { + ">": { dir: "parentNode", first: true }, + " ": { dir: "parentNode" }, + "+": { dir: "previousSibling", first: true }, + "~": { dir: "previousSibling" } + }, + + preFilter: { + "ATTR": function( match ) { + match[ 1 ] = match[ 1 ].replace( runescape, funescape ); + + // Move the given value to match[3] whether quoted or unquoted + match[ 3 ] = ( match[ 3 ] || match[ 4 ] || + match[ 5 ] || "" ).replace( runescape, funescape ); + + if ( match[ 2 ] === "~=" ) { + match[ 3 ] = " " + match[ 3 ] + " "; + } + + return match.slice( 0, 4 ); + }, + + "CHILD": function( match ) { + + /* matches from matchExpr["CHILD"] + 1 type (only|nth|...) + 2 what (child|of-type) + 3 argument (even|odd|\d*|\d*n([+-]\d+)?|...) + 4 xn-component of xn+y argument ([+-]?\d*n|) + 5 sign of xn-component + 6 x of xn-component + 7 sign of y-component + 8 y of y-component + */ + match[ 1 ] = match[ 1 ].toLowerCase(); + + if ( match[ 1 ].slice( 0, 3 ) === "nth" ) { + + // nth-* requires argument + if ( !match[ 3 ] ) { + Sizzle.error( match[ 0 ] ); + } + + // numeric x and y parameters for Expr.filter.CHILD + // remember that false/true cast respectively to 0/1 + match[ 4 ] = +( match[ 4 ] ? + match[ 5 ] + ( match[ 6 ] || 1 ) : + 2 * ( match[ 3 ] === "even" || match[ 3 ] === "odd" ) ); + match[ 5 ] = +( ( match[ 7 ] + match[ 8 ] ) || match[ 3 ] === "odd" ); + + // other types prohibit arguments + } else if ( match[ 3 ] ) { + Sizzle.error( match[ 0 ] ); + } + + return match; + }, + + "PSEUDO": function( match ) { + var excess, + unquoted = !match[ 6 ] && match[ 2 ]; + + if ( matchExpr[ "CHILD" ].test( match[ 0 ] ) ) { + return null; + } + + // Accept quoted arguments as-is + if ( match[ 3 ] ) { + match[ 2 ] = match[ 4 ] || match[ 5 ] || ""; + + // Strip excess characters from unquoted arguments + } else if ( unquoted && rpseudo.test( unquoted ) && + + // Get excess from tokenize (recursively) + ( excess = tokenize( unquoted, true ) ) && + + // advance to the next closing parenthesis + ( excess = unquoted.indexOf( ")", unquoted.length - excess ) - unquoted.length ) ) { + + // excess is a negative index + match[ 0 ] = match[ 0 ].slice( 0, excess ); + match[ 2 ] = unquoted.slice( 0, excess ); + } + + // Return only captures needed by the pseudo filter method (type and argument) + return match.slice( 0, 3 ); + } + }, + + filter: { + + "TAG": function( nodeNameSelector ) { + var nodeName = nodeNameSelector.replace( runescape, funescape ).toLowerCase(); + return nodeNameSelector === "*" ? + function() { + return true; + } : + function( elem ) { + return elem.nodeName && elem.nodeName.toLowerCase() === nodeName; + }; + }, + + "CLASS": function( className ) { + var pattern = classCache[ className + " " ]; + + return pattern || + ( pattern = new RegExp( "(^|" + whitespace + + ")" + className + "(" + whitespace + "|$)" ) ) && classCache( + className, function( elem ) { + return pattern.test( + typeof elem.className === "string" && elem.className || + typeof elem.getAttribute !== "undefined" && + elem.getAttribute( "class" ) || + "" + ); + } ); + }, + + "ATTR": function( name, operator, check ) { + return function( elem ) { + var result = Sizzle.attr( elem, name ); + + if ( result == null ) { + return operator === "!="; + } + if ( !operator ) { + return true; + } + + result += ""; + + /* eslint-disable max-len */ + + return operator === "=" ? result === check : + operator === "!=" ? result !== check : + operator === "^=" ? check && result.indexOf( check ) === 0 : + operator === "*=" ? check && result.indexOf( check ) > -1 : + operator === "$=" ? check && result.slice( -check.length ) === check : + operator === "~=" ? ( " " + result.replace( rwhitespace, " " ) + " " ).indexOf( check ) > -1 : + operator === "|=" ? result === check || result.slice( 0, check.length + 1 ) === check + "-" : + false; + /* eslint-enable max-len */ + + }; + }, + + "CHILD": function( type, what, _argument, first, last ) { + var simple = type.slice( 0, 3 ) !== "nth", + forward = type.slice( -4 ) !== "last", + ofType = what === "of-type"; + + return first === 1 && last === 0 ? + + // Shortcut for :nth-*(n) + function( elem ) { + return !!elem.parentNode; + } : + + function( elem, _context, xml ) { + var cache, uniqueCache, outerCache, node, nodeIndex, start, + dir = simple !== forward ? "nextSibling" : "previousSibling", + parent = elem.parentNode, + name = ofType && elem.nodeName.toLowerCase(), + useCache = !xml && !ofType, + diff = false; + + if ( parent ) { + + // :(first|last|only)-(child|of-type) + if ( simple ) { + while ( dir ) { + node = elem; + while ( ( node = node[ dir ] ) ) { + if ( ofType ? + node.nodeName.toLowerCase() === name : + node.nodeType === 1 ) { + + return false; + } + } + + // Reverse direction for :only-* (if we haven't yet done so) + start = dir = type === "only" && !start && "nextSibling"; + } + return true; + } + + start = [ forward ? parent.firstChild : parent.lastChild ]; + + // non-xml :nth-child(...) stores cache data on `parent` + if ( forward && useCache ) { + + // Seek `elem` from a previously-cached index + + // ...in a gzip-friendly way + node = parent; + outerCache = node[ expando ] || ( node[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ node.uniqueID ] || + ( outerCache[ node.uniqueID ] = {} ); + + cache = uniqueCache[ type ] || []; + nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; + diff = nodeIndex && cache[ 2 ]; + node = nodeIndex && parent.childNodes[ nodeIndex ]; + + while ( ( node = ++nodeIndex && node && node[ dir ] || + + // Fallback to seeking `elem` from the start + ( diff = nodeIndex = 0 ) || start.pop() ) ) { + + // When found, cache indexes on `parent` and break + if ( node.nodeType === 1 && ++diff && node === elem ) { + uniqueCache[ type ] = [ dirruns, nodeIndex, diff ]; + break; + } + } + + } else { + + // Use previously-cached element index if available + if ( useCache ) { + + // ...in a gzip-friendly way + node = elem; + outerCache = node[ expando ] || ( node[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ node.uniqueID ] || + ( outerCache[ node.uniqueID ] = {} ); + + cache = uniqueCache[ type ] || []; + nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; + diff = nodeIndex; + } + + // xml :nth-child(...) + // or :nth-last-child(...) or :nth(-last)?-of-type(...) + if ( diff === false ) { + + // Use the same loop as above to seek `elem` from the start + while ( ( node = ++nodeIndex && node && node[ dir ] || + ( diff = nodeIndex = 0 ) || start.pop() ) ) { + + if ( ( ofType ? + node.nodeName.toLowerCase() === name : + node.nodeType === 1 ) && + ++diff ) { + + // Cache the index of each encountered element + if ( useCache ) { + outerCache = node[ expando ] || + ( node[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ node.uniqueID ] || + ( outerCache[ node.uniqueID ] = {} ); + + uniqueCache[ type ] = [ dirruns, diff ]; + } + + if ( node === elem ) { + break; + } + } + } + } + } + + // Incorporate the offset, then check against cycle size + diff -= last; + return diff === first || ( diff % first === 0 && diff / first >= 0 ); + } + }; + }, + + "PSEUDO": function( pseudo, argument ) { + + // pseudo-class names are case-insensitive + // http://www.w3.org/TR/selectors/#pseudo-classes + // Prioritize by case sensitivity in case custom pseudos are added with uppercase letters + // Remember that setFilters inherits from pseudos + var args, + fn = Expr.pseudos[ pseudo ] || Expr.setFilters[ pseudo.toLowerCase() ] || + Sizzle.error( "unsupported pseudo: " + pseudo ); + + // The user may use createPseudo to indicate that + // arguments are needed to create the filter function + // just as Sizzle does + if ( fn[ expando ] ) { + return fn( argument ); + } + + // But maintain support for old signatures + if ( fn.length > 1 ) { + args = [ pseudo, pseudo, "", argument ]; + return Expr.setFilters.hasOwnProperty( pseudo.toLowerCase() ) ? + markFunction( function( seed, matches ) { + var idx, + matched = fn( seed, argument ), + i = matched.length; + while ( i-- ) { + idx = indexOf( seed, matched[ i ] ); + seed[ idx ] = !( matches[ idx ] = matched[ i ] ); + } + } ) : + function( elem ) { + return fn( elem, 0, args ); + }; + } + + return fn; + } + }, + + pseudos: { + + // Potentially complex pseudos + "not": markFunction( function( selector ) { + + // Trim the selector passed to compile + // to avoid treating leading and trailing + // spaces as combinators + var input = [], + results = [], + matcher = compile( selector.replace( rtrim, "$1" ) ); + + return matcher[ expando ] ? + markFunction( function( seed, matches, _context, xml ) { + var elem, + unmatched = matcher( seed, null, xml, [] ), + i = seed.length; + + // Match elements unmatched by `matcher` + while ( i-- ) { + if ( ( elem = unmatched[ i ] ) ) { + seed[ i ] = !( matches[ i ] = elem ); + } + } + } ) : + function( elem, _context, xml ) { + input[ 0 ] = elem; + matcher( input, null, xml, results ); + + // Don't keep the element (issue #299) + input[ 0 ] = null; + return !results.pop(); + }; + } ), + + "has": markFunction( function( selector ) { + return function( elem ) { + return Sizzle( selector, elem ).length > 0; + }; + } ), + + "contains": markFunction( function( text ) { + text = text.replace( runescape, funescape ); + return function( elem ) { + return ( elem.textContent || getText( elem ) ).indexOf( text ) > -1; + }; + } ), + + // "Whether an element is represented by a :lang() selector + // is based solely on the element's language value + // being equal to the identifier C, + // or beginning with the identifier C immediately followed by "-". + // The matching of C against the element's language value is performed case-insensitively. + // The identifier C does not have to be a valid language name." + // http://www.w3.org/TR/selectors/#lang-pseudo + "lang": markFunction( function( lang ) { + + // lang value must be a valid identifier + if ( !ridentifier.test( lang || "" ) ) { + Sizzle.error( "unsupported lang: " + lang ); + } + lang = lang.replace( runescape, funescape ).toLowerCase(); + return function( elem ) { + var elemLang; + do { + if ( ( elemLang = documentIsHTML ? + elem.lang : + elem.getAttribute( "xml:lang" ) || elem.getAttribute( "lang" ) ) ) { + + elemLang = elemLang.toLowerCase(); + return elemLang === lang || elemLang.indexOf( lang + "-" ) === 0; + } + } while ( ( elem = elem.parentNode ) && elem.nodeType === 1 ); + return false; + }; + } ), + + // Miscellaneous + "target": function( elem ) { + var hash = window.location && window.location.hash; + return hash && hash.slice( 1 ) === elem.id; + }, + + "root": function( elem ) { + return elem === docElem; + }, + + "focus": function( elem ) { + return elem === document.activeElement && + ( !document.hasFocus || document.hasFocus() ) && + !!( elem.type || elem.href || ~elem.tabIndex ); + }, + + // Boolean properties + "enabled": createDisabledPseudo( false ), + "disabled": createDisabledPseudo( true ), + + "checked": function( elem ) { + + // In CSS3, :checked should return both checked and selected elements + // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked + var nodeName = elem.nodeName.toLowerCase(); + return ( nodeName === "input" && !!elem.checked ) || + ( nodeName === "option" && !!elem.selected ); + }, + + "selected": function( elem ) { + + // Accessing this property makes selected-by-default + // options in Safari work properly + if ( elem.parentNode ) { + // eslint-disable-next-line no-unused-expressions + elem.parentNode.selectedIndex; + } + + return elem.selected === true; + }, + + // Contents + "empty": function( elem ) { + + // http://www.w3.org/TR/selectors/#empty-pseudo + // :empty is negated by element (1) or content nodes (text: 3; cdata: 4; entity ref: 5), + // but not by others (comment: 8; processing instruction: 7; etc.) + // nodeType < 6 works because attributes (2) do not appear as children + for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { + if ( elem.nodeType < 6 ) { + return false; + } + } + return true; + }, + + "parent": function( elem ) { + return !Expr.pseudos[ "empty" ]( elem ); + }, + + // Element/input types + "header": function( elem ) { + return rheader.test( elem.nodeName ); + }, + + "input": function( elem ) { + return rinputs.test( elem.nodeName ); + }, + + "button": function( elem ) { + var name = elem.nodeName.toLowerCase(); + return name === "input" && elem.type === "button" || name === "button"; + }, + + "text": function( elem ) { + var attr; + return elem.nodeName.toLowerCase() === "input" && + elem.type === "text" && + + // Support: IE<8 + // New HTML5 attribute values (e.g., "search") appear with elem.type === "text" + ( ( attr = elem.getAttribute( "type" ) ) == null || + attr.toLowerCase() === "text" ); + }, + + // Position-in-collection + "first": createPositionalPseudo( function() { + return [ 0 ]; + } ), + + "last": createPositionalPseudo( function( _matchIndexes, length ) { + return [ length - 1 ]; + } ), + + "eq": createPositionalPseudo( function( _matchIndexes, length, argument ) { + return [ argument < 0 ? argument + length : argument ]; + } ), + + "even": createPositionalPseudo( function( matchIndexes, length ) { + var i = 0; + for ( ; i < length; i += 2 ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ), + + "odd": createPositionalPseudo( function( matchIndexes, length ) { + var i = 1; + for ( ; i < length; i += 2 ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ), + + "lt": createPositionalPseudo( function( matchIndexes, length, argument ) { + var i = argument < 0 ? + argument + length : + argument > length ? + length : + argument; + for ( ; --i >= 0; ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ), + + "gt": createPositionalPseudo( function( matchIndexes, length, argument ) { + var i = argument < 0 ? argument + length : argument; + for ( ; ++i < length; ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ) + } +}; + +Expr.pseudos[ "nth" ] = Expr.pseudos[ "eq" ]; + +// Add button/input type pseudos +for ( i in { radio: true, checkbox: true, file: true, password: true, image: true } ) { + Expr.pseudos[ i ] = createInputPseudo( i ); +} +for ( i in { submit: true, reset: true } ) { + Expr.pseudos[ i ] = createButtonPseudo( i ); +} + +// Easy API for creating new setFilters +function setFilters() {} +setFilters.prototype = Expr.filters = Expr.pseudos; +Expr.setFilters = new setFilters(); + +tokenize = Sizzle.tokenize = function( selector, parseOnly ) { + var matched, match, tokens, type, + soFar, groups, preFilters, + cached = tokenCache[ selector + " " ]; + + if ( cached ) { + return parseOnly ? 0 : cached.slice( 0 ); + } + + soFar = selector; + groups = []; + preFilters = Expr.preFilter; + + while ( soFar ) { + + // Comma and first run + if ( !matched || ( match = rcomma.exec( soFar ) ) ) { + if ( match ) { + + // Don't consume trailing commas as valid + soFar = soFar.slice( match[ 0 ].length ) || soFar; + } + groups.push( ( tokens = [] ) ); + } + + matched = false; + + // Combinators + if ( ( match = rcombinators.exec( soFar ) ) ) { + matched = match.shift(); + tokens.push( { + value: matched, + + // Cast descendant combinators to space + type: match[ 0 ].replace( rtrim, " " ) + } ); + soFar = soFar.slice( matched.length ); + } + + // Filters + for ( type in Expr.filter ) { + if ( ( match = matchExpr[ type ].exec( soFar ) ) && ( !preFilters[ type ] || + ( match = preFilters[ type ]( match ) ) ) ) { + matched = match.shift(); + tokens.push( { + value: matched, + type: type, + matches: match + } ); + soFar = soFar.slice( matched.length ); + } + } + + if ( !matched ) { + break; + } + } + + // Return the length of the invalid excess + // if we're just parsing + // Otherwise, throw an error or return tokens + return parseOnly ? + soFar.length : + soFar ? + Sizzle.error( selector ) : + + // Cache the tokens + tokenCache( selector, groups ).slice( 0 ); +}; + +function toSelector( tokens ) { + var i = 0, + len = tokens.length, + selector = ""; + for ( ; i < len; i++ ) { + selector += tokens[ i ].value; + } + return selector; +} + +function addCombinator( matcher, combinator, base ) { + var dir = combinator.dir, + skip = combinator.next, + key = skip || dir, + checkNonElements = base && key === "parentNode", + doneName = done++; + + return combinator.first ? + + // Check against closest ancestor/preceding element + function( elem, context, xml ) { + while ( ( elem = elem[ dir ] ) ) { + if ( elem.nodeType === 1 || checkNonElements ) { + return matcher( elem, context, xml ); + } + } + return false; + } : + + // Check against all ancestor/preceding elements + function( elem, context, xml ) { + var oldCache, uniqueCache, outerCache, + newCache = [ dirruns, doneName ]; + + // We can't set arbitrary data on XML nodes, so they don't benefit from combinator caching + if ( xml ) { + while ( ( elem = elem[ dir ] ) ) { + if ( elem.nodeType === 1 || checkNonElements ) { + if ( matcher( elem, context, xml ) ) { + return true; + } + } + } + } else { + while ( ( elem = elem[ dir ] ) ) { + if ( elem.nodeType === 1 || checkNonElements ) { + outerCache = elem[ expando ] || ( elem[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ elem.uniqueID ] || + ( outerCache[ elem.uniqueID ] = {} ); + + if ( skip && skip === elem.nodeName.toLowerCase() ) { + elem = elem[ dir ] || elem; + } else if ( ( oldCache = uniqueCache[ key ] ) && + oldCache[ 0 ] === dirruns && oldCache[ 1 ] === doneName ) { + + // Assign to newCache so results back-propagate to previous elements + return ( newCache[ 2 ] = oldCache[ 2 ] ); + } else { + + // Reuse newcache so results back-propagate to previous elements + uniqueCache[ key ] = newCache; + + // A match means we're done; a fail means we have to keep checking + if ( ( newCache[ 2 ] = matcher( elem, context, xml ) ) ) { + return true; + } + } + } + } + } + return false; + }; +} + +function elementMatcher( matchers ) { + return matchers.length > 1 ? + function( elem, context, xml ) { + var i = matchers.length; + while ( i-- ) { + if ( !matchers[ i ]( elem, context, xml ) ) { + return false; + } + } + return true; + } : + matchers[ 0 ]; +} + +function multipleContexts( selector, contexts, results ) { + var i = 0, + len = contexts.length; + for ( ; i < len; i++ ) { + Sizzle( selector, contexts[ i ], results ); + } + return results; +} + +function condense( unmatched, map, filter, context, xml ) { + var elem, + newUnmatched = [], + i = 0, + len = unmatched.length, + mapped = map != null; + + for ( ; i < len; i++ ) { + if ( ( elem = unmatched[ i ] ) ) { + if ( !filter || filter( elem, context, xml ) ) { + newUnmatched.push( elem ); + if ( mapped ) { + map.push( i ); + } + } + } + } + + return newUnmatched; +} + +function setMatcher( preFilter, selector, matcher, postFilter, postFinder, postSelector ) { + if ( postFilter && !postFilter[ expando ] ) { + postFilter = setMatcher( postFilter ); + } + if ( postFinder && !postFinder[ expando ] ) { + postFinder = setMatcher( postFinder, postSelector ); + } + return markFunction( function( seed, results, context, xml ) { + var temp, i, elem, + preMap = [], + postMap = [], + preexisting = results.length, + + // Get initial elements from seed or context + elems = seed || multipleContexts( + selector || "*", + context.nodeType ? [ context ] : context, + [] + ), + + // Prefilter to get matcher input, preserving a map for seed-results synchronization + matcherIn = preFilter && ( seed || !selector ) ? + condense( elems, preMap, preFilter, context, xml ) : + elems, + + matcherOut = matcher ? + + // If we have a postFinder, or filtered seed, or non-seed postFilter or preexisting results, + postFinder || ( seed ? preFilter : preexisting || postFilter ) ? + + // ...intermediate processing is necessary + [] : + + // ...otherwise use results directly + results : + matcherIn; + + // Find primary matches + if ( matcher ) { + matcher( matcherIn, matcherOut, context, xml ); + } + + // Apply postFilter + if ( postFilter ) { + temp = condense( matcherOut, postMap ); + postFilter( temp, [], context, xml ); + + // Un-match failing elements by moving them back to matcherIn + i = temp.length; + while ( i-- ) { + if ( ( elem = temp[ i ] ) ) { + matcherOut[ postMap[ i ] ] = !( matcherIn[ postMap[ i ] ] = elem ); + } + } + } + + if ( seed ) { + if ( postFinder || preFilter ) { + if ( postFinder ) { + + // Get the final matcherOut by condensing this intermediate into postFinder contexts + temp = []; + i = matcherOut.length; + while ( i-- ) { + if ( ( elem = matcherOut[ i ] ) ) { + + // Restore matcherIn since elem is not yet a final match + temp.push( ( matcherIn[ i ] = elem ) ); + } + } + postFinder( null, ( matcherOut = [] ), temp, xml ); + } + + // Move matched elements from seed to results to keep them synchronized + i = matcherOut.length; + while ( i-- ) { + if ( ( elem = matcherOut[ i ] ) && + ( temp = postFinder ? indexOf( seed, elem ) : preMap[ i ] ) > -1 ) { + + seed[ temp ] = !( results[ temp ] = elem ); + } + } + } + + // Add elements to results, through postFinder if defined + } else { + matcherOut = condense( + matcherOut === results ? + matcherOut.splice( preexisting, matcherOut.length ) : + matcherOut + ); + if ( postFinder ) { + postFinder( null, results, matcherOut, xml ); + } else { + push.apply( results, matcherOut ); + } + } + } ); +} + +function matcherFromTokens( tokens ) { + var checkContext, matcher, j, + len = tokens.length, + leadingRelative = Expr.relative[ tokens[ 0 ].type ], + implicitRelative = leadingRelative || Expr.relative[ " " ], + i = leadingRelative ? 1 : 0, + + // The foundational matcher ensures that elements are reachable from top-level context(s) + matchContext = addCombinator( function( elem ) { + return elem === checkContext; + }, implicitRelative, true ), + matchAnyContext = addCombinator( function( elem ) { + return indexOf( checkContext, elem ) > -1; + }, implicitRelative, true ), + matchers = [ function( elem, context, xml ) { + var ret = ( !leadingRelative && ( xml || context !== outermostContext ) ) || ( + ( checkContext = context ).nodeType ? + matchContext( elem, context, xml ) : + matchAnyContext( elem, context, xml ) ); + + // Avoid hanging onto element (issue #299) + checkContext = null; + return ret; + } ]; + + for ( ; i < len; i++ ) { + if ( ( matcher = Expr.relative[ tokens[ i ].type ] ) ) { + matchers = [ addCombinator( elementMatcher( matchers ), matcher ) ]; + } else { + matcher = Expr.filter[ tokens[ i ].type ].apply( null, tokens[ i ].matches ); + + // Return special upon seeing a positional matcher + if ( matcher[ expando ] ) { + + // Find the next relative operator (if any) for proper handling + j = ++i; + for ( ; j < len; j++ ) { + if ( Expr.relative[ tokens[ j ].type ] ) { + break; + } + } + return setMatcher( + i > 1 && elementMatcher( matchers ), + i > 1 && toSelector( + + // If the preceding token was a descendant combinator, insert an implicit any-element `*` + tokens + .slice( 0, i - 1 ) + .concat( { value: tokens[ i - 2 ].type === " " ? "*" : "" } ) + ).replace( rtrim, "$1" ), + matcher, + i < j && matcherFromTokens( tokens.slice( i, j ) ), + j < len && matcherFromTokens( ( tokens = tokens.slice( j ) ) ), + j < len && toSelector( tokens ) + ); + } + matchers.push( matcher ); + } + } + + return elementMatcher( matchers ); +} + +function matcherFromGroupMatchers( elementMatchers, setMatchers ) { + var bySet = setMatchers.length > 0, + byElement = elementMatchers.length > 0, + superMatcher = function( seed, context, xml, results, outermost ) { + var elem, j, matcher, + matchedCount = 0, + i = "0", + unmatched = seed && [], + setMatched = [], + contextBackup = outermostContext, + + // We must always have either seed elements or outermost context + elems = seed || byElement && Expr.find[ "TAG" ]( "*", outermost ), + + // Use integer dirruns iff this is the outermost matcher + dirrunsUnique = ( dirruns += contextBackup == null ? 1 : Math.random() || 0.1 ), + len = elems.length; + + if ( outermost ) { + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + outermostContext = context == document || context || outermost; + } + + // Add elements passing elementMatchers directly to results + // Support: IE<9, Safari + // Tolerate NodeList properties (IE: "length"; Safari: ) matching elements by id + for ( ; i !== len && ( elem = elems[ i ] ) != null; i++ ) { + if ( byElement && elem ) { + j = 0; + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( !context && elem.ownerDocument != document ) { + setDocument( elem ); + xml = !documentIsHTML; + } + while ( ( matcher = elementMatchers[ j++ ] ) ) { + if ( matcher( elem, context || document, xml ) ) { + results.push( elem ); + break; + } + } + if ( outermost ) { + dirruns = dirrunsUnique; + } + } + + // Track unmatched elements for set filters + if ( bySet ) { + + // They will have gone through all possible matchers + if ( ( elem = !matcher && elem ) ) { + matchedCount--; + } + + // Lengthen the array for every element, matched or not + if ( seed ) { + unmatched.push( elem ); + } + } + } + + // `i` is now the count of elements visited above, and adding it to `matchedCount` + // makes the latter nonnegative. + matchedCount += i; + + // Apply set filters to unmatched elements + // NOTE: This can be skipped if there are no unmatched elements (i.e., `matchedCount` + // equals `i`), unless we didn't visit _any_ elements in the above loop because we have + // no element matchers and no seed. + // Incrementing an initially-string "0" `i` allows `i` to remain a string only in that + // case, which will result in a "00" `matchedCount` that differs from `i` but is also + // numerically zero. + if ( bySet && i !== matchedCount ) { + j = 0; + while ( ( matcher = setMatchers[ j++ ] ) ) { + matcher( unmatched, setMatched, context, xml ); + } + + if ( seed ) { + + // Reintegrate element matches to eliminate the need for sorting + if ( matchedCount > 0 ) { + while ( i-- ) { + if ( !( unmatched[ i ] || setMatched[ i ] ) ) { + setMatched[ i ] = pop.call( results ); + } + } + } + + // Discard index placeholder values to get only actual matches + setMatched = condense( setMatched ); + } + + // Add matches to results + push.apply( results, setMatched ); + + // Seedless set matches succeeding multiple successful matchers stipulate sorting + if ( outermost && !seed && setMatched.length > 0 && + ( matchedCount + setMatchers.length ) > 1 ) { + + Sizzle.uniqueSort( results ); + } + } + + // Override manipulation of globals by nested matchers + if ( outermost ) { + dirruns = dirrunsUnique; + outermostContext = contextBackup; + } + + return unmatched; + }; + + return bySet ? + markFunction( superMatcher ) : + superMatcher; +} + +compile = Sizzle.compile = function( selector, match /* Internal Use Only */ ) { + var i, + setMatchers = [], + elementMatchers = [], + cached = compilerCache[ selector + " " ]; + + if ( !cached ) { + + // Generate a function of recursive functions that can be used to check each element + if ( !match ) { + match = tokenize( selector ); + } + i = match.length; + while ( i-- ) { + cached = matcherFromTokens( match[ i ] ); + if ( cached[ expando ] ) { + setMatchers.push( cached ); + } else { + elementMatchers.push( cached ); + } + } + + // Cache the compiled function + cached = compilerCache( + selector, + matcherFromGroupMatchers( elementMatchers, setMatchers ) + ); + + // Save selector and tokenization + cached.selector = selector; + } + return cached; +}; + +/** + * A low-level selection function that works with Sizzle's compiled + * selector functions + * @param {String|Function} selector A selector or a pre-compiled + * selector function built with Sizzle.compile + * @param {Element} context + * @param {Array} [results] + * @param {Array} [seed] A set of elements to match against + */ +select = Sizzle.select = function( selector, context, results, seed ) { + var i, tokens, token, type, find, + compiled = typeof selector === "function" && selector, + match = !seed && tokenize( ( selector = compiled.selector || selector ) ); + + results = results || []; + + // Try to minimize operations if there is only one selector in the list and no seed + // (the latter of which guarantees us context) + if ( match.length === 1 ) { + + // Reduce context if the leading compound selector is an ID + tokens = match[ 0 ] = match[ 0 ].slice( 0 ); + if ( tokens.length > 2 && ( token = tokens[ 0 ] ).type === "ID" && + context.nodeType === 9 && documentIsHTML && Expr.relative[ tokens[ 1 ].type ] ) { + + context = ( Expr.find[ "ID" ]( token.matches[ 0 ] + .replace( runescape, funescape ), context ) || [] )[ 0 ]; + if ( !context ) { + return results; + + // Precompiled matchers will still verify ancestry, so step up a level + } else if ( compiled ) { + context = context.parentNode; + } + + selector = selector.slice( tokens.shift().value.length ); + } + + // Fetch a seed set for right-to-left matching + i = matchExpr[ "needsContext" ].test( selector ) ? 0 : tokens.length; + while ( i-- ) { + token = tokens[ i ]; + + // Abort if we hit a combinator + if ( Expr.relative[ ( type = token.type ) ] ) { + break; + } + if ( ( find = Expr.find[ type ] ) ) { + + // Search, expanding context for leading sibling combinators + if ( ( seed = find( + token.matches[ 0 ].replace( runescape, funescape ), + rsibling.test( tokens[ 0 ].type ) && testContext( context.parentNode ) || + context + ) ) ) { + + // If seed is empty or no tokens remain, we can return early + tokens.splice( i, 1 ); + selector = seed.length && toSelector( tokens ); + if ( !selector ) { + push.apply( results, seed ); + return results; + } + + break; + } + } + } + } + + // Compile and execute a filtering function if one is not provided + // Provide `match` to avoid retokenization if we modified the selector above + ( compiled || compile( selector, match ) )( + seed, + context, + !documentIsHTML, + results, + !context || rsibling.test( selector ) && testContext( context.parentNode ) || context + ); + return results; +}; + +// One-time assignments + +// Sort stability +support.sortStable = expando.split( "" ).sort( sortOrder ).join( "" ) === expando; + +// Support: Chrome 14-35+ +// Always assume duplicates if they aren't passed to the comparison function +support.detectDuplicates = !!hasDuplicate; + +// Initialize against the default document +setDocument(); + +// Support: Webkit<537.32 - Safari 6.0.3/Chrome 25 (fixed in Chrome 27) +// Detached nodes confoundingly follow *each other* +support.sortDetached = assert( function( el ) { + + // Should return 1, but returns 4 (following) + return el.compareDocumentPosition( document.createElement( "fieldset" ) ) & 1; +} ); + +// Support: IE<8 +// Prevent attribute/property "interpolation" +// https://msdn.microsoft.com/en-us/library/ms536429%28VS.85%29.aspx +if ( !assert( function( el ) { + el.innerHTML = ""; + return el.firstChild.getAttribute( "href" ) === "#"; +} ) ) { + addHandle( "type|href|height|width", function( elem, name, isXML ) { + if ( !isXML ) { + return elem.getAttribute( name, name.toLowerCase() === "type" ? 1 : 2 ); + } + } ); +} + +// Support: IE<9 +// Use defaultValue in place of getAttribute("value") +if ( !support.attributes || !assert( function( el ) { + el.innerHTML = ""; + el.firstChild.setAttribute( "value", "" ); + return el.firstChild.getAttribute( "value" ) === ""; +} ) ) { + addHandle( "value", function( elem, _name, isXML ) { + if ( !isXML && elem.nodeName.toLowerCase() === "input" ) { + return elem.defaultValue; + } + } ); +} + +// Support: IE<9 +// Use getAttributeNode to fetch booleans when getAttribute lies +if ( !assert( function( el ) { + return el.getAttribute( "disabled" ) == null; +} ) ) { + addHandle( booleans, function( elem, name, isXML ) { + var val; + if ( !isXML ) { + return elem[ name ] === true ? name.toLowerCase() : + ( val = elem.getAttributeNode( name ) ) && val.specified ? + val.value : + null; + } + } ); +} + +return Sizzle; + +} )( window ); + + + +jQuery.find = Sizzle; +jQuery.expr = Sizzle.selectors; + +// Deprecated +jQuery.expr[ ":" ] = jQuery.expr.pseudos; +jQuery.uniqueSort = jQuery.unique = Sizzle.uniqueSort; +jQuery.text = Sizzle.getText; +jQuery.isXMLDoc = Sizzle.isXML; +jQuery.contains = Sizzle.contains; +jQuery.escapeSelector = Sizzle.escape; + + + + +var dir = function( elem, dir, until ) { + var matched = [], + truncate = until !== undefined; + + while ( ( elem = elem[ dir ] ) && elem.nodeType !== 9 ) { + if ( elem.nodeType === 1 ) { + if ( truncate && jQuery( elem ).is( until ) ) { + break; + } + matched.push( elem ); + } + } + return matched; +}; + + +var siblings = function( n, elem ) { + var matched = []; + + for ( ; n; n = n.nextSibling ) { + if ( n.nodeType === 1 && n !== elem ) { + matched.push( n ); + } + } + + return matched; +}; + + +var rneedsContext = jQuery.expr.match.needsContext; + + + +function nodeName( elem, name ) { + + return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase(); + +}; +var rsingleTag = ( /^<([a-z][^\/\0>:\x20\t\r\n\f]*)[\x20\t\r\n\f]*\/?>(?:<\/\1>|)$/i ); + + + +// Implement the identical functionality for filter and not +function winnow( elements, qualifier, not ) { + if ( isFunction( qualifier ) ) { + return jQuery.grep( elements, function( elem, i ) { + return !!qualifier.call( elem, i, elem ) !== not; + } ); + } + + // Single element + if ( qualifier.nodeType ) { + return jQuery.grep( elements, function( elem ) { + return ( elem === qualifier ) !== not; + } ); + } + + // Arraylike of elements (jQuery, arguments, Array) + if ( typeof qualifier !== "string" ) { + return jQuery.grep( elements, function( elem ) { + return ( indexOf.call( qualifier, elem ) > -1 ) !== not; + } ); + } + + // Filtered directly for both simple and complex selectors + return jQuery.filter( qualifier, elements, not ); +} + +jQuery.filter = function( expr, elems, not ) { + var elem = elems[ 0 ]; + + if ( not ) { + expr = ":not(" + expr + ")"; + } + + if ( elems.length === 1 && elem.nodeType === 1 ) { + return jQuery.find.matchesSelector( elem, expr ) ? [ elem ] : []; + } + + return jQuery.find.matches( expr, jQuery.grep( elems, function( elem ) { + return elem.nodeType === 1; + } ) ); +}; + +jQuery.fn.extend( { + find: function( selector ) { + var i, ret, + len = this.length, + self = this; + + if ( typeof selector !== "string" ) { + return this.pushStack( jQuery( selector ).filter( function() { + for ( i = 0; i < len; i++ ) { + if ( jQuery.contains( self[ i ], this ) ) { + return true; + } + } + } ) ); + } + + ret = this.pushStack( [] ); + + for ( i = 0; i < len; i++ ) { + jQuery.find( selector, self[ i ], ret ); + } + + return len > 1 ? jQuery.uniqueSort( ret ) : ret; + }, + filter: function( selector ) { + return this.pushStack( winnow( this, selector || [], false ) ); + }, + not: function( selector ) { + return this.pushStack( winnow( this, selector || [], true ) ); + }, + is: function( selector ) { + return !!winnow( + this, + + // If this is a positional/relative selector, check membership in the returned set + // so $("p:first").is("p:last") won't return true for a doc with two "p". + typeof selector === "string" && rneedsContext.test( selector ) ? + jQuery( selector ) : + selector || [], + false + ).length; + } +} ); + + +// Initialize a jQuery object + + +// A central reference to the root jQuery(document) +var rootjQuery, + + // A simple way to check for HTML strings + // Prioritize #id over to avoid XSS via location.hash (#9521) + // Strict HTML recognition (#11290: must start with <) + // Shortcut simple #id case for speed + rquickExpr = /^(?:\s*(<[\w\W]+>)[^>]*|#([\w-]+))$/, + + init = jQuery.fn.init = function( selector, context, root ) { + var match, elem; + + // HANDLE: $(""), $(null), $(undefined), $(false) + if ( !selector ) { + return this; + } + + // Method init() accepts an alternate rootjQuery + // so migrate can support jQuery.sub (gh-2101) + root = root || rootjQuery; + + // Handle HTML strings + if ( typeof selector === "string" ) { + if ( selector[ 0 ] === "<" && + selector[ selector.length - 1 ] === ">" && + selector.length >= 3 ) { + + // Assume that strings that start and end with <> are HTML and skip the regex check + match = [ null, selector, null ]; + + } else { + match = rquickExpr.exec( selector ); + } + + // Match html or make sure no context is specified for #id + if ( match && ( match[ 1 ] || !context ) ) { + + // HANDLE: $(html) -> $(array) + if ( match[ 1 ] ) { + context = context instanceof jQuery ? context[ 0 ] : context; + + // Option to run scripts is true for back-compat + // Intentionally let the error be thrown if parseHTML is not present + jQuery.merge( this, jQuery.parseHTML( + match[ 1 ], + context && context.nodeType ? context.ownerDocument || context : document, + true + ) ); + + // HANDLE: $(html, props) + if ( rsingleTag.test( match[ 1 ] ) && jQuery.isPlainObject( context ) ) { + for ( match in context ) { + + // Properties of context are called as methods if possible + if ( isFunction( this[ match ] ) ) { + this[ match ]( context[ match ] ); + + // ...and otherwise set as attributes + } else { + this.attr( match, context[ match ] ); + } + } + } + + return this; + + // HANDLE: $(#id) + } else { + elem = document.getElementById( match[ 2 ] ); + + if ( elem ) { + + // Inject the element directly into the jQuery object + this[ 0 ] = elem; + this.length = 1; + } + return this; + } + + // HANDLE: $(expr, $(...)) + } else if ( !context || context.jquery ) { + return ( context || root ).find( selector ); + + // HANDLE: $(expr, context) + // (which is just equivalent to: $(context).find(expr) + } else { + return this.constructor( context ).find( selector ); + } + + // HANDLE: $(DOMElement) + } else if ( selector.nodeType ) { + this[ 0 ] = selector; + this.length = 1; + return this; + + // HANDLE: $(function) + // Shortcut for document ready + } else if ( isFunction( selector ) ) { + return root.ready !== undefined ? + root.ready( selector ) : + + // Execute immediately if ready is not present + selector( jQuery ); + } + + return jQuery.makeArray( selector, this ); + }; + +// Give the init function the jQuery prototype for later instantiation +init.prototype = jQuery.fn; + +// Initialize central reference +rootjQuery = jQuery( document ); + + +var rparentsprev = /^(?:parents|prev(?:Until|All))/, + + // Methods guaranteed to produce a unique set when starting from a unique set + guaranteedUnique = { + children: true, + contents: true, + next: true, + prev: true + }; + +jQuery.fn.extend( { + has: function( target ) { + var targets = jQuery( target, this ), + l = targets.length; + + return this.filter( function() { + var i = 0; + for ( ; i < l; i++ ) { + if ( jQuery.contains( this, targets[ i ] ) ) { + return true; + } + } + } ); + }, + + closest: function( selectors, context ) { + var cur, + i = 0, + l = this.length, + matched = [], + targets = typeof selectors !== "string" && jQuery( selectors ); + + // Positional selectors never match, since there's no _selection_ context + if ( !rneedsContext.test( selectors ) ) { + for ( ; i < l; i++ ) { + for ( cur = this[ i ]; cur && cur !== context; cur = cur.parentNode ) { + + // Always skip document fragments + if ( cur.nodeType < 11 && ( targets ? + targets.index( cur ) > -1 : + + // Don't pass non-elements to Sizzle + cur.nodeType === 1 && + jQuery.find.matchesSelector( cur, selectors ) ) ) { + + matched.push( cur ); + break; + } + } + } + } + + return this.pushStack( matched.length > 1 ? jQuery.uniqueSort( matched ) : matched ); + }, + + // Determine the position of an element within the set + index: function( elem ) { + + // No argument, return index in parent + if ( !elem ) { + return ( this[ 0 ] && this[ 0 ].parentNode ) ? this.first().prevAll().length : -1; + } + + // Index in selector + if ( typeof elem === "string" ) { + return indexOf.call( jQuery( elem ), this[ 0 ] ); + } + + // Locate the position of the desired element + return indexOf.call( this, + + // If it receives a jQuery object, the first element is used + elem.jquery ? elem[ 0 ] : elem + ); + }, + + add: function( selector, context ) { + return this.pushStack( + jQuery.uniqueSort( + jQuery.merge( this.get(), jQuery( selector, context ) ) + ) + ); + }, + + addBack: function( selector ) { + return this.add( selector == null ? + this.prevObject : this.prevObject.filter( selector ) + ); + } +} ); + +function sibling( cur, dir ) { + while ( ( cur = cur[ dir ] ) && cur.nodeType !== 1 ) {} + return cur; +} + +jQuery.each( { + parent: function( elem ) { + var parent = elem.parentNode; + return parent && parent.nodeType !== 11 ? parent : null; + }, + parents: function( elem ) { + return dir( elem, "parentNode" ); + }, + parentsUntil: function( elem, _i, until ) { + return dir( elem, "parentNode", until ); + }, + next: function( elem ) { + return sibling( elem, "nextSibling" ); + }, + prev: function( elem ) { + return sibling( elem, "previousSibling" ); + }, + nextAll: function( elem ) { + return dir( elem, "nextSibling" ); + }, + prevAll: function( elem ) { + return dir( elem, "previousSibling" ); + }, + nextUntil: function( elem, _i, until ) { + return dir( elem, "nextSibling", until ); + }, + prevUntil: function( elem, _i, until ) { + return dir( elem, "previousSibling", until ); + }, + siblings: function( elem ) { + return siblings( ( elem.parentNode || {} ).firstChild, elem ); + }, + children: function( elem ) { + return siblings( elem.firstChild ); + }, + contents: function( elem ) { + if ( elem.contentDocument != null && + + // Support: IE 11+ + // elements with no `data` attribute has an object + // `contentDocument` with a `null` prototype. + getProto( elem.contentDocument ) ) { + + return elem.contentDocument; + } + + // Support: IE 9 - 11 only, iOS 7 only, Android Browser <=4.3 only + // Treat the template element as a regular one in browsers that + // don't support it. + if ( nodeName( elem, "template" ) ) { + elem = elem.content || elem; + } + + return jQuery.merge( [], elem.childNodes ); + } +}, function( name, fn ) { + jQuery.fn[ name ] = function( until, selector ) { + var matched = jQuery.map( this, fn, until ); + + if ( name.slice( -5 ) !== "Until" ) { + selector = until; + } + + if ( selector && typeof selector === "string" ) { + matched = jQuery.filter( selector, matched ); + } + + if ( this.length > 1 ) { + + // Remove duplicates + if ( !guaranteedUnique[ name ] ) { + jQuery.uniqueSort( matched ); + } + + // Reverse order for parents* and prev-derivatives + if ( rparentsprev.test( name ) ) { + matched.reverse(); + } + } + + return this.pushStack( matched ); + }; +} ); +var rnothtmlwhite = ( /[^\x20\t\r\n\f]+/g ); + + + +// Convert String-formatted options into Object-formatted ones +function createOptions( options ) { + var object = {}; + jQuery.each( options.match( rnothtmlwhite ) || [], function( _, flag ) { + object[ flag ] = true; + } ); + return object; +} + +/* + * Create a callback list using the following parameters: + * + * options: an optional list of space-separated options that will change how + * the callback list behaves or a more traditional option object + * + * By default a callback list will act like an event callback list and can be + * "fired" multiple times. + * + * Possible options: + * + * once: will ensure the callback list can only be fired once (like a Deferred) + * + * memory: will keep track of previous values and will call any callback added + * after the list has been fired right away with the latest "memorized" + * values (like a Deferred) + * + * unique: will ensure a callback can only be added once (no duplicate in the list) + * + * stopOnFalse: interrupt callings when a callback returns false + * + */ +jQuery.Callbacks = function( options ) { + + // Convert options from String-formatted to Object-formatted if needed + // (we check in cache first) + options = typeof options === "string" ? + createOptions( options ) : + jQuery.extend( {}, options ); + + var // Flag to know if list is currently firing + firing, + + // Last fire value for non-forgettable lists + memory, + + // Flag to know if list was already fired + fired, + + // Flag to prevent firing + locked, + + // Actual callback list + list = [], + + // Queue of execution data for repeatable lists + queue = [], + + // Index of currently firing callback (modified by add/remove as needed) + firingIndex = -1, + + // Fire callbacks + fire = function() { + + // Enforce single-firing + locked = locked || options.once; + + // Execute callbacks for all pending executions, + // respecting firingIndex overrides and runtime changes + fired = firing = true; + for ( ; queue.length; firingIndex = -1 ) { + memory = queue.shift(); + while ( ++firingIndex < list.length ) { + + // Run callback and check for early termination + if ( list[ firingIndex ].apply( memory[ 0 ], memory[ 1 ] ) === false && + options.stopOnFalse ) { + + // Jump to end and forget the data so .add doesn't re-fire + firingIndex = list.length; + memory = false; + } + } + } + + // Forget the data if we're done with it + if ( !options.memory ) { + memory = false; + } + + firing = false; + + // Clean up if we're done firing for good + if ( locked ) { + + // Keep an empty list if we have data for future add calls + if ( memory ) { + list = []; + + // Otherwise, this object is spent + } else { + list = ""; + } + } + }, + + // Actual Callbacks object + self = { + + // Add a callback or a collection of callbacks to the list + add: function() { + if ( list ) { + + // If we have memory from a past run, we should fire after adding + if ( memory && !firing ) { + firingIndex = list.length - 1; + queue.push( memory ); + } + + ( function add( args ) { + jQuery.each( args, function( _, arg ) { + if ( isFunction( arg ) ) { + if ( !options.unique || !self.has( arg ) ) { + list.push( arg ); + } + } else if ( arg && arg.length && toType( arg ) !== "string" ) { + + // Inspect recursively + add( arg ); + } + } ); + } )( arguments ); + + if ( memory && !firing ) { + fire(); + } + } + return this; + }, + + // Remove a callback from the list + remove: function() { + jQuery.each( arguments, function( _, arg ) { + var index; + while ( ( index = jQuery.inArray( arg, list, index ) ) > -1 ) { + list.splice( index, 1 ); + + // Handle firing indexes + if ( index <= firingIndex ) { + firingIndex--; + } + } + } ); + return this; + }, + + // Check if a given callback is in the list. + // If no argument is given, return whether or not list has callbacks attached. + has: function( fn ) { + return fn ? + jQuery.inArray( fn, list ) > -1 : + list.length > 0; + }, + + // Remove all callbacks from the list + empty: function() { + if ( list ) { + list = []; + } + return this; + }, + + // Disable .fire and .add + // Abort any current/pending executions + // Clear all callbacks and values + disable: function() { + locked = queue = []; + list = memory = ""; + return this; + }, + disabled: function() { + return !list; + }, + + // Disable .fire + // Also disable .add unless we have memory (since it would have no effect) + // Abort any pending executions + lock: function() { + locked = queue = []; + if ( !memory && !firing ) { + list = memory = ""; + } + return this; + }, + locked: function() { + return !!locked; + }, + + // Call all callbacks with the given context and arguments + fireWith: function( context, args ) { + if ( !locked ) { + args = args || []; + args = [ context, args.slice ? args.slice() : args ]; + queue.push( args ); + if ( !firing ) { + fire(); + } + } + return this; + }, + + // Call all the callbacks with the given arguments + fire: function() { + self.fireWith( this, arguments ); + return this; + }, + + // To know if the callbacks have already been called at least once + fired: function() { + return !!fired; + } + }; + + return self; +}; + + +function Identity( v ) { + return v; +} +function Thrower( ex ) { + throw ex; +} + +function adoptValue( value, resolve, reject, noValue ) { + var method; + + try { + + // Check for promise aspect first to privilege synchronous behavior + if ( value && isFunction( ( method = value.promise ) ) ) { + method.call( value ).done( resolve ).fail( reject ); + + // Other thenables + } else if ( value && isFunction( ( method = value.then ) ) ) { + method.call( value, resolve, reject ); + + // Other non-thenables + } else { + + // Control `resolve` arguments by letting Array#slice cast boolean `noValue` to integer: + // * false: [ value ].slice( 0 ) => resolve( value ) + // * true: [ value ].slice( 1 ) => resolve() + resolve.apply( undefined, [ value ].slice( noValue ) ); + } + + // For Promises/A+, convert exceptions into rejections + // Since jQuery.when doesn't unwrap thenables, we can skip the extra checks appearing in + // Deferred#then to conditionally suppress rejection. + } catch ( value ) { + + // Support: Android 4.0 only + // Strict mode functions invoked without .call/.apply get global-object context + reject.apply( undefined, [ value ] ); + } +} + +jQuery.extend( { + + Deferred: function( func ) { + var tuples = [ + + // action, add listener, callbacks, + // ... .then handlers, argument index, [final state] + [ "notify", "progress", jQuery.Callbacks( "memory" ), + jQuery.Callbacks( "memory" ), 2 ], + [ "resolve", "done", jQuery.Callbacks( "once memory" ), + jQuery.Callbacks( "once memory" ), 0, "resolved" ], + [ "reject", "fail", jQuery.Callbacks( "once memory" ), + jQuery.Callbacks( "once memory" ), 1, "rejected" ] + ], + state = "pending", + promise = { + state: function() { + return state; + }, + always: function() { + deferred.done( arguments ).fail( arguments ); + return this; + }, + "catch": function( fn ) { + return promise.then( null, fn ); + }, + + // Keep pipe for back-compat + pipe: function( /* fnDone, fnFail, fnProgress */ ) { + var fns = arguments; + + return jQuery.Deferred( function( newDefer ) { + jQuery.each( tuples, function( _i, tuple ) { + + // Map tuples (progress, done, fail) to arguments (done, fail, progress) + var fn = isFunction( fns[ tuple[ 4 ] ] ) && fns[ tuple[ 4 ] ]; + + // deferred.progress(function() { bind to newDefer or newDefer.notify }) + // deferred.done(function() { bind to newDefer or newDefer.resolve }) + // deferred.fail(function() { bind to newDefer or newDefer.reject }) + deferred[ tuple[ 1 ] ]( function() { + var returned = fn && fn.apply( this, arguments ); + if ( returned && isFunction( returned.promise ) ) { + returned.promise() + .progress( newDefer.notify ) + .done( newDefer.resolve ) + .fail( newDefer.reject ); + } else { + newDefer[ tuple[ 0 ] + "With" ]( + this, + fn ? [ returned ] : arguments + ); + } + } ); + } ); + fns = null; + } ).promise(); + }, + then: function( onFulfilled, onRejected, onProgress ) { + var maxDepth = 0; + function resolve( depth, deferred, handler, special ) { + return function() { + var that = this, + args = arguments, + mightThrow = function() { + var returned, then; + + // Support: Promises/A+ section 2.3.3.3.3 + // https://promisesaplus.com/#point-59 + // Ignore double-resolution attempts + if ( depth < maxDepth ) { + return; + } + + returned = handler.apply( that, args ); + + // Support: Promises/A+ section 2.3.1 + // https://promisesaplus.com/#point-48 + if ( returned === deferred.promise() ) { + throw new TypeError( "Thenable self-resolution" ); + } + + // Support: Promises/A+ sections 2.3.3.1, 3.5 + // https://promisesaplus.com/#point-54 + // https://promisesaplus.com/#point-75 + // Retrieve `then` only once + then = returned && + + // Support: Promises/A+ section 2.3.4 + // https://promisesaplus.com/#point-64 + // Only check objects and functions for thenability + ( typeof returned === "object" || + typeof returned === "function" ) && + returned.then; + + // Handle a returned thenable + if ( isFunction( then ) ) { + + // Special processors (notify) just wait for resolution + if ( special ) { + then.call( + returned, + resolve( maxDepth, deferred, Identity, special ), + resolve( maxDepth, deferred, Thrower, special ) + ); + + // Normal processors (resolve) also hook into progress + } else { + + // ...and disregard older resolution values + maxDepth++; + + then.call( + returned, + resolve( maxDepth, deferred, Identity, special ), + resolve( maxDepth, deferred, Thrower, special ), + resolve( maxDepth, deferred, Identity, + deferred.notifyWith ) + ); + } + + // Handle all other returned values + } else { + + // Only substitute handlers pass on context + // and multiple values (non-spec behavior) + if ( handler !== Identity ) { + that = undefined; + args = [ returned ]; + } + + // Process the value(s) + // Default process is resolve + ( special || deferred.resolveWith )( that, args ); + } + }, + + // Only normal processors (resolve) catch and reject exceptions + process = special ? + mightThrow : + function() { + try { + mightThrow(); + } catch ( e ) { + + if ( jQuery.Deferred.exceptionHook ) { + jQuery.Deferred.exceptionHook( e, + process.stackTrace ); + } + + // Support: Promises/A+ section 2.3.3.3.4.1 + // https://promisesaplus.com/#point-61 + // Ignore post-resolution exceptions + if ( depth + 1 >= maxDepth ) { + + // Only substitute handlers pass on context + // and multiple values (non-spec behavior) + if ( handler !== Thrower ) { + that = undefined; + args = [ e ]; + } + + deferred.rejectWith( that, args ); + } + } + }; + + // Support: Promises/A+ section 2.3.3.3.1 + // https://promisesaplus.com/#point-57 + // Re-resolve promises immediately to dodge false rejection from + // subsequent errors + if ( depth ) { + process(); + } else { + + // Call an optional hook to record the stack, in case of exception + // since it's otherwise lost when execution goes async + if ( jQuery.Deferred.getStackHook ) { + process.stackTrace = jQuery.Deferred.getStackHook(); + } + window.setTimeout( process ); + } + }; + } + + return jQuery.Deferred( function( newDefer ) { + + // progress_handlers.add( ... ) + tuples[ 0 ][ 3 ].add( + resolve( + 0, + newDefer, + isFunction( onProgress ) ? + onProgress : + Identity, + newDefer.notifyWith + ) + ); + + // fulfilled_handlers.add( ... ) + tuples[ 1 ][ 3 ].add( + resolve( + 0, + newDefer, + isFunction( onFulfilled ) ? + onFulfilled : + Identity + ) + ); + + // rejected_handlers.add( ... ) + tuples[ 2 ][ 3 ].add( + resolve( + 0, + newDefer, + isFunction( onRejected ) ? + onRejected : + Thrower + ) + ); + } ).promise(); + }, + + // Get a promise for this deferred + // If obj is provided, the promise aspect is added to the object + promise: function( obj ) { + return obj != null ? jQuery.extend( obj, promise ) : promise; + } + }, + deferred = {}; + + // Add list-specific methods + jQuery.each( tuples, function( i, tuple ) { + var list = tuple[ 2 ], + stateString = tuple[ 5 ]; + + // promise.progress = list.add + // promise.done = list.add + // promise.fail = list.add + promise[ tuple[ 1 ] ] = list.add; + + // Handle state + if ( stateString ) { + list.add( + function() { + + // state = "resolved" (i.e., fulfilled) + // state = "rejected" + state = stateString; + }, + + // rejected_callbacks.disable + // fulfilled_callbacks.disable + tuples[ 3 - i ][ 2 ].disable, + + // rejected_handlers.disable + // fulfilled_handlers.disable + tuples[ 3 - i ][ 3 ].disable, + + // progress_callbacks.lock + tuples[ 0 ][ 2 ].lock, + + // progress_handlers.lock + tuples[ 0 ][ 3 ].lock + ); + } + + // progress_handlers.fire + // fulfilled_handlers.fire + // rejected_handlers.fire + list.add( tuple[ 3 ].fire ); + + // deferred.notify = function() { deferred.notifyWith(...) } + // deferred.resolve = function() { deferred.resolveWith(...) } + // deferred.reject = function() { deferred.rejectWith(...) } + deferred[ tuple[ 0 ] ] = function() { + deferred[ tuple[ 0 ] + "With" ]( this === deferred ? undefined : this, arguments ); + return this; + }; + + // deferred.notifyWith = list.fireWith + // deferred.resolveWith = list.fireWith + // deferred.rejectWith = list.fireWith + deferred[ tuple[ 0 ] + "With" ] = list.fireWith; + } ); + + // Make the deferred a promise + promise.promise( deferred ); + + // Call given func if any + if ( func ) { + func.call( deferred, deferred ); + } + + // All done! + return deferred; + }, + + // Deferred helper + when: function( singleValue ) { + var + + // count of uncompleted subordinates + remaining = arguments.length, + + // count of unprocessed arguments + i = remaining, + + // subordinate fulfillment data + resolveContexts = Array( i ), + resolveValues = slice.call( arguments ), + + // the master Deferred + master = jQuery.Deferred(), + + // subordinate callback factory + updateFunc = function( i ) { + return function( value ) { + resolveContexts[ i ] = this; + resolveValues[ i ] = arguments.length > 1 ? slice.call( arguments ) : value; + if ( !( --remaining ) ) { + master.resolveWith( resolveContexts, resolveValues ); + } + }; + }; + + // Single- and empty arguments are adopted like Promise.resolve + if ( remaining <= 1 ) { + adoptValue( singleValue, master.done( updateFunc( i ) ).resolve, master.reject, + !remaining ); + + // Use .then() to unwrap secondary thenables (cf. gh-3000) + if ( master.state() === "pending" || + isFunction( resolveValues[ i ] && resolveValues[ i ].then ) ) { + + return master.then(); + } + } + + // Multiple arguments are aggregated like Promise.all array elements + while ( i-- ) { + adoptValue( resolveValues[ i ], updateFunc( i ), master.reject ); + } + + return master.promise(); + } +} ); + + +// These usually indicate a programmer mistake during development, +// warn about them ASAP rather than swallowing them by default. +var rerrorNames = /^(Eval|Internal|Range|Reference|Syntax|Type|URI)Error$/; + +jQuery.Deferred.exceptionHook = function( error, stack ) { + + // Support: IE 8 - 9 only + // Console exists when dev tools are open, which can happen at any time + if ( window.console && window.console.warn && error && rerrorNames.test( error.name ) ) { + window.console.warn( "jQuery.Deferred exception: " + error.message, error.stack, stack ); + } +}; + + + + +jQuery.readyException = function( error ) { + window.setTimeout( function() { + throw error; + } ); +}; + + + + +// The deferred used on DOM ready +var readyList = jQuery.Deferred(); + +jQuery.fn.ready = function( fn ) { + + readyList + .then( fn ) + + // Wrap jQuery.readyException in a function so that the lookup + // happens at the time of error handling instead of callback + // registration. + .catch( function( error ) { + jQuery.readyException( error ); + } ); + + return this; +}; + +jQuery.extend( { + + // Is the DOM ready to be used? Set to true once it occurs. + isReady: false, + + // A counter to track how many items to wait for before + // the ready event fires. See #6781 + readyWait: 1, + + // Handle when the DOM is ready + ready: function( wait ) { + + // Abort if there are pending holds or we're already ready + if ( wait === true ? --jQuery.readyWait : jQuery.isReady ) { + return; + } + + // Remember that the DOM is ready + jQuery.isReady = true; + + // If a normal DOM Ready event fired, decrement, and wait if need be + if ( wait !== true && --jQuery.readyWait > 0 ) { + return; + } + + // If there are functions bound, to execute + readyList.resolveWith( document, [ jQuery ] ); + } +} ); + +jQuery.ready.then = readyList.then; + +// The ready event handler and self cleanup method +function completed() { + document.removeEventListener( "DOMContentLoaded", completed ); + window.removeEventListener( "load", completed ); + jQuery.ready(); +} + +// Catch cases where $(document).ready() is called +// after the browser event has already occurred. +// Support: IE <=9 - 10 only +// Older IE sometimes signals "interactive" too soon +if ( document.readyState === "complete" || + ( document.readyState !== "loading" && !document.documentElement.doScroll ) ) { + + // Handle it asynchronously to allow scripts the opportunity to delay ready + window.setTimeout( jQuery.ready ); + +} else { + + // Use the handy event callback + document.addEventListener( "DOMContentLoaded", completed ); + + // A fallback to window.onload, that will always work + window.addEventListener( "load", completed ); +} + + + + +// Multifunctional method to get and set values of a collection +// The value/s can optionally be executed if it's a function +var access = function( elems, fn, key, value, chainable, emptyGet, raw ) { + var i = 0, + len = elems.length, + bulk = key == null; + + // Sets many values + if ( toType( key ) === "object" ) { + chainable = true; + for ( i in key ) { + access( elems, fn, i, key[ i ], true, emptyGet, raw ); + } + + // Sets one value + } else if ( value !== undefined ) { + chainable = true; + + if ( !isFunction( value ) ) { + raw = true; + } + + if ( bulk ) { + + // Bulk operations run against the entire set + if ( raw ) { + fn.call( elems, value ); + fn = null; + + // ...except when executing function values + } else { + bulk = fn; + fn = function( elem, _key, value ) { + return bulk.call( jQuery( elem ), value ); + }; + } + } + + if ( fn ) { + for ( ; i < len; i++ ) { + fn( + elems[ i ], key, raw ? + value : + value.call( elems[ i ], i, fn( elems[ i ], key ) ) + ); + } + } + } + + if ( chainable ) { + return elems; + } + + // Gets + if ( bulk ) { + return fn.call( elems ); + } + + return len ? fn( elems[ 0 ], key ) : emptyGet; +}; + + +// Matches dashed string for camelizing +var rmsPrefix = /^-ms-/, + rdashAlpha = /-([a-z])/g; + +// Used by camelCase as callback to replace() +function fcamelCase( _all, letter ) { + return letter.toUpperCase(); +} + +// Convert dashed to camelCase; used by the css and data modules +// Support: IE <=9 - 11, Edge 12 - 15 +// Microsoft forgot to hump their vendor prefix (#9572) +function camelCase( string ) { + return string.replace( rmsPrefix, "ms-" ).replace( rdashAlpha, fcamelCase ); +} +var acceptData = function( owner ) { + + // Accepts only: + // - Node + // - Node.ELEMENT_NODE + // - Node.DOCUMENT_NODE + // - Object + // - Any + return owner.nodeType === 1 || owner.nodeType === 9 || !( +owner.nodeType ); +}; + + + + +function Data() { + this.expando = jQuery.expando + Data.uid++; +} + +Data.uid = 1; + +Data.prototype = { + + cache: function( owner ) { + + // Check if the owner object already has a cache + var value = owner[ this.expando ]; + + // If not, create one + if ( !value ) { + value = {}; + + // We can accept data for non-element nodes in modern browsers, + // but we should not, see #8335. + // Always return an empty object. + if ( acceptData( owner ) ) { + + // If it is a node unlikely to be stringify-ed or looped over + // use plain assignment + if ( owner.nodeType ) { + owner[ this.expando ] = value; + + // Otherwise secure it in a non-enumerable property + // configurable must be true to allow the property to be + // deleted when data is removed + } else { + Object.defineProperty( owner, this.expando, { + value: value, + configurable: true + } ); + } + } + } + + return value; + }, + set: function( owner, data, value ) { + var prop, + cache = this.cache( owner ); + + // Handle: [ owner, key, value ] args + // Always use camelCase key (gh-2257) + if ( typeof data === "string" ) { + cache[ camelCase( data ) ] = value; + + // Handle: [ owner, { properties } ] args + } else { + + // Copy the properties one-by-one to the cache object + for ( prop in data ) { + cache[ camelCase( prop ) ] = data[ prop ]; + } + } + return cache; + }, + get: function( owner, key ) { + return key === undefined ? + this.cache( owner ) : + + // Always use camelCase key (gh-2257) + owner[ this.expando ] && owner[ this.expando ][ camelCase( key ) ]; + }, + access: function( owner, key, value ) { + + // In cases where either: + // + // 1. No key was specified + // 2. A string key was specified, but no value provided + // + // Take the "read" path and allow the get method to determine + // which value to return, respectively either: + // + // 1. The entire cache object + // 2. The data stored at the key + // + if ( key === undefined || + ( ( key && typeof key === "string" ) && value === undefined ) ) { + + return this.get( owner, key ); + } + + // When the key is not a string, or both a key and value + // are specified, set or extend (existing objects) with either: + // + // 1. An object of properties + // 2. A key and value + // + this.set( owner, key, value ); + + // Since the "set" path can have two possible entry points + // return the expected data based on which path was taken[*] + return value !== undefined ? value : key; + }, + remove: function( owner, key ) { + var i, + cache = owner[ this.expando ]; + + if ( cache === undefined ) { + return; + } + + if ( key !== undefined ) { + + // Support array or space separated string of keys + if ( Array.isArray( key ) ) { + + // If key is an array of keys... + // We always set camelCase keys, so remove that. + key = key.map( camelCase ); + } else { + key = camelCase( key ); + + // If a key with the spaces exists, use it. + // Otherwise, create an array by matching non-whitespace + key = key in cache ? + [ key ] : + ( key.match( rnothtmlwhite ) || [] ); + } + + i = key.length; + + while ( i-- ) { + delete cache[ key[ i ] ]; + } + } + + // Remove the expando if there's no more data + if ( key === undefined || jQuery.isEmptyObject( cache ) ) { + + // Support: Chrome <=35 - 45 + // Webkit & Blink performance suffers when deleting properties + // from DOM nodes, so set to undefined instead + // https://bugs.chromium.org/p/chromium/issues/detail?id=378607 (bug restricted) + if ( owner.nodeType ) { + owner[ this.expando ] = undefined; + } else { + delete owner[ this.expando ]; + } + } + }, + hasData: function( owner ) { + var cache = owner[ this.expando ]; + return cache !== undefined && !jQuery.isEmptyObject( cache ); + } +}; +var dataPriv = new Data(); + +var dataUser = new Data(); + + + +// Implementation Summary +// +// 1. Enforce API surface and semantic compatibility with 1.9.x branch +// 2. Improve the module's maintainability by reducing the storage +// paths to a single mechanism. +// 3. Use the same single mechanism to support "private" and "user" data. +// 4. _Never_ expose "private" data to user code (TODO: Drop _data, _removeData) +// 5. Avoid exposing implementation details on user objects (eg. expando properties) +// 6. Provide a clear path for implementation upgrade to WeakMap in 2014 + +var rbrace = /^(?:\{[\w\W]*\}|\[[\w\W]*\])$/, + rmultiDash = /[A-Z]/g; + +function getData( data ) { + if ( data === "true" ) { + return true; + } + + if ( data === "false" ) { + return false; + } + + if ( data === "null" ) { + return null; + } + + // Only convert to a number if it doesn't change the string + if ( data === +data + "" ) { + return +data; + } + + if ( rbrace.test( data ) ) { + return JSON.parse( data ); + } + + return data; +} + +function dataAttr( elem, key, data ) { + var name; + + // If nothing was found internally, try to fetch any + // data from the HTML5 data-* attribute + if ( data === undefined && elem.nodeType === 1 ) { + name = "data-" + key.replace( rmultiDash, "-$&" ).toLowerCase(); + data = elem.getAttribute( name ); + + if ( typeof data === "string" ) { + try { + data = getData( data ); + } catch ( e ) {} + + // Make sure we set the data so it isn't changed later + dataUser.set( elem, key, data ); + } else { + data = undefined; + } + } + return data; +} + +jQuery.extend( { + hasData: function( elem ) { + return dataUser.hasData( elem ) || dataPriv.hasData( elem ); + }, + + data: function( elem, name, data ) { + return dataUser.access( elem, name, data ); + }, + + removeData: function( elem, name ) { + dataUser.remove( elem, name ); + }, + + // TODO: Now that all calls to _data and _removeData have been replaced + // with direct calls to dataPriv methods, these can be deprecated. + _data: function( elem, name, data ) { + return dataPriv.access( elem, name, data ); + }, + + _removeData: function( elem, name ) { + dataPriv.remove( elem, name ); + } +} ); + +jQuery.fn.extend( { + data: function( key, value ) { + var i, name, data, + elem = this[ 0 ], + attrs = elem && elem.attributes; + + // Gets all values + if ( key === undefined ) { + if ( this.length ) { + data = dataUser.get( elem ); + + if ( elem.nodeType === 1 && !dataPriv.get( elem, "hasDataAttrs" ) ) { + i = attrs.length; + while ( i-- ) { + + // Support: IE 11 only + // The attrs elements can be null (#14894) + if ( attrs[ i ] ) { + name = attrs[ i ].name; + if ( name.indexOf( "data-" ) === 0 ) { + name = camelCase( name.slice( 5 ) ); + dataAttr( elem, name, data[ name ] ); + } + } + } + dataPriv.set( elem, "hasDataAttrs", true ); + } + } + + return data; + } + + // Sets multiple values + if ( typeof key === "object" ) { + return this.each( function() { + dataUser.set( this, key ); + } ); + } + + return access( this, function( value ) { + var data; + + // The calling jQuery object (element matches) is not empty + // (and therefore has an element appears at this[ 0 ]) and the + // `value` parameter was not undefined. An empty jQuery object + // will result in `undefined` for elem = this[ 0 ] which will + // throw an exception if an attempt to read a data cache is made. + if ( elem && value === undefined ) { + + // Attempt to get data from the cache + // The key will always be camelCased in Data + data = dataUser.get( elem, key ); + if ( data !== undefined ) { + return data; + } + + // Attempt to "discover" the data in + // HTML5 custom data-* attrs + data = dataAttr( elem, key ); + if ( data !== undefined ) { + return data; + } + + // We tried really hard, but the data doesn't exist. + return; + } + + // Set the data... + this.each( function() { + + // We always store the camelCased key + dataUser.set( this, key, value ); + } ); + }, null, value, arguments.length > 1, null, true ); + }, + + removeData: function( key ) { + return this.each( function() { + dataUser.remove( this, key ); + } ); + } +} ); + + +jQuery.extend( { + queue: function( elem, type, data ) { + var queue; + + if ( elem ) { + type = ( type || "fx" ) + "queue"; + queue = dataPriv.get( elem, type ); + + // Speed up dequeue by getting out quickly if this is just a lookup + if ( data ) { + if ( !queue || Array.isArray( data ) ) { + queue = dataPriv.access( elem, type, jQuery.makeArray( data ) ); + } else { + queue.push( data ); + } + } + return queue || []; + } + }, + + dequeue: function( elem, type ) { + type = type || "fx"; + + var queue = jQuery.queue( elem, type ), + startLength = queue.length, + fn = queue.shift(), + hooks = jQuery._queueHooks( elem, type ), + next = function() { + jQuery.dequeue( elem, type ); + }; + + // If the fx queue is dequeued, always remove the progress sentinel + if ( fn === "inprogress" ) { + fn = queue.shift(); + startLength--; + } + + if ( fn ) { + + // Add a progress sentinel to prevent the fx queue from being + // automatically dequeued + if ( type === "fx" ) { + queue.unshift( "inprogress" ); + } + + // Clear up the last queue stop function + delete hooks.stop; + fn.call( elem, next, hooks ); + } + + if ( !startLength && hooks ) { + hooks.empty.fire(); + } + }, + + // Not public - generate a queueHooks object, or return the current one + _queueHooks: function( elem, type ) { + var key = type + "queueHooks"; + return dataPriv.get( elem, key ) || dataPriv.access( elem, key, { + empty: jQuery.Callbacks( "once memory" ).add( function() { + dataPriv.remove( elem, [ type + "queue", key ] ); + } ) + } ); + } +} ); + +jQuery.fn.extend( { + queue: function( type, data ) { + var setter = 2; + + if ( typeof type !== "string" ) { + data = type; + type = "fx"; + setter--; + } + + if ( arguments.length < setter ) { + return jQuery.queue( this[ 0 ], type ); + } + + return data === undefined ? + this : + this.each( function() { + var queue = jQuery.queue( this, type, data ); + + // Ensure a hooks for this queue + jQuery._queueHooks( this, type ); + + if ( type === "fx" && queue[ 0 ] !== "inprogress" ) { + jQuery.dequeue( this, type ); + } + } ); + }, + dequeue: function( type ) { + return this.each( function() { + jQuery.dequeue( this, type ); + } ); + }, + clearQueue: function( type ) { + return this.queue( type || "fx", [] ); + }, + + // Get a promise resolved when queues of a certain type + // are emptied (fx is the type by default) + promise: function( type, obj ) { + var tmp, + count = 1, + defer = jQuery.Deferred(), + elements = this, + i = this.length, + resolve = function() { + if ( !( --count ) ) { + defer.resolveWith( elements, [ elements ] ); + } + }; + + if ( typeof type !== "string" ) { + obj = type; + type = undefined; + } + type = type || "fx"; + + while ( i-- ) { + tmp = dataPriv.get( elements[ i ], type + "queueHooks" ); + if ( tmp && tmp.empty ) { + count++; + tmp.empty.add( resolve ); + } + } + resolve(); + return defer.promise( obj ); + } +} ); +var pnum = ( /[+-]?(?:\d*\.|)\d+(?:[eE][+-]?\d+|)/ ).source; + +var rcssNum = new RegExp( "^(?:([+-])=|)(" + pnum + ")([a-z%]*)$", "i" ); + + +var cssExpand = [ "Top", "Right", "Bottom", "Left" ]; + +var documentElement = document.documentElement; + + + + var isAttached = function( elem ) { + return jQuery.contains( elem.ownerDocument, elem ); + }, + composed = { composed: true }; + + // Support: IE 9 - 11+, Edge 12 - 18+, iOS 10.0 - 10.2 only + // Check attachment across shadow DOM boundaries when possible (gh-3504) + // Support: iOS 10.0-10.2 only + // Early iOS 10 versions support `attachShadow` but not `getRootNode`, + // leading to errors. We need to check for `getRootNode`. + if ( documentElement.getRootNode ) { + isAttached = function( elem ) { + return jQuery.contains( elem.ownerDocument, elem ) || + elem.getRootNode( composed ) === elem.ownerDocument; + }; + } +var isHiddenWithinTree = function( elem, el ) { + + // isHiddenWithinTree might be called from jQuery#filter function; + // in that case, element will be second argument + elem = el || elem; + + // Inline style trumps all + return elem.style.display === "none" || + elem.style.display === "" && + + // Otherwise, check computed style + // Support: Firefox <=43 - 45 + // Disconnected elements can have computed display: none, so first confirm that elem is + // in the document. + isAttached( elem ) && + + jQuery.css( elem, "display" ) === "none"; + }; + + + +function adjustCSS( elem, prop, valueParts, tween ) { + var adjusted, scale, + maxIterations = 20, + currentValue = tween ? + function() { + return tween.cur(); + } : + function() { + return jQuery.css( elem, prop, "" ); + }, + initial = currentValue(), + unit = valueParts && valueParts[ 3 ] || ( jQuery.cssNumber[ prop ] ? "" : "px" ), + + // Starting value computation is required for potential unit mismatches + initialInUnit = elem.nodeType && + ( jQuery.cssNumber[ prop ] || unit !== "px" && +initial ) && + rcssNum.exec( jQuery.css( elem, prop ) ); + + if ( initialInUnit && initialInUnit[ 3 ] !== unit ) { + + // Support: Firefox <=54 + // Halve the iteration target value to prevent interference from CSS upper bounds (gh-2144) + initial = initial / 2; + + // Trust units reported by jQuery.css + unit = unit || initialInUnit[ 3 ]; + + // Iteratively approximate from a nonzero starting point + initialInUnit = +initial || 1; + + while ( maxIterations-- ) { + + // Evaluate and update our best guess (doubling guesses that zero out). + // Finish if the scale equals or crosses 1 (making the old*new product non-positive). + jQuery.style( elem, prop, initialInUnit + unit ); + if ( ( 1 - scale ) * ( 1 - ( scale = currentValue() / initial || 0.5 ) ) <= 0 ) { + maxIterations = 0; + } + initialInUnit = initialInUnit / scale; + + } + + initialInUnit = initialInUnit * 2; + jQuery.style( elem, prop, initialInUnit + unit ); + + // Make sure we update the tween properties later on + valueParts = valueParts || []; + } + + if ( valueParts ) { + initialInUnit = +initialInUnit || +initial || 0; + + // Apply relative offset (+=/-=) if specified + adjusted = valueParts[ 1 ] ? + initialInUnit + ( valueParts[ 1 ] + 1 ) * valueParts[ 2 ] : + +valueParts[ 2 ]; + if ( tween ) { + tween.unit = unit; + tween.start = initialInUnit; + tween.end = adjusted; + } + } + return adjusted; +} + + +var defaultDisplayMap = {}; + +function getDefaultDisplay( elem ) { + var temp, + doc = elem.ownerDocument, + nodeName = elem.nodeName, + display = defaultDisplayMap[ nodeName ]; + + if ( display ) { + return display; + } + + temp = doc.body.appendChild( doc.createElement( nodeName ) ); + display = jQuery.css( temp, "display" ); + + temp.parentNode.removeChild( temp ); + + if ( display === "none" ) { + display = "block"; + } + defaultDisplayMap[ nodeName ] = display; + + return display; +} + +function showHide( elements, show ) { + var display, elem, + values = [], + index = 0, + length = elements.length; + + // Determine new display value for elements that need to change + for ( ; index < length; index++ ) { + elem = elements[ index ]; + if ( !elem.style ) { + continue; + } + + display = elem.style.display; + if ( show ) { + + // Since we force visibility upon cascade-hidden elements, an immediate (and slow) + // check is required in this first loop unless we have a nonempty display value (either + // inline or about-to-be-restored) + if ( display === "none" ) { + values[ index ] = dataPriv.get( elem, "display" ) || null; + if ( !values[ index ] ) { + elem.style.display = ""; + } + } + if ( elem.style.display === "" && isHiddenWithinTree( elem ) ) { + values[ index ] = getDefaultDisplay( elem ); + } + } else { + if ( display !== "none" ) { + values[ index ] = "none"; + + // Remember what we're overwriting + dataPriv.set( elem, "display", display ); + } + } + } + + // Set the display of the elements in a second loop to avoid constant reflow + for ( index = 0; index < length; index++ ) { + if ( values[ index ] != null ) { + elements[ index ].style.display = values[ index ]; + } + } + + return elements; +} + +jQuery.fn.extend( { + show: function() { + return showHide( this, true ); + }, + hide: function() { + return showHide( this ); + }, + toggle: function( state ) { + if ( typeof state === "boolean" ) { + return state ? this.show() : this.hide(); + } + + return this.each( function() { + if ( isHiddenWithinTree( this ) ) { + jQuery( this ).show(); + } else { + jQuery( this ).hide(); + } + } ); + } +} ); +var rcheckableType = ( /^(?:checkbox|radio)$/i ); + +var rtagName = ( /<([a-z][^\/\0>\x20\t\r\n\f]*)/i ); + +var rscriptType = ( /^$|^module$|\/(?:java|ecma)script/i ); + + + +( function() { + var fragment = document.createDocumentFragment(), + div = fragment.appendChild( document.createElement( "div" ) ), + input = document.createElement( "input" ); + + // Support: Android 4.0 - 4.3 only + // Check state lost if the name is set (#11217) + // Support: Windows Web Apps (WWA) + // `name` and `type` must use .setAttribute for WWA (#14901) + input.setAttribute( "type", "radio" ); + input.setAttribute( "checked", "checked" ); + input.setAttribute( "name", "t" ); + + div.appendChild( input ); + + // Support: Android <=4.1 only + // Older WebKit doesn't clone checked state correctly in fragments + support.checkClone = div.cloneNode( true ).cloneNode( true ).lastChild.checked; + + // Support: IE <=11 only + // Make sure textarea (and checkbox) defaultValue is properly cloned + div.innerHTML = ""; + support.noCloneChecked = !!div.cloneNode( true ).lastChild.defaultValue; + + // Support: IE <=9 only + // IE <=9 replaces "; + support.option = !!div.lastChild; +} )(); + + +// We have to close these tags to support XHTML (#13200) +var wrapMap = { + + // XHTML parsers do not magically insert elements in the + // same way that tag soup parsers do. So we cannot shorten + // this by omitting or other required elements. + thead: [ 1, "", "
" ], + col: [ 2, "", "
" ], + tr: [ 2, "", "
" ], + td: [ 3, "", "
" ], + + _default: [ 0, "", "" ] +}; + +wrapMap.tbody = wrapMap.tfoot = wrapMap.colgroup = wrapMap.caption = wrapMap.thead; +wrapMap.th = wrapMap.td; + +// Support: IE <=9 only +if ( !support.option ) { + wrapMap.optgroup = wrapMap.option = [ 1, "" ]; +} + + +function getAll( context, tag ) { + + // Support: IE <=9 - 11 only + // Use typeof to avoid zero-argument method invocation on host objects (#15151) + var ret; + + if ( typeof context.getElementsByTagName !== "undefined" ) { + ret = context.getElementsByTagName( tag || "*" ); + + } else if ( typeof context.querySelectorAll !== "undefined" ) { + ret = context.querySelectorAll( tag || "*" ); + + } else { + ret = []; + } + + if ( tag === undefined || tag && nodeName( context, tag ) ) { + return jQuery.merge( [ context ], ret ); + } + + return ret; +} + + +// Mark scripts as having already been evaluated +function setGlobalEval( elems, refElements ) { + var i = 0, + l = elems.length; + + for ( ; i < l; i++ ) { + dataPriv.set( + elems[ i ], + "globalEval", + !refElements || dataPriv.get( refElements[ i ], "globalEval" ) + ); + } +} + + +var rhtml = /<|&#?\w+;/; + +function buildFragment( elems, context, scripts, selection, ignored ) { + var elem, tmp, tag, wrap, attached, j, + fragment = context.createDocumentFragment(), + nodes = [], + i = 0, + l = elems.length; + + for ( ; i < l; i++ ) { + elem = elems[ i ]; + + if ( elem || elem === 0 ) { + + // Add nodes directly + if ( toType( elem ) === "object" ) { + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + jQuery.merge( nodes, elem.nodeType ? [ elem ] : elem ); + + // Convert non-html into a text node + } else if ( !rhtml.test( elem ) ) { + nodes.push( context.createTextNode( elem ) ); + + // Convert html into DOM nodes + } else { + tmp = tmp || fragment.appendChild( context.createElement( "div" ) ); + + // Deserialize a standard representation + tag = ( rtagName.exec( elem ) || [ "", "" ] )[ 1 ].toLowerCase(); + wrap = wrapMap[ tag ] || wrapMap._default; + tmp.innerHTML = wrap[ 1 ] + jQuery.htmlPrefilter( elem ) + wrap[ 2 ]; + + // Descend through wrappers to the right content + j = wrap[ 0 ]; + while ( j-- ) { + tmp = tmp.lastChild; + } + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + jQuery.merge( nodes, tmp.childNodes ); + + // Remember the top-level container + tmp = fragment.firstChild; + + // Ensure the created nodes are orphaned (#12392) + tmp.textContent = ""; + } + } + } + + // Remove wrapper from fragment + fragment.textContent = ""; + + i = 0; + while ( ( elem = nodes[ i++ ] ) ) { + + // Skip elements already in the context collection (trac-4087) + if ( selection && jQuery.inArray( elem, selection ) > -1 ) { + if ( ignored ) { + ignored.push( elem ); + } + continue; + } + + attached = isAttached( elem ); + + // Append to fragment + tmp = getAll( fragment.appendChild( elem ), "script" ); + + // Preserve script evaluation history + if ( attached ) { + setGlobalEval( tmp ); + } + + // Capture executables + if ( scripts ) { + j = 0; + while ( ( elem = tmp[ j++ ] ) ) { + if ( rscriptType.test( elem.type || "" ) ) { + scripts.push( elem ); + } + } + } + } + + return fragment; +} + + +var + rkeyEvent = /^key/, + rmouseEvent = /^(?:mouse|pointer|contextmenu|drag|drop)|click/, + rtypenamespace = /^([^.]*)(?:\.(.+)|)/; + +function returnTrue() { + return true; +} + +function returnFalse() { + return false; +} + +// Support: IE <=9 - 11+ +// focus() and blur() are asynchronous, except when they are no-op. +// So expect focus to be synchronous when the element is already active, +// and blur to be synchronous when the element is not already active. +// (focus and blur are always synchronous in other supported browsers, +// this just defines when we can count on it). +function expectSync( elem, type ) { + return ( elem === safeActiveElement() ) === ( type === "focus" ); +} + +// Support: IE <=9 only +// Accessing document.activeElement can throw unexpectedly +// https://bugs.jquery.com/ticket/13393 +function safeActiveElement() { + try { + return document.activeElement; + } catch ( err ) { } +} + +function on( elem, types, selector, data, fn, one ) { + var origFn, type; + + // Types can be a map of types/handlers + if ( typeof types === "object" ) { + + // ( types-Object, selector, data ) + if ( typeof selector !== "string" ) { + + // ( types-Object, data ) + data = data || selector; + selector = undefined; + } + for ( type in types ) { + on( elem, type, selector, data, types[ type ], one ); + } + return elem; + } + + if ( data == null && fn == null ) { + + // ( types, fn ) + fn = selector; + data = selector = undefined; + } else if ( fn == null ) { + if ( typeof selector === "string" ) { + + // ( types, selector, fn ) + fn = data; + data = undefined; + } else { + + // ( types, data, fn ) + fn = data; + data = selector; + selector = undefined; + } + } + if ( fn === false ) { + fn = returnFalse; + } else if ( !fn ) { + return elem; + } + + if ( one === 1 ) { + origFn = fn; + fn = function( event ) { + + // Can use an empty set, since event contains the info + jQuery().off( event ); + return origFn.apply( this, arguments ); + }; + + // Use same guid so caller can remove using origFn + fn.guid = origFn.guid || ( origFn.guid = jQuery.guid++ ); + } + return elem.each( function() { + jQuery.event.add( this, types, fn, data, selector ); + } ); +} + +/* + * Helper functions for managing events -- not part of the public interface. + * Props to Dean Edwards' addEvent library for many of the ideas. + */ +jQuery.event = { + + global: {}, + + add: function( elem, types, handler, data, selector ) { + + var handleObjIn, eventHandle, tmp, + events, t, handleObj, + special, handlers, type, namespaces, origType, + elemData = dataPriv.get( elem ); + + // Only attach events to objects that accept data + if ( !acceptData( elem ) ) { + return; + } + + // Caller can pass in an object of custom data in lieu of the handler + if ( handler.handler ) { + handleObjIn = handler; + handler = handleObjIn.handler; + selector = handleObjIn.selector; + } + + // Ensure that invalid selectors throw exceptions at attach time + // Evaluate against documentElement in case elem is a non-element node (e.g., document) + if ( selector ) { + jQuery.find.matchesSelector( documentElement, selector ); + } + + // Make sure that the handler has a unique ID, used to find/remove it later + if ( !handler.guid ) { + handler.guid = jQuery.guid++; + } + + // Init the element's event structure and main handler, if this is the first + if ( !( events = elemData.events ) ) { + events = elemData.events = Object.create( null ); + } + if ( !( eventHandle = elemData.handle ) ) { + eventHandle = elemData.handle = function( e ) { + + // Discard the second event of a jQuery.event.trigger() and + // when an event is called after a page has unloaded + return typeof jQuery !== "undefined" && jQuery.event.triggered !== e.type ? + jQuery.event.dispatch.apply( elem, arguments ) : undefined; + }; + } + + // Handle multiple events separated by a space + types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; + t = types.length; + while ( t-- ) { + tmp = rtypenamespace.exec( types[ t ] ) || []; + type = origType = tmp[ 1 ]; + namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); + + // There *must* be a type, no attaching namespace-only handlers + if ( !type ) { + continue; + } + + // If event changes its type, use the special event handlers for the changed type + special = jQuery.event.special[ type ] || {}; + + // If selector defined, determine special event api type, otherwise given type + type = ( selector ? special.delegateType : special.bindType ) || type; + + // Update special based on newly reset type + special = jQuery.event.special[ type ] || {}; + + // handleObj is passed to all event handlers + handleObj = jQuery.extend( { + type: type, + origType: origType, + data: data, + handler: handler, + guid: handler.guid, + selector: selector, + needsContext: selector && jQuery.expr.match.needsContext.test( selector ), + namespace: namespaces.join( "." ) + }, handleObjIn ); + + // Init the event handler queue if we're the first + if ( !( handlers = events[ type ] ) ) { + handlers = events[ type ] = []; + handlers.delegateCount = 0; + + // Only use addEventListener if the special events handler returns false + if ( !special.setup || + special.setup.call( elem, data, namespaces, eventHandle ) === false ) { + + if ( elem.addEventListener ) { + elem.addEventListener( type, eventHandle ); + } + } + } + + if ( special.add ) { + special.add.call( elem, handleObj ); + + if ( !handleObj.handler.guid ) { + handleObj.handler.guid = handler.guid; + } + } + + // Add to the element's handler list, delegates in front + if ( selector ) { + handlers.splice( handlers.delegateCount++, 0, handleObj ); + } else { + handlers.push( handleObj ); + } + + // Keep track of which events have ever been used, for event optimization + jQuery.event.global[ type ] = true; + } + + }, + + // Detach an event or set of events from an element + remove: function( elem, types, handler, selector, mappedTypes ) { + + var j, origCount, tmp, + events, t, handleObj, + special, handlers, type, namespaces, origType, + elemData = dataPriv.hasData( elem ) && dataPriv.get( elem ); + + if ( !elemData || !( events = elemData.events ) ) { + return; + } + + // Once for each type.namespace in types; type may be omitted + types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; + t = types.length; + while ( t-- ) { + tmp = rtypenamespace.exec( types[ t ] ) || []; + type = origType = tmp[ 1 ]; + namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); + + // Unbind all events (on this namespace, if provided) for the element + if ( !type ) { + for ( type in events ) { + jQuery.event.remove( elem, type + types[ t ], handler, selector, true ); + } + continue; + } + + special = jQuery.event.special[ type ] || {}; + type = ( selector ? special.delegateType : special.bindType ) || type; + handlers = events[ type ] || []; + tmp = tmp[ 2 ] && + new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ); + + // Remove matching events + origCount = j = handlers.length; + while ( j-- ) { + handleObj = handlers[ j ]; + + if ( ( mappedTypes || origType === handleObj.origType ) && + ( !handler || handler.guid === handleObj.guid ) && + ( !tmp || tmp.test( handleObj.namespace ) ) && + ( !selector || selector === handleObj.selector || + selector === "**" && handleObj.selector ) ) { + handlers.splice( j, 1 ); + + if ( handleObj.selector ) { + handlers.delegateCount--; + } + if ( special.remove ) { + special.remove.call( elem, handleObj ); + } + } + } + + // Remove generic event handler if we removed something and no more handlers exist + // (avoids potential for endless recursion during removal of special event handlers) + if ( origCount && !handlers.length ) { + if ( !special.teardown || + special.teardown.call( elem, namespaces, elemData.handle ) === false ) { + + jQuery.removeEvent( elem, type, elemData.handle ); + } + + delete events[ type ]; + } + } + + // Remove data and the expando if it's no longer used + if ( jQuery.isEmptyObject( events ) ) { + dataPriv.remove( elem, "handle events" ); + } + }, + + dispatch: function( nativeEvent ) { + + var i, j, ret, matched, handleObj, handlerQueue, + args = new Array( arguments.length ), + + // Make a writable jQuery.Event from the native event object + event = jQuery.event.fix( nativeEvent ), + + handlers = ( + dataPriv.get( this, "events" ) || Object.create( null ) + )[ event.type ] || [], + special = jQuery.event.special[ event.type ] || {}; + + // Use the fix-ed jQuery.Event rather than the (read-only) native event + args[ 0 ] = event; + + for ( i = 1; i < arguments.length; i++ ) { + args[ i ] = arguments[ i ]; + } + + event.delegateTarget = this; + + // Call the preDispatch hook for the mapped type, and let it bail if desired + if ( special.preDispatch && special.preDispatch.call( this, event ) === false ) { + return; + } + + // Determine handlers + handlerQueue = jQuery.event.handlers.call( this, event, handlers ); + + // Run delegates first; they may want to stop propagation beneath us + i = 0; + while ( ( matched = handlerQueue[ i++ ] ) && !event.isPropagationStopped() ) { + event.currentTarget = matched.elem; + + j = 0; + while ( ( handleObj = matched.handlers[ j++ ] ) && + !event.isImmediatePropagationStopped() ) { + + // If the event is namespaced, then each handler is only invoked if it is + // specially universal or its namespaces are a superset of the event's. + if ( !event.rnamespace || handleObj.namespace === false || + event.rnamespace.test( handleObj.namespace ) ) { + + event.handleObj = handleObj; + event.data = handleObj.data; + + ret = ( ( jQuery.event.special[ handleObj.origType ] || {} ).handle || + handleObj.handler ).apply( matched.elem, args ); + + if ( ret !== undefined ) { + if ( ( event.result = ret ) === false ) { + event.preventDefault(); + event.stopPropagation(); + } + } + } + } + } + + // Call the postDispatch hook for the mapped type + if ( special.postDispatch ) { + special.postDispatch.call( this, event ); + } + + return event.result; + }, + + handlers: function( event, handlers ) { + var i, handleObj, sel, matchedHandlers, matchedSelectors, + handlerQueue = [], + delegateCount = handlers.delegateCount, + cur = event.target; + + // Find delegate handlers + if ( delegateCount && + + // Support: IE <=9 + // Black-hole SVG instance trees (trac-13180) + cur.nodeType && + + // Support: Firefox <=42 + // Suppress spec-violating clicks indicating a non-primary pointer button (trac-3861) + // https://www.w3.org/TR/DOM-Level-3-Events/#event-type-click + // Support: IE 11 only + // ...but not arrow key "clicks" of radio inputs, which can have `button` -1 (gh-2343) + !( event.type === "click" && event.button >= 1 ) ) { + + for ( ; cur !== this; cur = cur.parentNode || this ) { + + // Don't check non-elements (#13208) + // Don't process clicks on disabled elements (#6911, #8165, #11382, #11764) + if ( cur.nodeType === 1 && !( event.type === "click" && cur.disabled === true ) ) { + matchedHandlers = []; + matchedSelectors = {}; + for ( i = 0; i < delegateCount; i++ ) { + handleObj = handlers[ i ]; + + // Don't conflict with Object.prototype properties (#13203) + sel = handleObj.selector + " "; + + if ( matchedSelectors[ sel ] === undefined ) { + matchedSelectors[ sel ] = handleObj.needsContext ? + jQuery( sel, this ).index( cur ) > -1 : + jQuery.find( sel, this, null, [ cur ] ).length; + } + if ( matchedSelectors[ sel ] ) { + matchedHandlers.push( handleObj ); + } + } + if ( matchedHandlers.length ) { + handlerQueue.push( { elem: cur, handlers: matchedHandlers } ); + } + } + } + } + + // Add the remaining (directly-bound) handlers + cur = this; + if ( delegateCount < handlers.length ) { + handlerQueue.push( { elem: cur, handlers: handlers.slice( delegateCount ) } ); + } + + return handlerQueue; + }, + + addProp: function( name, hook ) { + Object.defineProperty( jQuery.Event.prototype, name, { + enumerable: true, + configurable: true, + + get: isFunction( hook ) ? + function() { + if ( this.originalEvent ) { + return hook( this.originalEvent ); + } + } : + function() { + if ( this.originalEvent ) { + return this.originalEvent[ name ]; + } + }, + + set: function( value ) { + Object.defineProperty( this, name, { + enumerable: true, + configurable: true, + writable: true, + value: value + } ); + } + } ); + }, + + fix: function( originalEvent ) { + return originalEvent[ jQuery.expando ] ? + originalEvent : + new jQuery.Event( originalEvent ); + }, + + special: { + load: { + + // Prevent triggered image.load events from bubbling to window.load + noBubble: true + }, + click: { + + // Utilize native event to ensure correct state for checkable inputs + setup: function( data ) { + + // For mutual compressibility with _default, replace `this` access with a local var. + // `|| data` is dead code meant only to preserve the variable through minification. + var el = this || data; + + // Claim the first handler + if ( rcheckableType.test( el.type ) && + el.click && nodeName( el, "input" ) ) { + + // dataPriv.set( el, "click", ... ) + leverageNative( el, "click", returnTrue ); + } + + // Return false to allow normal processing in the caller + return false; + }, + trigger: function( data ) { + + // For mutual compressibility with _default, replace `this` access with a local var. + // `|| data` is dead code meant only to preserve the variable through minification. + var el = this || data; + + // Force setup before triggering a click + if ( rcheckableType.test( el.type ) && + el.click && nodeName( el, "input" ) ) { + + leverageNative( el, "click" ); + } + + // Return non-false to allow normal event-path propagation + return true; + }, + + // For cross-browser consistency, suppress native .click() on links + // Also prevent it if we're currently inside a leveraged native-event stack + _default: function( event ) { + var target = event.target; + return rcheckableType.test( target.type ) && + target.click && nodeName( target, "input" ) && + dataPriv.get( target, "click" ) || + nodeName( target, "a" ); + } + }, + + beforeunload: { + postDispatch: function( event ) { + + // Support: Firefox 20+ + // Firefox doesn't alert if the returnValue field is not set. + if ( event.result !== undefined && event.originalEvent ) { + event.originalEvent.returnValue = event.result; + } + } + } + } +}; + +// Ensure the presence of an event listener that handles manually-triggered +// synthetic events by interrupting progress until reinvoked in response to +// *native* events that it fires directly, ensuring that state changes have +// already occurred before other listeners are invoked. +function leverageNative( el, type, expectSync ) { + + // Missing expectSync indicates a trigger call, which must force setup through jQuery.event.add + if ( !expectSync ) { + if ( dataPriv.get( el, type ) === undefined ) { + jQuery.event.add( el, type, returnTrue ); + } + return; + } + + // Register the controller as a special universal handler for all event namespaces + dataPriv.set( el, type, false ); + jQuery.event.add( el, type, { + namespace: false, + handler: function( event ) { + var notAsync, result, + saved = dataPriv.get( this, type ); + + if ( ( event.isTrigger & 1 ) && this[ type ] ) { + + // Interrupt processing of the outer synthetic .trigger()ed event + // Saved data should be false in such cases, but might be a leftover capture object + // from an async native handler (gh-4350) + if ( !saved.length ) { + + // Store arguments for use when handling the inner native event + // There will always be at least one argument (an event object), so this array + // will not be confused with a leftover capture object. + saved = slice.call( arguments ); + dataPriv.set( this, type, saved ); + + // Trigger the native event and capture its result + // Support: IE <=9 - 11+ + // focus() and blur() are asynchronous + notAsync = expectSync( this, type ); + this[ type ](); + result = dataPriv.get( this, type ); + if ( saved !== result || notAsync ) { + dataPriv.set( this, type, false ); + } else { + result = {}; + } + if ( saved !== result ) { + + // Cancel the outer synthetic event + event.stopImmediatePropagation(); + event.preventDefault(); + return result.value; + } + + // If this is an inner synthetic event for an event with a bubbling surrogate + // (focus or blur), assume that the surrogate already propagated from triggering the + // native event and prevent that from happening again here. + // This technically gets the ordering wrong w.r.t. to `.trigger()` (in which the + // bubbling surrogate propagates *after* the non-bubbling base), but that seems + // less bad than duplication. + } else if ( ( jQuery.event.special[ type ] || {} ).delegateType ) { + event.stopPropagation(); + } + + // If this is a native event triggered above, everything is now in order + // Fire an inner synthetic event with the original arguments + } else if ( saved.length ) { + + // ...and capture the result + dataPriv.set( this, type, { + value: jQuery.event.trigger( + + // Support: IE <=9 - 11+ + // Extend with the prototype to reset the above stopImmediatePropagation() + jQuery.extend( saved[ 0 ], jQuery.Event.prototype ), + saved.slice( 1 ), + this + ) + } ); + + // Abort handling of the native event + event.stopImmediatePropagation(); + } + } + } ); +} + +jQuery.removeEvent = function( elem, type, handle ) { + + // This "if" is needed for plain objects + if ( elem.removeEventListener ) { + elem.removeEventListener( type, handle ); + } +}; + +jQuery.Event = function( src, props ) { + + // Allow instantiation without the 'new' keyword + if ( !( this instanceof jQuery.Event ) ) { + return new jQuery.Event( src, props ); + } + + // Event object + if ( src && src.type ) { + this.originalEvent = src; + this.type = src.type; + + // Events bubbling up the document may have been marked as prevented + // by a handler lower down the tree; reflect the correct value. + this.isDefaultPrevented = src.defaultPrevented || + src.defaultPrevented === undefined && + + // Support: Android <=2.3 only + src.returnValue === false ? + returnTrue : + returnFalse; + + // Create target properties + // Support: Safari <=6 - 7 only + // Target should not be a text node (#504, #13143) + this.target = ( src.target && src.target.nodeType === 3 ) ? + src.target.parentNode : + src.target; + + this.currentTarget = src.currentTarget; + this.relatedTarget = src.relatedTarget; + + // Event type + } else { + this.type = src; + } + + // Put explicitly provided properties onto the event object + if ( props ) { + jQuery.extend( this, props ); + } + + // Create a timestamp if incoming event doesn't have one + this.timeStamp = src && src.timeStamp || Date.now(); + + // Mark it as fixed + this[ jQuery.expando ] = true; +}; + +// jQuery.Event is based on DOM3 Events as specified by the ECMAScript Language Binding +// https://www.w3.org/TR/2003/WD-DOM-Level-3-Events-20030331/ecma-script-binding.html +jQuery.Event.prototype = { + constructor: jQuery.Event, + isDefaultPrevented: returnFalse, + isPropagationStopped: returnFalse, + isImmediatePropagationStopped: returnFalse, + isSimulated: false, + + preventDefault: function() { + var e = this.originalEvent; + + this.isDefaultPrevented = returnTrue; + + if ( e && !this.isSimulated ) { + e.preventDefault(); + } + }, + stopPropagation: function() { + var e = this.originalEvent; + + this.isPropagationStopped = returnTrue; + + if ( e && !this.isSimulated ) { + e.stopPropagation(); + } + }, + stopImmediatePropagation: function() { + var e = this.originalEvent; + + this.isImmediatePropagationStopped = returnTrue; + + if ( e && !this.isSimulated ) { + e.stopImmediatePropagation(); + } + + this.stopPropagation(); + } +}; + +// Includes all common event props including KeyEvent and MouseEvent specific props +jQuery.each( { + altKey: true, + bubbles: true, + cancelable: true, + changedTouches: true, + ctrlKey: true, + detail: true, + eventPhase: true, + metaKey: true, + pageX: true, + pageY: true, + shiftKey: true, + view: true, + "char": true, + code: true, + charCode: true, + key: true, + keyCode: true, + button: true, + buttons: true, + clientX: true, + clientY: true, + offsetX: true, + offsetY: true, + pointerId: true, + pointerType: true, + screenX: true, + screenY: true, + targetTouches: true, + toElement: true, + touches: true, + + which: function( event ) { + var button = event.button; + + // Add which for key events + if ( event.which == null && rkeyEvent.test( event.type ) ) { + return event.charCode != null ? event.charCode : event.keyCode; + } + + // Add which for click: 1 === left; 2 === middle; 3 === right + if ( !event.which && button !== undefined && rmouseEvent.test( event.type ) ) { + if ( button & 1 ) { + return 1; + } + + if ( button & 2 ) { + return 3; + } + + if ( button & 4 ) { + return 2; + } + + return 0; + } + + return event.which; + } +}, jQuery.event.addProp ); + +jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateType ) { + jQuery.event.special[ type ] = { + + // Utilize native event if possible so blur/focus sequence is correct + setup: function() { + + // Claim the first handler + // dataPriv.set( this, "focus", ... ) + // dataPriv.set( this, "blur", ... ) + leverageNative( this, type, expectSync ); + + // Return false to allow normal processing in the caller + return false; + }, + trigger: function() { + + // Force setup before trigger + leverageNative( this, type ); + + // Return non-false to allow normal event-path propagation + return true; + }, + + delegateType: delegateType + }; +} ); + +// Create mouseenter/leave events using mouseover/out and event-time checks +// so that event delegation works in jQuery. +// Do the same for pointerenter/pointerleave and pointerover/pointerout +// +// Support: Safari 7 only +// Safari sends mouseenter too often; see: +// https://bugs.chromium.org/p/chromium/issues/detail?id=470258 +// for the description of the bug (it existed in older Chrome versions as well). +jQuery.each( { + mouseenter: "mouseover", + mouseleave: "mouseout", + pointerenter: "pointerover", + pointerleave: "pointerout" +}, function( orig, fix ) { + jQuery.event.special[ orig ] = { + delegateType: fix, + bindType: fix, + + handle: function( event ) { + var ret, + target = this, + related = event.relatedTarget, + handleObj = event.handleObj; + + // For mouseenter/leave call the handler if related is outside the target. + // NB: No relatedTarget if the mouse left/entered the browser window + if ( !related || ( related !== target && !jQuery.contains( target, related ) ) ) { + event.type = handleObj.origType; + ret = handleObj.handler.apply( this, arguments ); + event.type = fix; + } + return ret; + } + }; +} ); + +jQuery.fn.extend( { + + on: function( types, selector, data, fn ) { + return on( this, types, selector, data, fn ); + }, + one: function( types, selector, data, fn ) { + return on( this, types, selector, data, fn, 1 ); + }, + off: function( types, selector, fn ) { + var handleObj, type; + if ( types && types.preventDefault && types.handleObj ) { + + // ( event ) dispatched jQuery.Event + handleObj = types.handleObj; + jQuery( types.delegateTarget ).off( + handleObj.namespace ? + handleObj.origType + "." + handleObj.namespace : + handleObj.origType, + handleObj.selector, + handleObj.handler + ); + return this; + } + if ( typeof types === "object" ) { + + // ( types-object [, selector] ) + for ( type in types ) { + this.off( type, selector, types[ type ] ); + } + return this; + } + if ( selector === false || typeof selector === "function" ) { + + // ( types [, fn] ) + fn = selector; + selector = undefined; + } + if ( fn === false ) { + fn = returnFalse; + } + return this.each( function() { + jQuery.event.remove( this, types, fn, selector ); + } ); + } +} ); + + +var + + // Support: IE <=10 - 11, Edge 12 - 13 only + // In IE/Edge using regex groups here causes severe slowdowns. + // See https://connect.microsoft.com/IE/feedback/details/1736512/ + rnoInnerhtml = /\s*$/g; + +// Prefer a tbody over its parent table for containing new rows +function manipulationTarget( elem, content ) { + if ( nodeName( elem, "table" ) && + nodeName( content.nodeType !== 11 ? content : content.firstChild, "tr" ) ) { + + return jQuery( elem ).children( "tbody" )[ 0 ] || elem; + } + + return elem; +} + +// Replace/restore the type attribute of script elements for safe DOM manipulation +function disableScript( elem ) { + elem.type = ( elem.getAttribute( "type" ) !== null ) + "/" + elem.type; + return elem; +} +function restoreScript( elem ) { + if ( ( elem.type || "" ).slice( 0, 5 ) === "true/" ) { + elem.type = elem.type.slice( 5 ); + } else { + elem.removeAttribute( "type" ); + } + + return elem; +} + +function cloneCopyEvent( src, dest ) { + var i, l, type, pdataOld, udataOld, udataCur, events; + + if ( dest.nodeType !== 1 ) { + return; + } + + // 1. Copy private data: events, handlers, etc. + if ( dataPriv.hasData( src ) ) { + pdataOld = dataPriv.get( src ); + events = pdataOld.events; + + if ( events ) { + dataPriv.remove( dest, "handle events" ); + + for ( type in events ) { + for ( i = 0, l = events[ type ].length; i < l; i++ ) { + jQuery.event.add( dest, type, events[ type ][ i ] ); + } + } + } + } + + // 2. Copy user data + if ( dataUser.hasData( src ) ) { + udataOld = dataUser.access( src ); + udataCur = jQuery.extend( {}, udataOld ); + + dataUser.set( dest, udataCur ); + } +} + +// Fix IE bugs, see support tests +function fixInput( src, dest ) { + var nodeName = dest.nodeName.toLowerCase(); + + // Fails to persist the checked state of a cloned checkbox or radio button. + if ( nodeName === "input" && rcheckableType.test( src.type ) ) { + dest.checked = src.checked; + + // Fails to return the selected option to the default selected state when cloning options + } else if ( nodeName === "input" || nodeName === "textarea" ) { + dest.defaultValue = src.defaultValue; + } +} + +function domManip( collection, args, callback, ignored ) { + + // Flatten any nested arrays + args = flat( args ); + + var fragment, first, scripts, hasScripts, node, doc, + i = 0, + l = collection.length, + iNoClone = l - 1, + value = args[ 0 ], + valueIsFunction = isFunction( value ); + + // We can't cloneNode fragments that contain checked, in WebKit + if ( valueIsFunction || + ( l > 1 && typeof value === "string" && + !support.checkClone && rchecked.test( value ) ) ) { + return collection.each( function( index ) { + var self = collection.eq( index ); + if ( valueIsFunction ) { + args[ 0 ] = value.call( this, index, self.html() ); + } + domManip( self, args, callback, ignored ); + } ); + } + + if ( l ) { + fragment = buildFragment( args, collection[ 0 ].ownerDocument, false, collection, ignored ); + first = fragment.firstChild; + + if ( fragment.childNodes.length === 1 ) { + fragment = first; + } + + // Require either new content or an interest in ignored elements to invoke the callback + if ( first || ignored ) { + scripts = jQuery.map( getAll( fragment, "script" ), disableScript ); + hasScripts = scripts.length; + + // Use the original fragment for the last item + // instead of the first because it can end up + // being emptied incorrectly in certain situations (#8070). + for ( ; i < l; i++ ) { + node = fragment; + + if ( i !== iNoClone ) { + node = jQuery.clone( node, true, true ); + + // Keep references to cloned scripts for later restoration + if ( hasScripts ) { + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + jQuery.merge( scripts, getAll( node, "script" ) ); + } + } + + callback.call( collection[ i ], node, i ); + } + + if ( hasScripts ) { + doc = scripts[ scripts.length - 1 ].ownerDocument; + + // Reenable scripts + jQuery.map( scripts, restoreScript ); + + // Evaluate executable scripts on first document insertion + for ( i = 0; i < hasScripts; i++ ) { + node = scripts[ i ]; + if ( rscriptType.test( node.type || "" ) && + !dataPriv.access( node, "globalEval" ) && + jQuery.contains( doc, node ) ) { + + if ( node.src && ( node.type || "" ).toLowerCase() !== "module" ) { + + // Optional AJAX dependency, but won't run scripts if not present + if ( jQuery._evalUrl && !node.noModule ) { + jQuery._evalUrl( node.src, { + nonce: node.nonce || node.getAttribute( "nonce" ) + }, doc ); + } + } else { + DOMEval( node.textContent.replace( rcleanScript, "" ), node, doc ); + } + } + } + } + } + } + + return collection; +} + +function remove( elem, selector, keepData ) { + var node, + nodes = selector ? jQuery.filter( selector, elem ) : elem, + i = 0; + + for ( ; ( node = nodes[ i ] ) != null; i++ ) { + if ( !keepData && node.nodeType === 1 ) { + jQuery.cleanData( getAll( node ) ); + } + + if ( node.parentNode ) { + if ( keepData && isAttached( node ) ) { + setGlobalEval( getAll( node, "script" ) ); + } + node.parentNode.removeChild( node ); + } + } + + return elem; +} + +jQuery.extend( { + htmlPrefilter: function( html ) { + return html; + }, + + clone: function( elem, dataAndEvents, deepDataAndEvents ) { + var i, l, srcElements, destElements, + clone = elem.cloneNode( true ), + inPage = isAttached( elem ); + + // Fix IE cloning issues + if ( !support.noCloneChecked && ( elem.nodeType === 1 || elem.nodeType === 11 ) && + !jQuery.isXMLDoc( elem ) ) { + + // We eschew Sizzle here for performance reasons: https://jsperf.com/getall-vs-sizzle/2 + destElements = getAll( clone ); + srcElements = getAll( elem ); + + for ( i = 0, l = srcElements.length; i < l; i++ ) { + fixInput( srcElements[ i ], destElements[ i ] ); + } + } + + // Copy the events from the original to the clone + if ( dataAndEvents ) { + if ( deepDataAndEvents ) { + srcElements = srcElements || getAll( elem ); + destElements = destElements || getAll( clone ); + + for ( i = 0, l = srcElements.length; i < l; i++ ) { + cloneCopyEvent( srcElements[ i ], destElements[ i ] ); + } + } else { + cloneCopyEvent( elem, clone ); + } + } + + // Preserve script evaluation history + destElements = getAll( clone, "script" ); + if ( destElements.length > 0 ) { + setGlobalEval( destElements, !inPage && getAll( elem, "script" ) ); + } + + // Return the cloned set + return clone; + }, + + cleanData: function( elems ) { + var data, elem, type, + special = jQuery.event.special, + i = 0; + + for ( ; ( elem = elems[ i ] ) !== undefined; i++ ) { + if ( acceptData( elem ) ) { + if ( ( data = elem[ dataPriv.expando ] ) ) { + if ( data.events ) { + for ( type in data.events ) { + if ( special[ type ] ) { + jQuery.event.remove( elem, type ); + + // This is a shortcut to avoid jQuery.event.remove's overhead + } else { + jQuery.removeEvent( elem, type, data.handle ); + } + } + } + + // Support: Chrome <=35 - 45+ + // Assign undefined instead of using delete, see Data#remove + elem[ dataPriv.expando ] = undefined; + } + if ( elem[ dataUser.expando ] ) { + + // Support: Chrome <=35 - 45+ + // Assign undefined instead of using delete, see Data#remove + elem[ dataUser.expando ] = undefined; + } + } + } + } +} ); + +jQuery.fn.extend( { + detach: function( selector ) { + return remove( this, selector, true ); + }, + + remove: function( selector ) { + return remove( this, selector ); + }, + + text: function( value ) { + return access( this, function( value ) { + return value === undefined ? + jQuery.text( this ) : + this.empty().each( function() { + if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { + this.textContent = value; + } + } ); + }, null, value, arguments.length ); + }, + + append: function() { + return domManip( this, arguments, function( elem ) { + if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { + var target = manipulationTarget( this, elem ); + target.appendChild( elem ); + } + } ); + }, + + prepend: function() { + return domManip( this, arguments, function( elem ) { + if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { + var target = manipulationTarget( this, elem ); + target.insertBefore( elem, target.firstChild ); + } + } ); + }, + + before: function() { + return domManip( this, arguments, function( elem ) { + if ( this.parentNode ) { + this.parentNode.insertBefore( elem, this ); + } + } ); + }, + + after: function() { + return domManip( this, arguments, function( elem ) { + if ( this.parentNode ) { + this.parentNode.insertBefore( elem, this.nextSibling ); + } + } ); + }, + + empty: function() { + var elem, + i = 0; + + for ( ; ( elem = this[ i ] ) != null; i++ ) { + if ( elem.nodeType === 1 ) { + + // Prevent memory leaks + jQuery.cleanData( getAll( elem, false ) ); + + // Remove any remaining nodes + elem.textContent = ""; + } + } + + return this; + }, + + clone: function( dataAndEvents, deepDataAndEvents ) { + dataAndEvents = dataAndEvents == null ? false : dataAndEvents; + deepDataAndEvents = deepDataAndEvents == null ? dataAndEvents : deepDataAndEvents; + + return this.map( function() { + return jQuery.clone( this, dataAndEvents, deepDataAndEvents ); + } ); + }, + + html: function( value ) { + return access( this, function( value ) { + var elem = this[ 0 ] || {}, + i = 0, + l = this.length; + + if ( value === undefined && elem.nodeType === 1 ) { + return elem.innerHTML; + } + + // See if we can take a shortcut and just use innerHTML + if ( typeof value === "string" && !rnoInnerhtml.test( value ) && + !wrapMap[ ( rtagName.exec( value ) || [ "", "" ] )[ 1 ].toLowerCase() ] ) { + + value = jQuery.htmlPrefilter( value ); + + try { + for ( ; i < l; i++ ) { + elem = this[ i ] || {}; + + // Remove element nodes and prevent memory leaks + if ( elem.nodeType === 1 ) { + jQuery.cleanData( getAll( elem, false ) ); + elem.innerHTML = value; + } + } + + elem = 0; + + // If using innerHTML throws an exception, use the fallback method + } catch ( e ) {} + } + + if ( elem ) { + this.empty().append( value ); + } + }, null, value, arguments.length ); + }, + + replaceWith: function() { + var ignored = []; + + // Make the changes, replacing each non-ignored context element with the new content + return domManip( this, arguments, function( elem ) { + var parent = this.parentNode; + + if ( jQuery.inArray( this, ignored ) < 0 ) { + jQuery.cleanData( getAll( this ) ); + if ( parent ) { + parent.replaceChild( elem, this ); + } + } + + // Force callback invocation + }, ignored ); + } +} ); + +jQuery.each( { + appendTo: "append", + prependTo: "prepend", + insertBefore: "before", + insertAfter: "after", + replaceAll: "replaceWith" +}, function( name, original ) { + jQuery.fn[ name ] = function( selector ) { + var elems, + ret = [], + insert = jQuery( selector ), + last = insert.length - 1, + i = 0; + + for ( ; i <= last; i++ ) { + elems = i === last ? this : this.clone( true ); + jQuery( insert[ i ] )[ original ]( elems ); + + // Support: Android <=4.0 only, PhantomJS 1 only + // .get() because push.apply(_, arraylike) throws on ancient WebKit + push.apply( ret, elems.get() ); + } + + return this.pushStack( ret ); + }; +} ); +var rnumnonpx = new RegExp( "^(" + pnum + ")(?!px)[a-z%]+$", "i" ); + +var getStyles = function( elem ) { + + // Support: IE <=11 only, Firefox <=30 (#15098, #14150) + // IE throws on elements created in popups + // FF meanwhile throws on frame elements through "defaultView.getComputedStyle" + var view = elem.ownerDocument.defaultView; + + if ( !view || !view.opener ) { + view = window; + } + + return view.getComputedStyle( elem ); + }; + +var swap = function( elem, options, callback ) { + var ret, name, + old = {}; + + // Remember the old values, and insert the new ones + for ( name in options ) { + old[ name ] = elem.style[ name ]; + elem.style[ name ] = options[ name ]; + } + + ret = callback.call( elem ); + + // Revert the old values + for ( name in options ) { + elem.style[ name ] = old[ name ]; + } + + return ret; +}; + + +var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" ); + + + +( function() { + + // Executing both pixelPosition & boxSizingReliable tests require only one layout + // so they're executed at the same time to save the second computation. + function computeStyleTests() { + + // This is a singleton, we need to execute it only once + if ( !div ) { + return; + } + + container.style.cssText = "position:absolute;left:-11111px;width:60px;" + + "margin-top:1px;padding:0;border:0"; + div.style.cssText = + "position:relative;display:block;box-sizing:border-box;overflow:scroll;" + + "margin:auto;border:1px;padding:1px;" + + "width:60%;top:1%"; + documentElement.appendChild( container ).appendChild( div ); + + var divStyle = window.getComputedStyle( div ); + pixelPositionVal = divStyle.top !== "1%"; + + // Support: Android 4.0 - 4.3 only, Firefox <=3 - 44 + reliableMarginLeftVal = roundPixelMeasures( divStyle.marginLeft ) === 12; + + // Support: Android 4.0 - 4.3 only, Safari <=9.1 - 10.1, iOS <=7.0 - 9.3 + // Some styles come back with percentage values, even though they shouldn't + div.style.right = "60%"; + pixelBoxStylesVal = roundPixelMeasures( divStyle.right ) === 36; + + // Support: IE 9 - 11 only + // Detect misreporting of content dimensions for box-sizing:border-box elements + boxSizingReliableVal = roundPixelMeasures( divStyle.width ) === 36; + + // Support: IE 9 only + // Detect overflow:scroll screwiness (gh-3699) + // Support: Chrome <=64 + // Don't get tricked when zoom affects offsetWidth (gh-4029) + div.style.position = "absolute"; + scrollboxSizeVal = roundPixelMeasures( div.offsetWidth / 3 ) === 12; + + documentElement.removeChild( container ); + + // Nullify the div so it wouldn't be stored in the memory and + // it will also be a sign that checks already performed + div = null; + } + + function roundPixelMeasures( measure ) { + return Math.round( parseFloat( measure ) ); + } + + var pixelPositionVal, boxSizingReliableVal, scrollboxSizeVal, pixelBoxStylesVal, + reliableTrDimensionsVal, reliableMarginLeftVal, + container = document.createElement( "div" ), + div = document.createElement( "div" ); + + // Finish early in limited (non-browser) environments + if ( !div.style ) { + return; + } + + // Support: IE <=9 - 11 only + // Style of cloned element affects source element cloned (#8908) + div.style.backgroundClip = "content-box"; + div.cloneNode( true ).style.backgroundClip = ""; + support.clearCloneStyle = div.style.backgroundClip === "content-box"; + + jQuery.extend( support, { + boxSizingReliable: function() { + computeStyleTests(); + return boxSizingReliableVal; + }, + pixelBoxStyles: function() { + computeStyleTests(); + return pixelBoxStylesVal; + }, + pixelPosition: function() { + computeStyleTests(); + return pixelPositionVal; + }, + reliableMarginLeft: function() { + computeStyleTests(); + return reliableMarginLeftVal; + }, + scrollboxSize: function() { + computeStyleTests(); + return scrollboxSizeVal; + }, + + // Support: IE 9 - 11+, Edge 15 - 18+ + // IE/Edge misreport `getComputedStyle` of table rows with width/height + // set in CSS while `offset*` properties report correct values. + // Behavior in IE 9 is more subtle than in newer versions & it passes + // some versions of this test; make sure not to make it pass there! + reliableTrDimensions: function() { + var table, tr, trChild, trStyle; + if ( reliableTrDimensionsVal == null ) { + table = document.createElement( "table" ); + tr = document.createElement( "tr" ); + trChild = document.createElement( "div" ); + + table.style.cssText = "position:absolute;left:-11111px"; + tr.style.height = "1px"; + trChild.style.height = "9px"; + + documentElement + .appendChild( table ) + .appendChild( tr ) + .appendChild( trChild ); + + trStyle = window.getComputedStyle( tr ); + reliableTrDimensionsVal = parseInt( trStyle.height ) > 3; + + documentElement.removeChild( table ); + } + return reliableTrDimensionsVal; + } + } ); +} )(); + + +function curCSS( elem, name, computed ) { + var width, minWidth, maxWidth, ret, + + // Support: Firefox 51+ + // Retrieving style before computed somehow + // fixes an issue with getting wrong values + // on detached elements + style = elem.style; + + computed = computed || getStyles( elem ); + + // getPropertyValue is needed for: + // .css('filter') (IE 9 only, #12537) + // .css('--customProperty) (#3144) + if ( computed ) { + ret = computed.getPropertyValue( name ) || computed[ name ]; + + if ( ret === "" && !isAttached( elem ) ) { + ret = jQuery.style( elem, name ); + } + + // A tribute to the "awesome hack by Dean Edwards" + // Android Browser returns percentage for some values, + // but width seems to be reliably pixels. + // This is against the CSSOM draft spec: + // https://drafts.csswg.org/cssom/#resolved-values + if ( !support.pixelBoxStyles() && rnumnonpx.test( ret ) && rboxStyle.test( name ) ) { + + // Remember the original values + width = style.width; + minWidth = style.minWidth; + maxWidth = style.maxWidth; + + // Put in the new values to get a computed value out + style.minWidth = style.maxWidth = style.width = ret; + ret = computed.width; + + // Revert the changed values + style.width = width; + style.minWidth = minWidth; + style.maxWidth = maxWidth; + } + } + + return ret !== undefined ? + + // Support: IE <=9 - 11 only + // IE returns zIndex value as an integer. + ret + "" : + ret; +} + + +function addGetHookIf( conditionFn, hookFn ) { + + // Define the hook, we'll check on the first run if it's really needed. + return { + get: function() { + if ( conditionFn() ) { + + // Hook not needed (or it's not possible to use it due + // to missing dependency), remove it. + delete this.get; + return; + } + + // Hook needed; redefine it so that the support test is not executed again. + return ( this.get = hookFn ).apply( this, arguments ); + } + }; +} + + +var cssPrefixes = [ "Webkit", "Moz", "ms" ], + emptyStyle = document.createElement( "div" ).style, + vendorProps = {}; + +// Return a vendor-prefixed property or undefined +function vendorPropName( name ) { + + // Check for vendor prefixed names + var capName = name[ 0 ].toUpperCase() + name.slice( 1 ), + i = cssPrefixes.length; + + while ( i-- ) { + name = cssPrefixes[ i ] + capName; + if ( name in emptyStyle ) { + return name; + } + } +} + +// Return a potentially-mapped jQuery.cssProps or vendor prefixed property +function finalPropName( name ) { + var final = jQuery.cssProps[ name ] || vendorProps[ name ]; + + if ( final ) { + return final; + } + if ( name in emptyStyle ) { + return name; + } + return vendorProps[ name ] = vendorPropName( name ) || name; +} + + +var + + // Swappable if display is none or starts with table + // except "table", "table-cell", or "table-caption" + // See here for display values: https://developer.mozilla.org/en-US/docs/CSS/display + rdisplayswap = /^(none|table(?!-c[ea]).+)/, + rcustomProp = /^--/, + cssShow = { position: "absolute", visibility: "hidden", display: "block" }, + cssNormalTransform = { + letterSpacing: "0", + fontWeight: "400" + }; + +function setPositiveNumber( _elem, value, subtract ) { + + // Any relative (+/-) values have already been + // normalized at this point + var matches = rcssNum.exec( value ); + return matches ? + + // Guard against undefined "subtract", e.g., when used as in cssHooks + Math.max( 0, matches[ 2 ] - ( subtract || 0 ) ) + ( matches[ 3 ] || "px" ) : + value; +} + +function boxModelAdjustment( elem, dimension, box, isBorderBox, styles, computedVal ) { + var i = dimension === "width" ? 1 : 0, + extra = 0, + delta = 0; + + // Adjustment may not be necessary + if ( box === ( isBorderBox ? "border" : "content" ) ) { + return 0; + } + + for ( ; i < 4; i += 2 ) { + + // Both box models exclude margin + if ( box === "margin" ) { + delta += jQuery.css( elem, box + cssExpand[ i ], true, styles ); + } + + // If we get here with a content-box, we're seeking "padding" or "border" or "margin" + if ( !isBorderBox ) { + + // Add padding + delta += jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); + + // For "border" or "margin", add border + if ( box !== "padding" ) { + delta += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); + + // But still keep track of it otherwise + } else { + extra += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); + } + + // If we get here with a border-box (content + padding + border), we're seeking "content" or + // "padding" or "margin" + } else { + + // For "content", subtract padding + if ( box === "content" ) { + delta -= jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); + } + + // For "content" or "padding", subtract border + if ( box !== "margin" ) { + delta -= jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); + } + } + } + + // Account for positive content-box scroll gutter when requested by providing computedVal + if ( !isBorderBox && computedVal >= 0 ) { + + // offsetWidth/offsetHeight is a rounded sum of content, padding, scroll gutter, and border + // Assuming integer scroll gutter, subtract the rest and round down + delta += Math.max( 0, Math.ceil( + elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - + computedVal - + delta - + extra - + 0.5 + + // If offsetWidth/offsetHeight is unknown, then we can't determine content-box scroll gutter + // Use an explicit zero to avoid NaN (gh-3964) + ) ) || 0; + } + + return delta; +} + +function getWidthOrHeight( elem, dimension, extra ) { + + // Start with computed style + var styles = getStyles( elem ), + + // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-4322). + // Fake content-box until we know it's needed to know the true value. + boxSizingNeeded = !support.boxSizingReliable() || extra, + isBorderBox = boxSizingNeeded && + jQuery.css( elem, "boxSizing", false, styles ) === "border-box", + valueIsBorderBox = isBorderBox, + + val = curCSS( elem, dimension, styles ), + offsetProp = "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ); + + // Support: Firefox <=54 + // Return a confounding non-pixel value or feign ignorance, as appropriate. + if ( rnumnonpx.test( val ) ) { + if ( !extra ) { + return val; + } + val = "auto"; + } + + + // Support: IE 9 - 11 only + // Use offsetWidth/offsetHeight for when box sizing is unreliable. + // In those cases, the computed value can be trusted to be border-box. + if ( ( !support.boxSizingReliable() && isBorderBox || + + // Support: IE 10 - 11+, Edge 15 - 18+ + // IE/Edge misreport `getComputedStyle` of table rows with width/height + // set in CSS while `offset*` properties report correct values. + // Interestingly, in some cases IE 9 doesn't suffer from this issue. + !support.reliableTrDimensions() && nodeName( elem, "tr" ) || + + // Fall back to offsetWidth/offsetHeight when value is "auto" + // This happens for inline elements with no explicit setting (gh-3571) + val === "auto" || + + // Support: Android <=4.1 - 4.3 only + // Also use offsetWidth/offsetHeight for misreported inline dimensions (gh-3602) + !parseFloat( val ) && jQuery.css( elem, "display", false, styles ) === "inline" ) && + + // Make sure the element is visible & connected + elem.getClientRects().length ) { + + isBorderBox = jQuery.css( elem, "boxSizing", false, styles ) === "border-box"; + + // Where available, offsetWidth/offsetHeight approximate border box dimensions. + // Where not available (e.g., SVG), assume unreliable box-sizing and interpret the + // retrieved value as a content box dimension. + valueIsBorderBox = offsetProp in elem; + if ( valueIsBorderBox ) { + val = elem[ offsetProp ]; + } + } + + // Normalize "" and auto + val = parseFloat( val ) || 0; + + // Adjust for the element's box model + return ( val + + boxModelAdjustment( + elem, + dimension, + extra || ( isBorderBox ? "border" : "content" ), + valueIsBorderBox, + styles, + + // Provide the current computed size to request scroll gutter calculation (gh-3589) + val + ) + ) + "px"; +} + +jQuery.extend( { + + // Add in style property hooks for overriding the default + // behavior of getting and setting a style property + cssHooks: { + opacity: { + get: function( elem, computed ) { + if ( computed ) { + + // We should always get a number back from opacity + var ret = curCSS( elem, "opacity" ); + return ret === "" ? "1" : ret; + } + } + } + }, + + // Don't automatically add "px" to these possibly-unitless properties + cssNumber: { + "animationIterationCount": true, + "columnCount": true, + "fillOpacity": true, + "flexGrow": true, + "flexShrink": true, + "fontWeight": true, + "gridArea": true, + "gridColumn": true, + "gridColumnEnd": true, + "gridColumnStart": true, + "gridRow": true, + "gridRowEnd": true, + "gridRowStart": true, + "lineHeight": true, + "opacity": true, + "order": true, + "orphans": true, + "widows": true, + "zIndex": true, + "zoom": true + }, + + // Add in properties whose names you wish to fix before + // setting or getting the value + cssProps: {}, + + // Get and set the style property on a DOM Node + style: function( elem, name, value, extra ) { + + // Don't set styles on text and comment nodes + if ( !elem || elem.nodeType === 3 || elem.nodeType === 8 || !elem.style ) { + return; + } + + // Make sure that we're working with the right name + var ret, type, hooks, + origName = camelCase( name ), + isCustomProp = rcustomProp.test( name ), + style = elem.style; + + // Make sure that we're working with the right name. We don't + // want to query the value if it is a CSS custom property + // since they are user-defined. + if ( !isCustomProp ) { + name = finalPropName( origName ); + } + + // Gets hook for the prefixed version, then unprefixed version + hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; + + // Check if we're setting a value + if ( value !== undefined ) { + type = typeof value; + + // Convert "+=" or "-=" to relative numbers (#7345) + if ( type === "string" && ( ret = rcssNum.exec( value ) ) && ret[ 1 ] ) { + value = adjustCSS( elem, name, ret ); + + // Fixes bug #9237 + type = "number"; + } + + // Make sure that null and NaN values aren't set (#7116) + if ( value == null || value !== value ) { + return; + } + + // If a number was passed in, add the unit (except for certain CSS properties) + // The isCustomProp check can be removed in jQuery 4.0 when we only auto-append + // "px" to a few hardcoded values. + if ( type === "number" && !isCustomProp ) { + value += ret && ret[ 3 ] || ( jQuery.cssNumber[ origName ] ? "" : "px" ); + } + + // background-* props affect original clone's values + if ( !support.clearCloneStyle && value === "" && name.indexOf( "background" ) === 0 ) { + style[ name ] = "inherit"; + } + + // If a hook was provided, use that value, otherwise just set the specified value + if ( !hooks || !( "set" in hooks ) || + ( value = hooks.set( elem, value, extra ) ) !== undefined ) { + + if ( isCustomProp ) { + style.setProperty( name, value ); + } else { + style[ name ] = value; + } + } + + } else { + + // If a hook was provided get the non-computed value from there + if ( hooks && "get" in hooks && + ( ret = hooks.get( elem, false, extra ) ) !== undefined ) { + + return ret; + } + + // Otherwise just get the value from the style object + return style[ name ]; + } + }, + + css: function( elem, name, extra, styles ) { + var val, num, hooks, + origName = camelCase( name ), + isCustomProp = rcustomProp.test( name ); + + // Make sure that we're working with the right name. We don't + // want to modify the value if it is a CSS custom property + // since they are user-defined. + if ( !isCustomProp ) { + name = finalPropName( origName ); + } + + // Try prefixed name followed by the unprefixed name + hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; + + // If a hook was provided get the computed value from there + if ( hooks && "get" in hooks ) { + val = hooks.get( elem, true, extra ); + } + + // Otherwise, if a way to get the computed value exists, use that + if ( val === undefined ) { + val = curCSS( elem, name, styles ); + } + + // Convert "normal" to computed value + if ( val === "normal" && name in cssNormalTransform ) { + val = cssNormalTransform[ name ]; + } + + // Make numeric if forced or a qualifier was provided and val looks numeric + if ( extra === "" || extra ) { + num = parseFloat( val ); + return extra === true || isFinite( num ) ? num || 0 : val; + } + + return val; + } +} ); + +jQuery.each( [ "height", "width" ], function( _i, dimension ) { + jQuery.cssHooks[ dimension ] = { + get: function( elem, computed, extra ) { + if ( computed ) { + + // Certain elements can have dimension info if we invisibly show them + // but it must have a current display style that would benefit + return rdisplayswap.test( jQuery.css( elem, "display" ) ) && + + // Support: Safari 8+ + // Table columns in Safari have non-zero offsetWidth & zero + // getBoundingClientRect().width unless display is changed. + // Support: IE <=11 only + // Running getBoundingClientRect on a disconnected node + // in IE throws an error. + ( !elem.getClientRects().length || !elem.getBoundingClientRect().width ) ? + swap( elem, cssShow, function() { + return getWidthOrHeight( elem, dimension, extra ); + } ) : + getWidthOrHeight( elem, dimension, extra ); + } + }, + + set: function( elem, value, extra ) { + var matches, + styles = getStyles( elem ), + + // Only read styles.position if the test has a chance to fail + // to avoid forcing a reflow. + scrollboxSizeBuggy = !support.scrollboxSize() && + styles.position === "absolute", + + // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-3991) + boxSizingNeeded = scrollboxSizeBuggy || extra, + isBorderBox = boxSizingNeeded && + jQuery.css( elem, "boxSizing", false, styles ) === "border-box", + subtract = extra ? + boxModelAdjustment( + elem, + dimension, + extra, + isBorderBox, + styles + ) : + 0; + + // Account for unreliable border-box dimensions by comparing offset* to computed and + // faking a content-box to get border and padding (gh-3699) + if ( isBorderBox && scrollboxSizeBuggy ) { + subtract -= Math.ceil( + elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - + parseFloat( styles[ dimension ] ) - + boxModelAdjustment( elem, dimension, "border", false, styles ) - + 0.5 + ); + } + + // Convert to pixels if value adjustment is needed + if ( subtract && ( matches = rcssNum.exec( value ) ) && + ( matches[ 3 ] || "px" ) !== "px" ) { + + elem.style[ dimension ] = value; + value = jQuery.css( elem, dimension ); + } + + return setPositiveNumber( elem, value, subtract ); + } + }; +} ); + +jQuery.cssHooks.marginLeft = addGetHookIf( support.reliableMarginLeft, + function( elem, computed ) { + if ( computed ) { + return ( parseFloat( curCSS( elem, "marginLeft" ) ) || + elem.getBoundingClientRect().left - + swap( elem, { marginLeft: 0 }, function() { + return elem.getBoundingClientRect().left; + } ) + ) + "px"; + } + } +); + +// These hooks are used by animate to expand properties +jQuery.each( { + margin: "", + padding: "", + border: "Width" +}, function( prefix, suffix ) { + jQuery.cssHooks[ prefix + suffix ] = { + expand: function( value ) { + var i = 0, + expanded = {}, + + // Assumes a single number if not a string + parts = typeof value === "string" ? value.split( " " ) : [ value ]; + + for ( ; i < 4; i++ ) { + expanded[ prefix + cssExpand[ i ] + suffix ] = + parts[ i ] || parts[ i - 2 ] || parts[ 0 ]; + } + + return expanded; + } + }; + + if ( prefix !== "margin" ) { + jQuery.cssHooks[ prefix + suffix ].set = setPositiveNumber; + } +} ); + +jQuery.fn.extend( { + css: function( name, value ) { + return access( this, function( elem, name, value ) { + var styles, len, + map = {}, + i = 0; + + if ( Array.isArray( name ) ) { + styles = getStyles( elem ); + len = name.length; + + for ( ; i < len; i++ ) { + map[ name[ i ] ] = jQuery.css( elem, name[ i ], false, styles ); + } + + return map; + } + + return value !== undefined ? + jQuery.style( elem, name, value ) : + jQuery.css( elem, name ); + }, name, value, arguments.length > 1 ); + } +} ); + + +function Tween( elem, options, prop, end, easing ) { + return new Tween.prototype.init( elem, options, prop, end, easing ); +} +jQuery.Tween = Tween; + +Tween.prototype = { + constructor: Tween, + init: function( elem, options, prop, end, easing, unit ) { + this.elem = elem; + this.prop = prop; + this.easing = easing || jQuery.easing._default; + this.options = options; + this.start = this.now = this.cur(); + this.end = end; + this.unit = unit || ( jQuery.cssNumber[ prop ] ? "" : "px" ); + }, + cur: function() { + var hooks = Tween.propHooks[ this.prop ]; + + return hooks && hooks.get ? + hooks.get( this ) : + Tween.propHooks._default.get( this ); + }, + run: function( percent ) { + var eased, + hooks = Tween.propHooks[ this.prop ]; + + if ( this.options.duration ) { + this.pos = eased = jQuery.easing[ this.easing ]( + percent, this.options.duration * percent, 0, 1, this.options.duration + ); + } else { + this.pos = eased = percent; + } + this.now = ( this.end - this.start ) * eased + this.start; + + if ( this.options.step ) { + this.options.step.call( this.elem, this.now, this ); + } + + if ( hooks && hooks.set ) { + hooks.set( this ); + } else { + Tween.propHooks._default.set( this ); + } + return this; + } +}; + +Tween.prototype.init.prototype = Tween.prototype; + +Tween.propHooks = { + _default: { + get: function( tween ) { + var result; + + // Use a property on the element directly when it is not a DOM element, + // or when there is no matching style property that exists. + if ( tween.elem.nodeType !== 1 || + tween.elem[ tween.prop ] != null && tween.elem.style[ tween.prop ] == null ) { + return tween.elem[ tween.prop ]; + } + + // Passing an empty string as a 3rd parameter to .css will automatically + // attempt a parseFloat and fallback to a string if the parse fails. + // Simple values such as "10px" are parsed to Float; + // complex values such as "rotate(1rad)" are returned as-is. + result = jQuery.css( tween.elem, tween.prop, "" ); + + // Empty strings, null, undefined and "auto" are converted to 0. + return !result || result === "auto" ? 0 : result; + }, + set: function( tween ) { + + // Use step hook for back compat. + // Use cssHook if its there. + // Use .style if available and use plain properties where available. + if ( jQuery.fx.step[ tween.prop ] ) { + jQuery.fx.step[ tween.prop ]( tween ); + } else if ( tween.elem.nodeType === 1 && ( + jQuery.cssHooks[ tween.prop ] || + tween.elem.style[ finalPropName( tween.prop ) ] != null ) ) { + jQuery.style( tween.elem, tween.prop, tween.now + tween.unit ); + } else { + tween.elem[ tween.prop ] = tween.now; + } + } + } +}; + +// Support: IE <=9 only +// Panic based approach to setting things on disconnected nodes +Tween.propHooks.scrollTop = Tween.propHooks.scrollLeft = { + set: function( tween ) { + if ( tween.elem.nodeType && tween.elem.parentNode ) { + tween.elem[ tween.prop ] = tween.now; + } + } +}; + +jQuery.easing = { + linear: function( p ) { + return p; + }, + swing: function( p ) { + return 0.5 - Math.cos( p * Math.PI ) / 2; + }, + _default: "swing" +}; + +jQuery.fx = Tween.prototype.init; + +// Back compat <1.8 extension point +jQuery.fx.step = {}; + + + + +var + fxNow, inProgress, + rfxtypes = /^(?:toggle|show|hide)$/, + rrun = /queueHooks$/; + +function schedule() { + if ( inProgress ) { + if ( document.hidden === false && window.requestAnimationFrame ) { + window.requestAnimationFrame( schedule ); + } else { + window.setTimeout( schedule, jQuery.fx.interval ); + } + + jQuery.fx.tick(); + } +} + +// Animations created synchronously will run synchronously +function createFxNow() { + window.setTimeout( function() { + fxNow = undefined; + } ); + return ( fxNow = Date.now() ); +} + +// Generate parameters to create a standard animation +function genFx( type, includeWidth ) { + var which, + i = 0, + attrs = { height: type }; + + // If we include width, step value is 1 to do all cssExpand values, + // otherwise step value is 2 to skip over Left and Right + includeWidth = includeWidth ? 1 : 0; + for ( ; i < 4; i += 2 - includeWidth ) { + which = cssExpand[ i ]; + attrs[ "margin" + which ] = attrs[ "padding" + which ] = type; + } + + if ( includeWidth ) { + attrs.opacity = attrs.width = type; + } + + return attrs; +} + +function createTween( value, prop, animation ) { + var tween, + collection = ( Animation.tweeners[ prop ] || [] ).concat( Animation.tweeners[ "*" ] ), + index = 0, + length = collection.length; + for ( ; index < length; index++ ) { + if ( ( tween = collection[ index ].call( animation, prop, value ) ) ) { + + // We're done with this property + return tween; + } + } +} + +function defaultPrefilter( elem, props, opts ) { + var prop, value, toggle, hooks, oldfire, propTween, restoreDisplay, display, + isBox = "width" in props || "height" in props, + anim = this, + orig = {}, + style = elem.style, + hidden = elem.nodeType && isHiddenWithinTree( elem ), + dataShow = dataPriv.get( elem, "fxshow" ); + + // Queue-skipping animations hijack the fx hooks + if ( !opts.queue ) { + hooks = jQuery._queueHooks( elem, "fx" ); + if ( hooks.unqueued == null ) { + hooks.unqueued = 0; + oldfire = hooks.empty.fire; + hooks.empty.fire = function() { + if ( !hooks.unqueued ) { + oldfire(); + } + }; + } + hooks.unqueued++; + + anim.always( function() { + + // Ensure the complete handler is called before this completes + anim.always( function() { + hooks.unqueued--; + if ( !jQuery.queue( elem, "fx" ).length ) { + hooks.empty.fire(); + } + } ); + } ); + } + + // Detect show/hide animations + for ( prop in props ) { + value = props[ prop ]; + if ( rfxtypes.test( value ) ) { + delete props[ prop ]; + toggle = toggle || value === "toggle"; + if ( value === ( hidden ? "hide" : "show" ) ) { + + // Pretend to be hidden if this is a "show" and + // there is still data from a stopped show/hide + if ( value === "show" && dataShow && dataShow[ prop ] !== undefined ) { + hidden = true; + + // Ignore all other no-op show/hide data + } else { + continue; + } + } + orig[ prop ] = dataShow && dataShow[ prop ] || jQuery.style( elem, prop ); + } + } + + // Bail out if this is a no-op like .hide().hide() + propTween = !jQuery.isEmptyObject( props ); + if ( !propTween && jQuery.isEmptyObject( orig ) ) { + return; + } + + // Restrict "overflow" and "display" styles during box animations + if ( isBox && elem.nodeType === 1 ) { + + // Support: IE <=9 - 11, Edge 12 - 15 + // Record all 3 overflow attributes because IE does not infer the shorthand + // from identically-valued overflowX and overflowY and Edge just mirrors + // the overflowX value there. + opts.overflow = [ style.overflow, style.overflowX, style.overflowY ]; + + // Identify a display type, preferring old show/hide data over the CSS cascade + restoreDisplay = dataShow && dataShow.display; + if ( restoreDisplay == null ) { + restoreDisplay = dataPriv.get( elem, "display" ); + } + display = jQuery.css( elem, "display" ); + if ( display === "none" ) { + if ( restoreDisplay ) { + display = restoreDisplay; + } else { + + // Get nonempty value(s) by temporarily forcing visibility + showHide( [ elem ], true ); + restoreDisplay = elem.style.display || restoreDisplay; + display = jQuery.css( elem, "display" ); + showHide( [ elem ] ); + } + } + + // Animate inline elements as inline-block + if ( display === "inline" || display === "inline-block" && restoreDisplay != null ) { + if ( jQuery.css( elem, "float" ) === "none" ) { + + // Restore the original display value at the end of pure show/hide animations + if ( !propTween ) { + anim.done( function() { + style.display = restoreDisplay; + } ); + if ( restoreDisplay == null ) { + display = style.display; + restoreDisplay = display === "none" ? "" : display; + } + } + style.display = "inline-block"; + } + } + } + + if ( opts.overflow ) { + style.overflow = "hidden"; + anim.always( function() { + style.overflow = opts.overflow[ 0 ]; + style.overflowX = opts.overflow[ 1 ]; + style.overflowY = opts.overflow[ 2 ]; + } ); + } + + // Implement show/hide animations + propTween = false; + for ( prop in orig ) { + + // General show/hide setup for this element animation + if ( !propTween ) { + if ( dataShow ) { + if ( "hidden" in dataShow ) { + hidden = dataShow.hidden; + } + } else { + dataShow = dataPriv.access( elem, "fxshow", { display: restoreDisplay } ); + } + + // Store hidden/visible for toggle so `.stop().toggle()` "reverses" + if ( toggle ) { + dataShow.hidden = !hidden; + } + + // Show elements before animating them + if ( hidden ) { + showHide( [ elem ], true ); + } + + /* eslint-disable no-loop-func */ + + anim.done( function() { + + /* eslint-enable no-loop-func */ + + // The final step of a "hide" animation is actually hiding the element + if ( !hidden ) { + showHide( [ elem ] ); + } + dataPriv.remove( elem, "fxshow" ); + for ( prop in orig ) { + jQuery.style( elem, prop, orig[ prop ] ); + } + } ); + } + + // Per-property setup + propTween = createTween( hidden ? dataShow[ prop ] : 0, prop, anim ); + if ( !( prop in dataShow ) ) { + dataShow[ prop ] = propTween.start; + if ( hidden ) { + propTween.end = propTween.start; + propTween.start = 0; + } + } + } +} + +function propFilter( props, specialEasing ) { + var index, name, easing, value, hooks; + + // camelCase, specialEasing and expand cssHook pass + for ( index in props ) { + name = camelCase( index ); + easing = specialEasing[ name ]; + value = props[ index ]; + if ( Array.isArray( value ) ) { + easing = value[ 1 ]; + value = props[ index ] = value[ 0 ]; + } + + if ( index !== name ) { + props[ name ] = value; + delete props[ index ]; + } + + hooks = jQuery.cssHooks[ name ]; + if ( hooks && "expand" in hooks ) { + value = hooks.expand( value ); + delete props[ name ]; + + // Not quite $.extend, this won't overwrite existing keys. + // Reusing 'index' because we have the correct "name" + for ( index in value ) { + if ( !( index in props ) ) { + props[ index ] = value[ index ]; + specialEasing[ index ] = easing; + } + } + } else { + specialEasing[ name ] = easing; + } + } +} + +function Animation( elem, properties, options ) { + var result, + stopped, + index = 0, + length = Animation.prefilters.length, + deferred = jQuery.Deferred().always( function() { + + // Don't match elem in the :animated selector + delete tick.elem; + } ), + tick = function() { + if ( stopped ) { + return false; + } + var currentTime = fxNow || createFxNow(), + remaining = Math.max( 0, animation.startTime + animation.duration - currentTime ), + + // Support: Android 2.3 only + // Archaic crash bug won't allow us to use `1 - ( 0.5 || 0 )` (#12497) + temp = remaining / animation.duration || 0, + percent = 1 - temp, + index = 0, + length = animation.tweens.length; + + for ( ; index < length; index++ ) { + animation.tweens[ index ].run( percent ); + } + + deferred.notifyWith( elem, [ animation, percent, remaining ] ); + + // If there's more to do, yield + if ( percent < 1 && length ) { + return remaining; + } + + // If this was an empty animation, synthesize a final progress notification + if ( !length ) { + deferred.notifyWith( elem, [ animation, 1, 0 ] ); + } + + // Resolve the animation and report its conclusion + deferred.resolveWith( elem, [ animation ] ); + return false; + }, + animation = deferred.promise( { + elem: elem, + props: jQuery.extend( {}, properties ), + opts: jQuery.extend( true, { + specialEasing: {}, + easing: jQuery.easing._default + }, options ), + originalProperties: properties, + originalOptions: options, + startTime: fxNow || createFxNow(), + duration: options.duration, + tweens: [], + createTween: function( prop, end ) { + var tween = jQuery.Tween( elem, animation.opts, prop, end, + animation.opts.specialEasing[ prop ] || animation.opts.easing ); + animation.tweens.push( tween ); + return tween; + }, + stop: function( gotoEnd ) { + var index = 0, + + // If we are going to the end, we want to run all the tweens + // otherwise we skip this part + length = gotoEnd ? animation.tweens.length : 0; + if ( stopped ) { + return this; + } + stopped = true; + for ( ; index < length; index++ ) { + animation.tweens[ index ].run( 1 ); + } + + // Resolve when we played the last frame; otherwise, reject + if ( gotoEnd ) { + deferred.notifyWith( elem, [ animation, 1, 0 ] ); + deferred.resolveWith( elem, [ animation, gotoEnd ] ); + } else { + deferred.rejectWith( elem, [ animation, gotoEnd ] ); + } + return this; + } + } ), + props = animation.props; + + propFilter( props, animation.opts.specialEasing ); + + for ( ; index < length; index++ ) { + result = Animation.prefilters[ index ].call( animation, elem, props, animation.opts ); + if ( result ) { + if ( isFunction( result.stop ) ) { + jQuery._queueHooks( animation.elem, animation.opts.queue ).stop = + result.stop.bind( result ); + } + return result; + } + } + + jQuery.map( props, createTween, animation ); + + if ( isFunction( animation.opts.start ) ) { + animation.opts.start.call( elem, animation ); + } + + // Attach callbacks from options + animation + .progress( animation.opts.progress ) + .done( animation.opts.done, animation.opts.complete ) + .fail( animation.opts.fail ) + .always( animation.opts.always ); + + jQuery.fx.timer( + jQuery.extend( tick, { + elem: elem, + anim: animation, + queue: animation.opts.queue + } ) + ); + + return animation; +} + +jQuery.Animation = jQuery.extend( Animation, { + + tweeners: { + "*": [ function( prop, value ) { + var tween = this.createTween( prop, value ); + adjustCSS( tween.elem, prop, rcssNum.exec( value ), tween ); + return tween; + } ] + }, + + tweener: function( props, callback ) { + if ( isFunction( props ) ) { + callback = props; + props = [ "*" ]; + } else { + props = props.match( rnothtmlwhite ); + } + + var prop, + index = 0, + length = props.length; + + for ( ; index < length; index++ ) { + prop = props[ index ]; + Animation.tweeners[ prop ] = Animation.tweeners[ prop ] || []; + Animation.tweeners[ prop ].unshift( callback ); + } + }, + + prefilters: [ defaultPrefilter ], + + prefilter: function( callback, prepend ) { + if ( prepend ) { + Animation.prefilters.unshift( callback ); + } else { + Animation.prefilters.push( callback ); + } + } +} ); + +jQuery.speed = function( speed, easing, fn ) { + var opt = speed && typeof speed === "object" ? jQuery.extend( {}, speed ) : { + complete: fn || !fn && easing || + isFunction( speed ) && speed, + duration: speed, + easing: fn && easing || easing && !isFunction( easing ) && easing + }; + + // Go to the end state if fx are off + if ( jQuery.fx.off ) { + opt.duration = 0; + + } else { + if ( typeof opt.duration !== "number" ) { + if ( opt.duration in jQuery.fx.speeds ) { + opt.duration = jQuery.fx.speeds[ opt.duration ]; + + } else { + opt.duration = jQuery.fx.speeds._default; + } + } + } + + // Normalize opt.queue - true/undefined/null -> "fx" + if ( opt.queue == null || opt.queue === true ) { + opt.queue = "fx"; + } + + // Queueing + opt.old = opt.complete; + + opt.complete = function() { + if ( isFunction( opt.old ) ) { + opt.old.call( this ); + } + + if ( opt.queue ) { + jQuery.dequeue( this, opt.queue ); + } + }; + + return opt; +}; + +jQuery.fn.extend( { + fadeTo: function( speed, to, easing, callback ) { + + // Show any hidden elements after setting opacity to 0 + return this.filter( isHiddenWithinTree ).css( "opacity", 0 ).show() + + // Animate to the value specified + .end().animate( { opacity: to }, speed, easing, callback ); + }, + animate: function( prop, speed, easing, callback ) { + var empty = jQuery.isEmptyObject( prop ), + optall = jQuery.speed( speed, easing, callback ), + doAnimation = function() { + + // Operate on a copy of prop so per-property easing won't be lost + var anim = Animation( this, jQuery.extend( {}, prop ), optall ); + + // Empty animations, or finishing resolves immediately + if ( empty || dataPriv.get( this, "finish" ) ) { + anim.stop( true ); + } + }; + doAnimation.finish = doAnimation; + + return empty || optall.queue === false ? + this.each( doAnimation ) : + this.queue( optall.queue, doAnimation ); + }, + stop: function( type, clearQueue, gotoEnd ) { + var stopQueue = function( hooks ) { + var stop = hooks.stop; + delete hooks.stop; + stop( gotoEnd ); + }; + + if ( typeof type !== "string" ) { + gotoEnd = clearQueue; + clearQueue = type; + type = undefined; + } + if ( clearQueue ) { + this.queue( type || "fx", [] ); + } + + return this.each( function() { + var dequeue = true, + index = type != null && type + "queueHooks", + timers = jQuery.timers, + data = dataPriv.get( this ); + + if ( index ) { + if ( data[ index ] && data[ index ].stop ) { + stopQueue( data[ index ] ); + } + } else { + for ( index in data ) { + if ( data[ index ] && data[ index ].stop && rrun.test( index ) ) { + stopQueue( data[ index ] ); + } + } + } + + for ( index = timers.length; index--; ) { + if ( timers[ index ].elem === this && + ( type == null || timers[ index ].queue === type ) ) { + + timers[ index ].anim.stop( gotoEnd ); + dequeue = false; + timers.splice( index, 1 ); + } + } + + // Start the next in the queue if the last step wasn't forced. + // Timers currently will call their complete callbacks, which + // will dequeue but only if they were gotoEnd. + if ( dequeue || !gotoEnd ) { + jQuery.dequeue( this, type ); + } + } ); + }, + finish: function( type ) { + if ( type !== false ) { + type = type || "fx"; + } + return this.each( function() { + var index, + data = dataPriv.get( this ), + queue = data[ type + "queue" ], + hooks = data[ type + "queueHooks" ], + timers = jQuery.timers, + length = queue ? queue.length : 0; + + // Enable finishing flag on private data + data.finish = true; + + // Empty the queue first + jQuery.queue( this, type, [] ); + + if ( hooks && hooks.stop ) { + hooks.stop.call( this, true ); + } + + // Look for any active animations, and finish them + for ( index = timers.length; index--; ) { + if ( timers[ index ].elem === this && timers[ index ].queue === type ) { + timers[ index ].anim.stop( true ); + timers.splice( index, 1 ); + } + } + + // Look for any animations in the old queue and finish them + for ( index = 0; index < length; index++ ) { + if ( queue[ index ] && queue[ index ].finish ) { + queue[ index ].finish.call( this ); + } + } + + // Turn off finishing flag + delete data.finish; + } ); + } +} ); + +jQuery.each( [ "toggle", "show", "hide" ], function( _i, name ) { + var cssFn = jQuery.fn[ name ]; + jQuery.fn[ name ] = function( speed, easing, callback ) { + return speed == null || typeof speed === "boolean" ? + cssFn.apply( this, arguments ) : + this.animate( genFx( name, true ), speed, easing, callback ); + }; +} ); + +// Generate shortcuts for custom animations +jQuery.each( { + slideDown: genFx( "show" ), + slideUp: genFx( "hide" ), + slideToggle: genFx( "toggle" ), + fadeIn: { opacity: "show" }, + fadeOut: { opacity: "hide" }, + fadeToggle: { opacity: "toggle" } +}, function( name, props ) { + jQuery.fn[ name ] = function( speed, easing, callback ) { + return this.animate( props, speed, easing, callback ); + }; +} ); + +jQuery.timers = []; +jQuery.fx.tick = function() { + var timer, + i = 0, + timers = jQuery.timers; + + fxNow = Date.now(); + + for ( ; i < timers.length; i++ ) { + timer = timers[ i ]; + + // Run the timer and safely remove it when done (allowing for external removal) + if ( !timer() && timers[ i ] === timer ) { + timers.splice( i--, 1 ); + } + } + + if ( !timers.length ) { + jQuery.fx.stop(); + } + fxNow = undefined; +}; + +jQuery.fx.timer = function( timer ) { + jQuery.timers.push( timer ); + jQuery.fx.start(); +}; + +jQuery.fx.interval = 13; +jQuery.fx.start = function() { + if ( inProgress ) { + return; + } + + inProgress = true; + schedule(); +}; + +jQuery.fx.stop = function() { + inProgress = null; +}; + +jQuery.fx.speeds = { + slow: 600, + fast: 200, + + // Default speed + _default: 400 +}; + + +// Based off of the plugin by Clint Helfers, with permission. +// https://web.archive.org/web/20100324014747/http://blindsignals.com/index.php/2009/07/jquery-delay/ +jQuery.fn.delay = function( time, type ) { + time = jQuery.fx ? jQuery.fx.speeds[ time ] || time : time; + type = type || "fx"; + + return this.queue( type, function( next, hooks ) { + var timeout = window.setTimeout( next, time ); + hooks.stop = function() { + window.clearTimeout( timeout ); + }; + } ); +}; + + +( function() { + var input = document.createElement( "input" ), + select = document.createElement( "select" ), + opt = select.appendChild( document.createElement( "option" ) ); + + input.type = "checkbox"; + + // Support: Android <=4.3 only + // Default value for a checkbox should be "on" + support.checkOn = input.value !== ""; + + // Support: IE <=11 only + // Must access selectedIndex to make default options select + support.optSelected = opt.selected; + + // Support: IE <=11 only + // An input loses its value after becoming a radio + input = document.createElement( "input" ); + input.value = "t"; + input.type = "radio"; + support.radioValue = input.value === "t"; +} )(); + + +var boolHook, + attrHandle = jQuery.expr.attrHandle; + +jQuery.fn.extend( { + attr: function( name, value ) { + return access( this, jQuery.attr, name, value, arguments.length > 1 ); + }, + + removeAttr: function( name ) { + return this.each( function() { + jQuery.removeAttr( this, name ); + } ); + } +} ); + +jQuery.extend( { + attr: function( elem, name, value ) { + var ret, hooks, + nType = elem.nodeType; + + // Don't get/set attributes on text, comment and attribute nodes + if ( nType === 3 || nType === 8 || nType === 2 ) { + return; + } + + // Fallback to prop when attributes are not supported + if ( typeof elem.getAttribute === "undefined" ) { + return jQuery.prop( elem, name, value ); + } + + // Attribute hooks are determined by the lowercase version + // Grab necessary hook if one is defined + if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { + hooks = jQuery.attrHooks[ name.toLowerCase() ] || + ( jQuery.expr.match.bool.test( name ) ? boolHook : undefined ); + } + + if ( value !== undefined ) { + if ( value === null ) { + jQuery.removeAttr( elem, name ); + return; + } + + if ( hooks && "set" in hooks && + ( ret = hooks.set( elem, value, name ) ) !== undefined ) { + return ret; + } + + elem.setAttribute( name, value + "" ); + return value; + } + + if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { + return ret; + } + + ret = jQuery.find.attr( elem, name ); + + // Non-existent attributes return null, we normalize to undefined + return ret == null ? undefined : ret; + }, + + attrHooks: { + type: { + set: function( elem, value ) { + if ( !support.radioValue && value === "radio" && + nodeName( elem, "input" ) ) { + var val = elem.value; + elem.setAttribute( "type", value ); + if ( val ) { + elem.value = val; + } + return value; + } + } + } + }, + + removeAttr: function( elem, value ) { + var name, + i = 0, + + // Attribute names can contain non-HTML whitespace characters + // https://html.spec.whatwg.org/multipage/syntax.html#attributes-2 + attrNames = value && value.match( rnothtmlwhite ); + + if ( attrNames && elem.nodeType === 1 ) { + while ( ( name = attrNames[ i++ ] ) ) { + elem.removeAttribute( name ); + } + } + } +} ); + +// Hooks for boolean attributes +boolHook = { + set: function( elem, value, name ) { + if ( value === false ) { + + // Remove boolean attributes when set to false + jQuery.removeAttr( elem, name ); + } else { + elem.setAttribute( name, name ); + } + return name; + } +}; + +jQuery.each( jQuery.expr.match.bool.source.match( /\w+/g ), function( _i, name ) { + var getter = attrHandle[ name ] || jQuery.find.attr; + + attrHandle[ name ] = function( elem, name, isXML ) { + var ret, handle, + lowercaseName = name.toLowerCase(); + + if ( !isXML ) { + + // Avoid an infinite loop by temporarily removing this function from the getter + handle = attrHandle[ lowercaseName ]; + attrHandle[ lowercaseName ] = ret; + ret = getter( elem, name, isXML ) != null ? + lowercaseName : + null; + attrHandle[ lowercaseName ] = handle; + } + return ret; + }; +} ); + + + + +var rfocusable = /^(?:input|select|textarea|button)$/i, + rclickable = /^(?:a|area)$/i; + +jQuery.fn.extend( { + prop: function( name, value ) { + return access( this, jQuery.prop, name, value, arguments.length > 1 ); + }, + + removeProp: function( name ) { + return this.each( function() { + delete this[ jQuery.propFix[ name ] || name ]; + } ); + } +} ); + +jQuery.extend( { + prop: function( elem, name, value ) { + var ret, hooks, + nType = elem.nodeType; + + // Don't get/set properties on text, comment and attribute nodes + if ( nType === 3 || nType === 8 || nType === 2 ) { + return; + } + + if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { + + // Fix name and attach hooks + name = jQuery.propFix[ name ] || name; + hooks = jQuery.propHooks[ name ]; + } + + if ( value !== undefined ) { + if ( hooks && "set" in hooks && + ( ret = hooks.set( elem, value, name ) ) !== undefined ) { + return ret; + } + + return ( elem[ name ] = value ); + } + + if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { + return ret; + } + + return elem[ name ]; + }, + + propHooks: { + tabIndex: { + get: function( elem ) { + + // Support: IE <=9 - 11 only + // elem.tabIndex doesn't always return the + // correct value when it hasn't been explicitly set + // https://web.archive.org/web/20141116233347/http://fluidproject.org/blog/2008/01/09/getting-setting-and-removing-tabindex-values-with-javascript/ + // Use proper attribute retrieval(#12072) + var tabindex = jQuery.find.attr( elem, "tabindex" ); + + if ( tabindex ) { + return parseInt( tabindex, 10 ); + } + + if ( + rfocusable.test( elem.nodeName ) || + rclickable.test( elem.nodeName ) && + elem.href + ) { + return 0; + } + + return -1; + } + } + }, + + propFix: { + "for": "htmlFor", + "class": "className" + } +} ); + +// Support: IE <=11 only +// Accessing the selectedIndex property +// forces the browser to respect setting selected +// on the option +// The getter ensures a default option is selected +// when in an optgroup +// eslint rule "no-unused-expressions" is disabled for this code +// since it considers such accessions noop +if ( !support.optSelected ) { + jQuery.propHooks.selected = { + get: function( elem ) { + + /* eslint no-unused-expressions: "off" */ + + var parent = elem.parentNode; + if ( parent && parent.parentNode ) { + parent.parentNode.selectedIndex; + } + return null; + }, + set: function( elem ) { + + /* eslint no-unused-expressions: "off" */ + + var parent = elem.parentNode; + if ( parent ) { + parent.selectedIndex; + + if ( parent.parentNode ) { + parent.parentNode.selectedIndex; + } + } + } + }; +} + +jQuery.each( [ + "tabIndex", + "readOnly", + "maxLength", + "cellSpacing", + "cellPadding", + "rowSpan", + "colSpan", + "useMap", + "frameBorder", + "contentEditable" +], function() { + jQuery.propFix[ this.toLowerCase() ] = this; +} ); + + + + + // Strip and collapse whitespace according to HTML spec + // https://infra.spec.whatwg.org/#strip-and-collapse-ascii-whitespace + function stripAndCollapse( value ) { + var tokens = value.match( rnothtmlwhite ) || []; + return tokens.join( " " ); + } + + +function getClass( elem ) { + return elem.getAttribute && elem.getAttribute( "class" ) || ""; +} + +function classesToArray( value ) { + if ( Array.isArray( value ) ) { + return value; + } + if ( typeof value === "string" ) { + return value.match( rnothtmlwhite ) || []; + } + return []; +} + +jQuery.fn.extend( { + addClass: function( value ) { + var classes, elem, cur, curValue, clazz, j, finalValue, + i = 0; + + if ( isFunction( value ) ) { + return this.each( function( j ) { + jQuery( this ).addClass( value.call( this, j, getClass( this ) ) ); + } ); + } + + classes = classesToArray( value ); + + if ( classes.length ) { + while ( ( elem = this[ i++ ] ) ) { + curValue = getClass( elem ); + cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); + + if ( cur ) { + j = 0; + while ( ( clazz = classes[ j++ ] ) ) { + if ( cur.indexOf( " " + clazz + " " ) < 0 ) { + cur += clazz + " "; + } + } + + // Only assign if different to avoid unneeded rendering. + finalValue = stripAndCollapse( cur ); + if ( curValue !== finalValue ) { + elem.setAttribute( "class", finalValue ); + } + } + } + } + + return this; + }, + + removeClass: function( value ) { + var classes, elem, cur, curValue, clazz, j, finalValue, + i = 0; + + if ( isFunction( value ) ) { + return this.each( function( j ) { + jQuery( this ).removeClass( value.call( this, j, getClass( this ) ) ); + } ); + } + + if ( !arguments.length ) { + return this.attr( "class", "" ); + } + + classes = classesToArray( value ); + + if ( classes.length ) { + while ( ( elem = this[ i++ ] ) ) { + curValue = getClass( elem ); + + // This expression is here for better compressibility (see addClass) + cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); + + if ( cur ) { + j = 0; + while ( ( clazz = classes[ j++ ] ) ) { + + // Remove *all* instances + while ( cur.indexOf( " " + clazz + " " ) > -1 ) { + cur = cur.replace( " " + clazz + " ", " " ); + } + } + + // Only assign if different to avoid unneeded rendering. + finalValue = stripAndCollapse( cur ); + if ( curValue !== finalValue ) { + elem.setAttribute( "class", finalValue ); + } + } + } + } + + return this; + }, + + toggleClass: function( value, stateVal ) { + var type = typeof value, + isValidValue = type === "string" || Array.isArray( value ); + + if ( typeof stateVal === "boolean" && isValidValue ) { + return stateVal ? this.addClass( value ) : this.removeClass( value ); + } + + if ( isFunction( value ) ) { + return this.each( function( i ) { + jQuery( this ).toggleClass( + value.call( this, i, getClass( this ), stateVal ), + stateVal + ); + } ); + } + + return this.each( function() { + var className, i, self, classNames; + + if ( isValidValue ) { + + // Toggle individual class names + i = 0; + self = jQuery( this ); + classNames = classesToArray( value ); + + while ( ( className = classNames[ i++ ] ) ) { + + // Check each className given, space separated list + if ( self.hasClass( className ) ) { + self.removeClass( className ); + } else { + self.addClass( className ); + } + } + + // Toggle whole class name + } else if ( value === undefined || type === "boolean" ) { + className = getClass( this ); + if ( className ) { + + // Store className if set + dataPriv.set( this, "__className__", className ); + } + + // If the element has a class name or if we're passed `false`, + // then remove the whole classname (if there was one, the above saved it). + // Otherwise bring back whatever was previously saved (if anything), + // falling back to the empty string if nothing was stored. + if ( this.setAttribute ) { + this.setAttribute( "class", + className || value === false ? + "" : + dataPriv.get( this, "__className__" ) || "" + ); + } + } + } ); + }, + + hasClass: function( selector ) { + var className, elem, + i = 0; + + className = " " + selector + " "; + while ( ( elem = this[ i++ ] ) ) { + if ( elem.nodeType === 1 && + ( " " + stripAndCollapse( getClass( elem ) ) + " " ).indexOf( className ) > -1 ) { + return true; + } + } + + return false; + } +} ); + + + + +var rreturn = /\r/g; + +jQuery.fn.extend( { + val: function( value ) { + var hooks, ret, valueIsFunction, + elem = this[ 0 ]; + + if ( !arguments.length ) { + if ( elem ) { + hooks = jQuery.valHooks[ elem.type ] || + jQuery.valHooks[ elem.nodeName.toLowerCase() ]; + + if ( hooks && + "get" in hooks && + ( ret = hooks.get( elem, "value" ) ) !== undefined + ) { + return ret; + } + + ret = elem.value; + + // Handle most common string cases + if ( typeof ret === "string" ) { + return ret.replace( rreturn, "" ); + } + + // Handle cases where value is null/undef or number + return ret == null ? "" : ret; + } + + return; + } + + valueIsFunction = isFunction( value ); + + return this.each( function( i ) { + var val; + + if ( this.nodeType !== 1 ) { + return; + } + + if ( valueIsFunction ) { + val = value.call( this, i, jQuery( this ).val() ); + } else { + val = value; + } + + // Treat null/undefined as ""; convert numbers to string + if ( val == null ) { + val = ""; + + } else if ( typeof val === "number" ) { + val += ""; + + } else if ( Array.isArray( val ) ) { + val = jQuery.map( val, function( value ) { + return value == null ? "" : value + ""; + } ); + } + + hooks = jQuery.valHooks[ this.type ] || jQuery.valHooks[ this.nodeName.toLowerCase() ]; + + // If set returns undefined, fall back to normal setting + if ( !hooks || !( "set" in hooks ) || hooks.set( this, val, "value" ) === undefined ) { + this.value = val; + } + } ); + } +} ); + +jQuery.extend( { + valHooks: { + option: { + get: function( elem ) { + + var val = jQuery.find.attr( elem, "value" ); + return val != null ? + val : + + // Support: IE <=10 - 11 only + // option.text throws exceptions (#14686, #14858) + // Strip and collapse whitespace + // https://html.spec.whatwg.org/#strip-and-collapse-whitespace + stripAndCollapse( jQuery.text( elem ) ); + } + }, + select: { + get: function( elem ) { + var value, option, i, + options = elem.options, + index = elem.selectedIndex, + one = elem.type === "select-one", + values = one ? null : [], + max = one ? index + 1 : options.length; + + if ( index < 0 ) { + i = max; + + } else { + i = one ? index : 0; + } + + // Loop through all the selected options + for ( ; i < max; i++ ) { + option = options[ i ]; + + // Support: IE <=9 only + // IE8-9 doesn't update selected after form reset (#2551) + if ( ( option.selected || i === index ) && + + // Don't return options that are disabled or in a disabled optgroup + !option.disabled && + ( !option.parentNode.disabled || + !nodeName( option.parentNode, "optgroup" ) ) ) { + + // Get the specific value for the option + value = jQuery( option ).val(); + + // We don't need an array for one selects + if ( one ) { + return value; + } + + // Multi-Selects return an array + values.push( value ); + } + } + + return values; + }, + + set: function( elem, value ) { + var optionSet, option, + options = elem.options, + values = jQuery.makeArray( value ), + i = options.length; + + while ( i-- ) { + option = options[ i ]; + + /* eslint-disable no-cond-assign */ + + if ( option.selected = + jQuery.inArray( jQuery.valHooks.option.get( option ), values ) > -1 + ) { + optionSet = true; + } + + /* eslint-enable no-cond-assign */ + } + + // Force browsers to behave consistently when non-matching value is set + if ( !optionSet ) { + elem.selectedIndex = -1; + } + return values; + } + } + } +} ); + +// Radios and checkboxes getter/setter +jQuery.each( [ "radio", "checkbox" ], function() { + jQuery.valHooks[ this ] = { + set: function( elem, value ) { + if ( Array.isArray( value ) ) { + return ( elem.checked = jQuery.inArray( jQuery( elem ).val(), value ) > -1 ); + } + } + }; + if ( !support.checkOn ) { + jQuery.valHooks[ this ].get = function( elem ) { + return elem.getAttribute( "value" ) === null ? "on" : elem.value; + }; + } +} ); + + + + +// Return jQuery for attributes-only inclusion + + +support.focusin = "onfocusin" in window; + + +var rfocusMorph = /^(?:focusinfocus|focusoutblur)$/, + stopPropagationCallback = function( e ) { + e.stopPropagation(); + }; + +jQuery.extend( jQuery.event, { + + trigger: function( event, data, elem, onlyHandlers ) { + + var i, cur, tmp, bubbleType, ontype, handle, special, lastElement, + eventPath = [ elem || document ], + type = hasOwn.call( event, "type" ) ? event.type : event, + namespaces = hasOwn.call( event, "namespace" ) ? event.namespace.split( "." ) : []; + + cur = lastElement = tmp = elem = elem || document; + + // Don't do events on text and comment nodes + if ( elem.nodeType === 3 || elem.nodeType === 8 ) { + return; + } + + // focus/blur morphs to focusin/out; ensure we're not firing them right now + if ( rfocusMorph.test( type + jQuery.event.triggered ) ) { + return; + } + + if ( type.indexOf( "." ) > -1 ) { + + // Namespaced trigger; create a regexp to match event type in handle() + namespaces = type.split( "." ); + type = namespaces.shift(); + namespaces.sort(); + } + ontype = type.indexOf( ":" ) < 0 && "on" + type; + + // Caller can pass in a jQuery.Event object, Object, or just an event type string + event = event[ jQuery.expando ] ? + event : + new jQuery.Event( type, typeof event === "object" && event ); + + // Trigger bitmask: & 1 for native handlers; & 2 for jQuery (always true) + event.isTrigger = onlyHandlers ? 2 : 3; + event.namespace = namespaces.join( "." ); + event.rnamespace = event.namespace ? + new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ) : + null; + + // Clean up the event in case it is being reused + event.result = undefined; + if ( !event.target ) { + event.target = elem; + } + + // Clone any incoming data and prepend the event, creating the handler arg list + data = data == null ? + [ event ] : + jQuery.makeArray( data, [ event ] ); + + // Allow special events to draw outside the lines + special = jQuery.event.special[ type ] || {}; + if ( !onlyHandlers && special.trigger && special.trigger.apply( elem, data ) === false ) { + return; + } + + // Determine event propagation path in advance, per W3C events spec (#9951) + // Bubble up to document, then to window; watch for a global ownerDocument var (#9724) + if ( !onlyHandlers && !special.noBubble && !isWindow( elem ) ) { + + bubbleType = special.delegateType || type; + if ( !rfocusMorph.test( bubbleType + type ) ) { + cur = cur.parentNode; + } + for ( ; cur; cur = cur.parentNode ) { + eventPath.push( cur ); + tmp = cur; + } + + // Only add window if we got to document (e.g., not plain obj or detached DOM) + if ( tmp === ( elem.ownerDocument || document ) ) { + eventPath.push( tmp.defaultView || tmp.parentWindow || window ); + } + } + + // Fire handlers on the event path + i = 0; + while ( ( cur = eventPath[ i++ ] ) && !event.isPropagationStopped() ) { + lastElement = cur; + event.type = i > 1 ? + bubbleType : + special.bindType || type; + + // jQuery handler + handle = ( + dataPriv.get( cur, "events" ) || Object.create( null ) + )[ event.type ] && + dataPriv.get( cur, "handle" ); + if ( handle ) { + handle.apply( cur, data ); + } + + // Native handler + handle = ontype && cur[ ontype ]; + if ( handle && handle.apply && acceptData( cur ) ) { + event.result = handle.apply( cur, data ); + if ( event.result === false ) { + event.preventDefault(); + } + } + } + event.type = type; + + // If nobody prevented the default action, do it now + if ( !onlyHandlers && !event.isDefaultPrevented() ) { + + if ( ( !special._default || + special._default.apply( eventPath.pop(), data ) === false ) && + acceptData( elem ) ) { + + // Call a native DOM method on the target with the same name as the event. + // Don't do default actions on window, that's where global variables be (#6170) + if ( ontype && isFunction( elem[ type ] ) && !isWindow( elem ) ) { + + // Don't re-trigger an onFOO event when we call its FOO() method + tmp = elem[ ontype ]; + + if ( tmp ) { + elem[ ontype ] = null; + } + + // Prevent re-triggering of the same event, since we already bubbled it above + jQuery.event.triggered = type; + + if ( event.isPropagationStopped() ) { + lastElement.addEventListener( type, stopPropagationCallback ); + } + + elem[ type ](); + + if ( event.isPropagationStopped() ) { + lastElement.removeEventListener( type, stopPropagationCallback ); + } + + jQuery.event.triggered = undefined; + + if ( tmp ) { + elem[ ontype ] = tmp; + } + } + } + } + + return event.result; + }, + + // Piggyback on a donor event to simulate a different one + // Used only for `focus(in | out)` events + simulate: function( type, elem, event ) { + var e = jQuery.extend( + new jQuery.Event(), + event, + { + type: type, + isSimulated: true + } + ); + + jQuery.event.trigger( e, null, elem ); + } + +} ); + +jQuery.fn.extend( { + + trigger: function( type, data ) { + return this.each( function() { + jQuery.event.trigger( type, data, this ); + } ); + }, + triggerHandler: function( type, data ) { + var elem = this[ 0 ]; + if ( elem ) { + return jQuery.event.trigger( type, data, elem, true ); + } + } +} ); + + +// Support: Firefox <=44 +// Firefox doesn't have focus(in | out) events +// Related ticket - https://bugzilla.mozilla.org/show_bug.cgi?id=687787 +// +// Support: Chrome <=48 - 49, Safari <=9.0 - 9.1 +// focus(in | out) events fire after focus & blur events, +// which is spec violation - http://www.w3.org/TR/DOM-Level-3-Events/#events-focusevent-event-order +// Related ticket - https://bugs.chromium.org/p/chromium/issues/detail?id=449857 +if ( !support.focusin ) { + jQuery.each( { focus: "focusin", blur: "focusout" }, function( orig, fix ) { + + // Attach a single capturing handler on the document while someone wants focusin/focusout + var handler = function( event ) { + jQuery.event.simulate( fix, event.target, jQuery.event.fix( event ) ); + }; + + jQuery.event.special[ fix ] = { + setup: function() { + + // Handle: regular nodes (via `this.ownerDocument`), window + // (via `this.document`) & document (via `this`). + var doc = this.ownerDocument || this.document || this, + attaches = dataPriv.access( doc, fix ); + + if ( !attaches ) { + doc.addEventListener( orig, handler, true ); + } + dataPriv.access( doc, fix, ( attaches || 0 ) + 1 ); + }, + teardown: function() { + var doc = this.ownerDocument || this.document || this, + attaches = dataPriv.access( doc, fix ) - 1; + + if ( !attaches ) { + doc.removeEventListener( orig, handler, true ); + dataPriv.remove( doc, fix ); + + } else { + dataPriv.access( doc, fix, attaches ); + } + } + }; + } ); +} +var location = window.location; + +var nonce = { guid: Date.now() }; + +var rquery = ( /\?/ ); + + + +// Cross-browser xml parsing +jQuery.parseXML = function( data ) { + var xml; + if ( !data || typeof data !== "string" ) { + return null; + } + + // Support: IE 9 - 11 only + // IE throws on parseFromString with invalid input. + try { + xml = ( new window.DOMParser() ).parseFromString( data, "text/xml" ); + } catch ( e ) { + xml = undefined; + } + + if ( !xml || xml.getElementsByTagName( "parsererror" ).length ) { + jQuery.error( "Invalid XML: " + data ); + } + return xml; +}; + + +var + rbracket = /\[\]$/, + rCRLF = /\r?\n/g, + rsubmitterTypes = /^(?:submit|button|image|reset|file)$/i, + rsubmittable = /^(?:input|select|textarea|keygen)/i; + +function buildParams( prefix, obj, traditional, add ) { + var name; + + if ( Array.isArray( obj ) ) { + + // Serialize array item. + jQuery.each( obj, function( i, v ) { + if ( traditional || rbracket.test( prefix ) ) { + + // Treat each array item as a scalar. + add( prefix, v ); + + } else { + + // Item is non-scalar (array or object), encode its numeric index. + buildParams( + prefix + "[" + ( typeof v === "object" && v != null ? i : "" ) + "]", + v, + traditional, + add + ); + } + } ); + + } else if ( !traditional && toType( obj ) === "object" ) { + + // Serialize object item. + for ( name in obj ) { + buildParams( prefix + "[" + name + "]", obj[ name ], traditional, add ); + } + + } else { + + // Serialize scalar item. + add( prefix, obj ); + } +} + +// Serialize an array of form elements or a set of +// key/values into a query string +jQuery.param = function( a, traditional ) { + var prefix, + s = [], + add = function( key, valueOrFunction ) { + + // If value is a function, invoke it and use its return value + var value = isFunction( valueOrFunction ) ? + valueOrFunction() : + valueOrFunction; + + s[ s.length ] = encodeURIComponent( key ) + "=" + + encodeURIComponent( value == null ? "" : value ); + }; + + if ( a == null ) { + return ""; + } + + // If an array was passed in, assume that it is an array of form elements. + if ( Array.isArray( a ) || ( a.jquery && !jQuery.isPlainObject( a ) ) ) { + + // Serialize the form elements + jQuery.each( a, function() { + add( this.name, this.value ); + } ); + + } else { + + // If traditional, encode the "old" way (the way 1.3.2 or older + // did it), otherwise encode params recursively. + for ( prefix in a ) { + buildParams( prefix, a[ prefix ], traditional, add ); + } + } + + // Return the resulting serialization + return s.join( "&" ); +}; + +jQuery.fn.extend( { + serialize: function() { + return jQuery.param( this.serializeArray() ); + }, + serializeArray: function() { + return this.map( function() { + + // Can add propHook for "elements" to filter or add form elements + var elements = jQuery.prop( this, "elements" ); + return elements ? jQuery.makeArray( elements ) : this; + } ) + .filter( function() { + var type = this.type; + + // Use .is( ":disabled" ) so that fieldset[disabled] works + return this.name && !jQuery( this ).is( ":disabled" ) && + rsubmittable.test( this.nodeName ) && !rsubmitterTypes.test( type ) && + ( this.checked || !rcheckableType.test( type ) ); + } ) + .map( function( _i, elem ) { + var val = jQuery( this ).val(); + + if ( val == null ) { + return null; + } + + if ( Array.isArray( val ) ) { + return jQuery.map( val, function( val ) { + return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; + } ); + } + + return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; + } ).get(); + } +} ); + + +var + r20 = /%20/g, + rhash = /#.*$/, + rantiCache = /([?&])_=[^&]*/, + rheaders = /^(.*?):[ \t]*([^\r\n]*)$/mg, + + // #7653, #8125, #8152: local protocol detection + rlocalProtocol = /^(?:about|app|app-storage|.+-extension|file|res|widget):$/, + rnoContent = /^(?:GET|HEAD)$/, + rprotocol = /^\/\//, + + /* Prefilters + * 1) They are useful to introduce custom dataTypes (see ajax/jsonp.js for an example) + * 2) These are called: + * - BEFORE asking for a transport + * - AFTER param serialization (s.data is a string if s.processData is true) + * 3) key is the dataType + * 4) the catchall symbol "*" can be used + * 5) execution will start with transport dataType and THEN continue down to "*" if needed + */ + prefilters = {}, + + /* Transports bindings + * 1) key is the dataType + * 2) the catchall symbol "*" can be used + * 3) selection will start with transport dataType and THEN go to "*" if needed + */ + transports = {}, + + // Avoid comment-prolog char sequence (#10098); must appease lint and evade compression + allTypes = "*/".concat( "*" ), + + // Anchor tag for parsing the document origin + originAnchor = document.createElement( "a" ); + originAnchor.href = location.href; + +// Base "constructor" for jQuery.ajaxPrefilter and jQuery.ajaxTransport +function addToPrefiltersOrTransports( structure ) { + + // dataTypeExpression is optional and defaults to "*" + return function( dataTypeExpression, func ) { + + if ( typeof dataTypeExpression !== "string" ) { + func = dataTypeExpression; + dataTypeExpression = "*"; + } + + var dataType, + i = 0, + dataTypes = dataTypeExpression.toLowerCase().match( rnothtmlwhite ) || []; + + if ( isFunction( func ) ) { + + // For each dataType in the dataTypeExpression + while ( ( dataType = dataTypes[ i++ ] ) ) { + + // Prepend if requested + if ( dataType[ 0 ] === "+" ) { + dataType = dataType.slice( 1 ) || "*"; + ( structure[ dataType ] = structure[ dataType ] || [] ).unshift( func ); + + // Otherwise append + } else { + ( structure[ dataType ] = structure[ dataType ] || [] ).push( func ); + } + } + } + }; +} + +// Base inspection function for prefilters and transports +function inspectPrefiltersOrTransports( structure, options, originalOptions, jqXHR ) { + + var inspected = {}, + seekingTransport = ( structure === transports ); + + function inspect( dataType ) { + var selected; + inspected[ dataType ] = true; + jQuery.each( structure[ dataType ] || [], function( _, prefilterOrFactory ) { + var dataTypeOrTransport = prefilterOrFactory( options, originalOptions, jqXHR ); + if ( typeof dataTypeOrTransport === "string" && + !seekingTransport && !inspected[ dataTypeOrTransport ] ) { + + options.dataTypes.unshift( dataTypeOrTransport ); + inspect( dataTypeOrTransport ); + return false; + } else if ( seekingTransport ) { + return !( selected = dataTypeOrTransport ); + } + } ); + return selected; + } + + return inspect( options.dataTypes[ 0 ] ) || !inspected[ "*" ] && inspect( "*" ); +} + +// A special extend for ajax options +// that takes "flat" options (not to be deep extended) +// Fixes #9887 +function ajaxExtend( target, src ) { + var key, deep, + flatOptions = jQuery.ajaxSettings.flatOptions || {}; + + for ( key in src ) { + if ( src[ key ] !== undefined ) { + ( flatOptions[ key ] ? target : ( deep || ( deep = {} ) ) )[ key ] = src[ key ]; + } + } + if ( deep ) { + jQuery.extend( true, target, deep ); + } + + return target; +} + +/* Handles responses to an ajax request: + * - finds the right dataType (mediates between content-type and expected dataType) + * - returns the corresponding response + */ +function ajaxHandleResponses( s, jqXHR, responses ) { + + var ct, type, finalDataType, firstDataType, + contents = s.contents, + dataTypes = s.dataTypes; + + // Remove auto dataType and get content-type in the process + while ( dataTypes[ 0 ] === "*" ) { + dataTypes.shift(); + if ( ct === undefined ) { + ct = s.mimeType || jqXHR.getResponseHeader( "Content-Type" ); + } + } + + // Check if we're dealing with a known content-type + if ( ct ) { + for ( type in contents ) { + if ( contents[ type ] && contents[ type ].test( ct ) ) { + dataTypes.unshift( type ); + break; + } + } + } + + // Check to see if we have a response for the expected dataType + if ( dataTypes[ 0 ] in responses ) { + finalDataType = dataTypes[ 0 ]; + } else { + + // Try convertible dataTypes + for ( type in responses ) { + if ( !dataTypes[ 0 ] || s.converters[ type + " " + dataTypes[ 0 ] ] ) { + finalDataType = type; + break; + } + if ( !firstDataType ) { + firstDataType = type; + } + } + + // Or just use first one + finalDataType = finalDataType || firstDataType; + } + + // If we found a dataType + // We add the dataType to the list if needed + // and return the corresponding response + if ( finalDataType ) { + if ( finalDataType !== dataTypes[ 0 ] ) { + dataTypes.unshift( finalDataType ); + } + return responses[ finalDataType ]; + } +} + +/* Chain conversions given the request and the original response + * Also sets the responseXXX fields on the jqXHR instance + */ +function ajaxConvert( s, response, jqXHR, isSuccess ) { + var conv2, current, conv, tmp, prev, + converters = {}, + + // Work with a copy of dataTypes in case we need to modify it for conversion + dataTypes = s.dataTypes.slice(); + + // Create converters map with lowercased keys + if ( dataTypes[ 1 ] ) { + for ( conv in s.converters ) { + converters[ conv.toLowerCase() ] = s.converters[ conv ]; + } + } + + current = dataTypes.shift(); + + // Convert to each sequential dataType + while ( current ) { + + if ( s.responseFields[ current ] ) { + jqXHR[ s.responseFields[ current ] ] = response; + } + + // Apply the dataFilter if provided + if ( !prev && isSuccess && s.dataFilter ) { + response = s.dataFilter( response, s.dataType ); + } + + prev = current; + current = dataTypes.shift(); + + if ( current ) { + + // There's only work to do if current dataType is non-auto + if ( current === "*" ) { + + current = prev; + + // Convert response if prev dataType is non-auto and differs from current + } else if ( prev !== "*" && prev !== current ) { + + // Seek a direct converter + conv = converters[ prev + " " + current ] || converters[ "* " + current ]; + + // If none found, seek a pair + if ( !conv ) { + for ( conv2 in converters ) { + + // If conv2 outputs current + tmp = conv2.split( " " ); + if ( tmp[ 1 ] === current ) { + + // If prev can be converted to accepted input + conv = converters[ prev + " " + tmp[ 0 ] ] || + converters[ "* " + tmp[ 0 ] ]; + if ( conv ) { + + // Condense equivalence converters + if ( conv === true ) { + conv = converters[ conv2 ]; + + // Otherwise, insert the intermediate dataType + } else if ( converters[ conv2 ] !== true ) { + current = tmp[ 0 ]; + dataTypes.unshift( tmp[ 1 ] ); + } + break; + } + } + } + } + + // Apply converter (if not an equivalence) + if ( conv !== true ) { + + // Unless errors are allowed to bubble, catch and return them + if ( conv && s.throws ) { + response = conv( response ); + } else { + try { + response = conv( response ); + } catch ( e ) { + return { + state: "parsererror", + error: conv ? e : "No conversion from " + prev + " to " + current + }; + } + } + } + } + } + } + + return { state: "success", data: response }; +} + +jQuery.extend( { + + // Counter for holding the number of active queries + active: 0, + + // Last-Modified header cache for next request + lastModified: {}, + etag: {}, + + ajaxSettings: { + url: location.href, + type: "GET", + isLocal: rlocalProtocol.test( location.protocol ), + global: true, + processData: true, + async: true, + contentType: "application/x-www-form-urlencoded; charset=UTF-8", + + /* + timeout: 0, + data: null, + dataType: null, + username: null, + password: null, + cache: null, + throws: false, + traditional: false, + headers: {}, + */ + + accepts: { + "*": allTypes, + text: "text/plain", + html: "text/html", + xml: "application/xml, text/xml", + json: "application/json, text/javascript" + }, + + contents: { + xml: /\bxml\b/, + html: /\bhtml/, + json: /\bjson\b/ + }, + + responseFields: { + xml: "responseXML", + text: "responseText", + json: "responseJSON" + }, + + // Data converters + // Keys separate source (or catchall "*") and destination types with a single space + converters: { + + // Convert anything to text + "* text": String, + + // Text to html (true = no transformation) + "text html": true, + + // Evaluate text as a json expression + "text json": JSON.parse, + + // Parse text as xml + "text xml": jQuery.parseXML + }, + + // For options that shouldn't be deep extended: + // you can add your own custom options here if + // and when you create one that shouldn't be + // deep extended (see ajaxExtend) + flatOptions: { + url: true, + context: true + } + }, + + // Creates a full fledged settings object into target + // with both ajaxSettings and settings fields. + // If target is omitted, writes into ajaxSettings. + ajaxSetup: function( target, settings ) { + return settings ? + + // Building a settings object + ajaxExtend( ajaxExtend( target, jQuery.ajaxSettings ), settings ) : + + // Extending ajaxSettings + ajaxExtend( jQuery.ajaxSettings, target ); + }, + + ajaxPrefilter: addToPrefiltersOrTransports( prefilters ), + ajaxTransport: addToPrefiltersOrTransports( transports ), + + // Main method + ajax: function( url, options ) { + + // If url is an object, simulate pre-1.5 signature + if ( typeof url === "object" ) { + options = url; + url = undefined; + } + + // Force options to be an object + options = options || {}; + + var transport, + + // URL without anti-cache param + cacheURL, + + // Response headers + responseHeadersString, + responseHeaders, + + // timeout handle + timeoutTimer, + + // Url cleanup var + urlAnchor, + + // Request state (becomes false upon send and true upon completion) + completed, + + // To know if global events are to be dispatched + fireGlobals, + + // Loop variable + i, + + // uncached part of the url + uncached, + + // Create the final options object + s = jQuery.ajaxSetup( {}, options ), + + // Callbacks context + callbackContext = s.context || s, + + // Context for global events is callbackContext if it is a DOM node or jQuery collection + globalEventContext = s.context && + ( callbackContext.nodeType || callbackContext.jquery ) ? + jQuery( callbackContext ) : + jQuery.event, + + // Deferreds + deferred = jQuery.Deferred(), + completeDeferred = jQuery.Callbacks( "once memory" ), + + // Status-dependent callbacks + statusCode = s.statusCode || {}, + + // Headers (they are sent all at once) + requestHeaders = {}, + requestHeadersNames = {}, + + // Default abort message + strAbort = "canceled", + + // Fake xhr + jqXHR = { + readyState: 0, + + // Builds headers hashtable if needed + getResponseHeader: function( key ) { + var match; + if ( completed ) { + if ( !responseHeaders ) { + responseHeaders = {}; + while ( ( match = rheaders.exec( responseHeadersString ) ) ) { + responseHeaders[ match[ 1 ].toLowerCase() + " " ] = + ( responseHeaders[ match[ 1 ].toLowerCase() + " " ] || [] ) + .concat( match[ 2 ] ); + } + } + match = responseHeaders[ key.toLowerCase() + " " ]; + } + return match == null ? null : match.join( ", " ); + }, + + // Raw string + getAllResponseHeaders: function() { + return completed ? responseHeadersString : null; + }, + + // Caches the header + setRequestHeader: function( name, value ) { + if ( completed == null ) { + name = requestHeadersNames[ name.toLowerCase() ] = + requestHeadersNames[ name.toLowerCase() ] || name; + requestHeaders[ name ] = value; + } + return this; + }, + + // Overrides response content-type header + overrideMimeType: function( type ) { + if ( completed == null ) { + s.mimeType = type; + } + return this; + }, + + // Status-dependent callbacks + statusCode: function( map ) { + var code; + if ( map ) { + if ( completed ) { + + // Execute the appropriate callbacks + jqXHR.always( map[ jqXHR.status ] ); + } else { + + // Lazy-add the new callbacks in a way that preserves old ones + for ( code in map ) { + statusCode[ code ] = [ statusCode[ code ], map[ code ] ]; + } + } + } + return this; + }, + + // Cancel the request + abort: function( statusText ) { + var finalText = statusText || strAbort; + if ( transport ) { + transport.abort( finalText ); + } + done( 0, finalText ); + return this; + } + }; + + // Attach deferreds + deferred.promise( jqXHR ); + + // Add protocol if not provided (prefilters might expect it) + // Handle falsy url in the settings object (#10093: consistency with old signature) + // We also use the url parameter if available + s.url = ( ( url || s.url || location.href ) + "" ) + .replace( rprotocol, location.protocol + "//" ); + + // Alias method option to type as per ticket #12004 + s.type = options.method || options.type || s.method || s.type; + + // Extract dataTypes list + s.dataTypes = ( s.dataType || "*" ).toLowerCase().match( rnothtmlwhite ) || [ "" ]; + + // A cross-domain request is in order when the origin doesn't match the current origin. + if ( s.crossDomain == null ) { + urlAnchor = document.createElement( "a" ); + + // Support: IE <=8 - 11, Edge 12 - 15 + // IE throws exception on accessing the href property if url is malformed, + // e.g. http://example.com:80x/ + try { + urlAnchor.href = s.url; + + // Support: IE <=8 - 11 only + // Anchor's host property isn't correctly set when s.url is relative + urlAnchor.href = urlAnchor.href; + s.crossDomain = originAnchor.protocol + "//" + originAnchor.host !== + urlAnchor.protocol + "//" + urlAnchor.host; + } catch ( e ) { + + // If there is an error parsing the URL, assume it is crossDomain, + // it can be rejected by the transport if it is invalid + s.crossDomain = true; + } + } + + // Convert data if not already a string + if ( s.data && s.processData && typeof s.data !== "string" ) { + s.data = jQuery.param( s.data, s.traditional ); + } + + // Apply prefilters + inspectPrefiltersOrTransports( prefilters, s, options, jqXHR ); + + // If request was aborted inside a prefilter, stop there + if ( completed ) { + return jqXHR; + } + + // We can fire global events as of now if asked to + // Don't fire events if jQuery.event is undefined in an AMD-usage scenario (#15118) + fireGlobals = jQuery.event && s.global; + + // Watch for a new set of requests + if ( fireGlobals && jQuery.active++ === 0 ) { + jQuery.event.trigger( "ajaxStart" ); + } + + // Uppercase the type + s.type = s.type.toUpperCase(); + + // Determine if request has content + s.hasContent = !rnoContent.test( s.type ); + + // Save the URL in case we're toying with the If-Modified-Since + // and/or If-None-Match header later on + // Remove hash to simplify url manipulation + cacheURL = s.url.replace( rhash, "" ); + + // More options handling for requests with no content + if ( !s.hasContent ) { + + // Remember the hash so we can put it back + uncached = s.url.slice( cacheURL.length ); + + // If data is available and should be processed, append data to url + if ( s.data && ( s.processData || typeof s.data === "string" ) ) { + cacheURL += ( rquery.test( cacheURL ) ? "&" : "?" ) + s.data; + + // #9682: remove data so that it's not used in an eventual retry + delete s.data; + } + + // Add or update anti-cache param if needed + if ( s.cache === false ) { + cacheURL = cacheURL.replace( rantiCache, "$1" ); + uncached = ( rquery.test( cacheURL ) ? "&" : "?" ) + "_=" + ( nonce.guid++ ) + + uncached; + } + + // Put hash and anti-cache on the URL that will be requested (gh-1732) + s.url = cacheURL + uncached; + + // Change '%20' to '+' if this is encoded form body content (gh-2658) + } else if ( s.data && s.processData && + ( s.contentType || "" ).indexOf( "application/x-www-form-urlencoded" ) === 0 ) { + s.data = s.data.replace( r20, "+" ); + } + + // Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode. + if ( s.ifModified ) { + if ( jQuery.lastModified[ cacheURL ] ) { + jqXHR.setRequestHeader( "If-Modified-Since", jQuery.lastModified[ cacheURL ] ); + } + if ( jQuery.etag[ cacheURL ] ) { + jqXHR.setRequestHeader( "If-None-Match", jQuery.etag[ cacheURL ] ); + } + } + + // Set the correct header, if data is being sent + if ( s.data && s.hasContent && s.contentType !== false || options.contentType ) { + jqXHR.setRequestHeader( "Content-Type", s.contentType ); + } + + // Set the Accepts header for the server, depending on the dataType + jqXHR.setRequestHeader( + "Accept", + s.dataTypes[ 0 ] && s.accepts[ s.dataTypes[ 0 ] ] ? + s.accepts[ s.dataTypes[ 0 ] ] + + ( s.dataTypes[ 0 ] !== "*" ? 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+ + + + + + \ No newline at end of file diff --git a/docs/content/bvps/boundary-value-problems.html b/docs/content/bvps/boundary-value-problems.html new file mode 100644 index 0000000..ab64c72 --- /dev/null +++ b/docs/content/bvps/boundary-value-problems.html @@ -0,0 +1,435 @@ + + + + + + + + 4. Boundary Value Problems — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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4. Boundary Value Problems

+

This chapter focuses on methods for solving 2nd-order ODEs constrained by boundary conditions: boundary-value problems (BVPs).

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+ + + + + + \ No newline at end of file diff --git a/docs/content/bvps/eigenvalue.html b/docs/content/bvps/eigenvalue.html new file mode 100644 index 0000000..f86062f --- /dev/null +++ b/docs/content/bvps/eigenvalue.html @@ -0,0 +1,941 @@ + + + + + + + + 4.3. Eigenvalue problems — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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4.3. Eigenvalue problems

+

“Eigenvalue” means characteristic value. These types of problems show up in many areas involving boundary-value problems, where we may not be able to obtain an analytical solution, but we can identify certain characteristic values that tell us important information about the system: the eigenvalues.

+
+

4.3.1. Example: beam buckling

+

Let’s consider deflection in a simply supported (static) vertical beam: \(y(x)\), with boundary conditions \(y(0) = 0\) and \(y(L) = 0\). To get the governing equation, start with considering the sum of moments around the upper pin:

+
+(4.46)\[\begin{align} +\sum M &= M_z + P y = 0 \\ +M_z &= -P y +\end{align}\]
+

We also know that \(M_z = E I y''\), so we can obtain

+
+(4.47)\[\begin{align} +M_z = E I \frac{d^2 y}{dx^2} &= -P y \\ +y'' + \frac{P}{EI} y &= 0 +\end{align}\]
+

This equation governs the stability of a beam, considering small deflections. +To simplify things, let’s define \(\lambda^2 = \frac{P}{EI}\), which gives us the ODE

+
+(4.48)\[\begin{equation} +y'' + \lambda^2 y = 0 +\end{equation}\]
+

We can get the general solution to this:

+
+(4.49)\[\begin{equation} +y(x) = A \cos (\lambda x) + B \sin (\lambda x) +\end{equation}\]
+

To find the coefficients, let’s apply the boundary conditions, starting with \(x=0\):

+
+(4.50)\[\begin{align} +y(x=0) &= 0 = A \cos 0 + B \sin 0 \\ +\rightarrow A &= 0 \\ +y(x=L) &= 0 = B \sin (\lambda L) +\end{align}\]
+

Now what? \(B \neq 0\), because otherwise we would have the trivial solution \(y(x) = 0\). Instead, to satisfy the boundary condition, we need

+
+(4.51)\[\begin{align} +B \neq 0 \rightarrow \sin (\lambda L) &= 0 \\ +\text{so} \quad \lambda L &= n \pi \quad n = 1, 2, 3, \ldots, \infty \\ +\lambda &= \frac{n \pi}{L} \quad n = 1, 2, 3, \ldots, \infty +\end{align}\]
+

\(\lambda\) give the the eigenvalues for this problem; as you can see, there are an infinite number, that correspond to eigenfunctions:

+
+(4.52)\[\begin{equation} +y_n = B \sin \left( \frac{n \pi x}{L} \right) \quad n = 1, 2, 3, \ldots, \infty +\end{equation}\]
+

The eigenvalues and associated eigenfunctions physically represent different modes of deflection. +For example, consider the first three modes (corresponding to \(n = 1, 2, 3\)):

+
+
+
clear all; clc
+
+L = 1.0;
+x = linspace(0, L);
+subplot(1,3,1);
+y = sin(pi * x / L);
+plot(y, x); title('n = 1')
+subplot(1,3,2);
+y = sin(2 * pi * x / L);
+plot(y, x); title('n = 2')
+subplot(1,3,3);
+y = sin(3* pi * x / L);
+plot(y, x); title('n = 3')
+
+
+
+
+../../_images/eigenvalue_1_0.png +
+
+

Here we see different modes of how the beam will buckle. How do we know when this happens?

+

Recall that the eigenvalue is connected to the physical properties of the beam:

+
+(4.53)\[\begin{gather} +\lambda^2 = \frac{P}{EI} \rightarrow \lambda = \sqrt{\frac{P}{EI}} = \frac{n \pi}{L} \\ +P = \frac{EI}{L} n^2 \pi^2 +\end{gather}\]
+

This means that when the combination of load force and beam properties match certain values, the beam will deflect—and buckle—in one of the modes corresponding to the associated eigenfunction.

+

In particular, the first mode (\(n=1\)) is interesting, because this is the first one that will be encountered if a load starts at zero and increases. This is the Euler critical load of buckling, \(P_{cr}\):

+
+(4.54)\[\begin{gather} +\lambda_1 = \frac{\pi}{L} \rightarrow \lambda_1^2 = \frac{P}{EI} = \frac{\pi^2}{L^2} \\ +P_{cr} = \frac{\pi^2 E I}{L^2} +\end{gather}\]
+
+
+

4.3.2. Example: beam buckling with different boundary conditions

+

Let’s consider a slightly different case, where at \(x=0\) the beam is supported such that \(y'(0) = 0\). How does the beam buckle in this case?

+

The governing equation and general solution are the same:

+
+(4.55)\[\begin{align} +y'' + \lambda^2 y &= 0 \\ +y(x) &= A \cos (\lambda x) + B \sin (\lambda x) +\end{align}\]
+

but our boundary conditions are now different:

+
+(4.56)\[\begin{align} +y'(0) = 0 = -\lambda A \sin(0) + \lambda B\cos(0) \\ +\rightarrow B &= 0 \\ +y &= A \cos (\lambda x) \\ +y(L) &= 0 = A \cos (\lambda L) \\ +A \neq 0 \rightarrow \cos(\lambda L) &= 0 \\ +\text{so} \quad \lambda L &= \frac{(2n-1) \pi}{2} \quad n = 1,2,3,\ldots, \infty \\ +\lambda &= \frac{(2n-1) \pi}{2 L} \quad n = 1,2,3,\ldots, \infty +\end{align}\]
+

Then, the critical buckling load, again corresponding to \(n=1\), is

+
+(4.57)\[\begin{equation} +P_{cr} = \frac{\pi^2 EI}{4 L^2} +\end{equation}\]
+
+
+

4.3.3. Getting eigenvalues numerically

+

We can only get the eigenvalues analytically if we can obtain an analytical solution of the ODE, but we might want to get eigenvalues for more complex problems too. In that case, we can use an approach based on finite differences to find the eigenvalues.

+

Consider the same problem as above, for deflection of a simply supported beam:

+
+(4.58)\[\begin{equation} +y'' + \lambda^2 y = 0 +\end{equation}\]
+

with boundary conditions \(y(0) = 0\) and \(y(L) = 0\). Let’s represent this using finite differences, for a case where \(L=3\) and \(\Delta x = 1\), so we have four points in our solution grid.

+

The finite difference representation of the ODE is:

+
+(4.59)\[\begin{align} +\frac{y_{i-1} - 2y_i + y_{i+1}}{\Delta x^2} + \lambda^2 y_i &= 0 \\ +y_{i-1} + \left( \lambda^2 \Delta x^2 - 2 \right) y_i + y_{i+1} &= 0 +\end{align}\]
+

However, in this case, we are not solving for the values of deflection (\(y_i\)), but instead the eigenvalues \(\lambda\).

+

Then, we can write the system of equations using the above recursion formula and our two boundary conditions:

+
+(4.60)\[\begin{align} +y_1 &= 0 \\ +y_1 + y_2 \left( \lambda^2 \Delta x^2 - 2 \right) + y_3 &= 0 \\ +y_2 + y_3 \left( \lambda^2 \Delta x^2 - 2 \right) + y_4 &= 0 \\ +y_4 &= 0 +\end{align}\]
+

which we can simplify down to two equations by incorporating the boundary conditions into the equations for the two middle points, and also letting \(k = \lambda^2 \Delta x^2\):

+
+(4.61)\[\begin{align} +y_2 (k-2) + y_3 &= 0 \\ +y_2 + y_3 (k-2) &= 0 +\end{align}\]
+

Let’s modify this once more by multiplying both equations by \(-1\):

+
+(4.62)\[\begin{align} +y_2 (2-k) - y_3 &= 0 \\ +-y_2 + y_3 (2-k) &= 0 +\end{align}\]
+

Now we can represent this system of equations as a matrix equation \(A \mathbf{y} = \mathbf{b} = \mathbf{0}\):

+
+(4.63)\[\begin{equation} +\begin{bmatrix} 2-k & -1 \\ -1 & 2-k \end{bmatrix} +\begin{bmatrix} y_2 \\ y_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} +\end{equation}\]
+

\(\mathbf{y} = \mathbf{0}\) is a trivial solution to this, so instead \(\det(A) = 0\) satisfies this equation. +For our \(2\times 2\) matrix, that looks like:

+
+(4.64)\[\begin{align} +\det(A) = \begin{vmatrix} 2-k & -1 \\ -1 & 2-k \end{vmatrix} = (2-k)^2 - 1 &= 0 \\ +k^2 - 4k + 3 &= 0 \\ +(k-3)(k-1) &= 0 +\end{align}\]
+

so the roots of this equation are \(k_1 = 1\) and \(k_2 = 3\). Recall that \(k\) is directly related to the eigenvalue: \(k = \lambda^2 \Delta x^2\), and \(\Delta x = 1\) for this case, so we can calculate the two associated eigenvalues:

+
+(4.65)\[\begin{align} +k_1 &= \lambda_1^2 \Delta x^2 = 1 \rightarrow \lambda_1 = 1 \\ +k_2 &= \lambda_2^2 \Delta x^2 = 3 \rightarrow \lambda_2 = \sqrt{3} = 1.732 +\end{align}\]
+

Our work has given us approximations for the first two eigenvalues. We can compare these against the exact values, given in general by \(\lambda = n \pi / L\) (which we determined above):

+
+(4.66)\[\begin{align} +n=1: \quad \lambda_1 &= \frac{\pi}{L} = \frac{\pi}{3} = 1.0472 \\ +n=2: \quad \lambda_2 &= \frac{2\pi}{L} = \frac{2\pi}{3} = 2.0944 +\end{align}\]
+

So, our approximations are close, but with some obvious error. This is because we used a fairly crude step size of \(\Delta x = 1\), dividing the domain into just three segments. By using a finer resolution, we can get more-accurate eigenvalues and also more of them (remember, there are actually an infinite number!).

+

To do that, we will need to use Matlab, which offers the eig() function for calculating eigenvalues—essentially it is finding the roots to the polynomial given by \(\det(A) = 0\). We need to modify this slightly, though, to use the function:

+
+(4.67)\[\begin{align} +\det(A) &= 0 \\ +\det \left( A^* - k I \right) = 0 +\end{align}\]
+

where the new matrix is

+
+(4.68)\[\begin{equation} +A^* = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} +\end{equation}\]
+

Then, eig(A*) will provide the values of \(k\), which we can use to find the \(\lambda\)s:

+
+
+
clear all; clc
+
+dx = 1.0;
+L = 3.0;
+
+Astar = [2 -1; -1 2];
+k = eig(Astar);
+
+lambda = sqrt(k) / dx;
+
+fprintf('lambda_1: %6.3f\n', lambda(1));
+fprintf('lambda_2: %6.3f', lambda(2));
+
+
+
+
+
lambda_1:  1.000
+lambda_2:  1.732
+
+
+
+
+

As expected, this matches with our manual calculation above. But, we might want to calculate these eigenvalues more accurately, so let’s generalize this a bit and then try using \(\Delta x= 0.1\):

+
+
+
clear all; clc
+
+dx = 0.1;
+L = 3.0;
+x = 0 : dx : L;
+n = length(x) - 2;
+
+Astar = zeros(n,n);
+for i = 1 : n
+    if i == 1
+        Astar(1,1) = 2;
+        Astar(1,2) = -1;
+    elseif i == n
+        Astar(n,n-1) = -1;
+        Astar(n,n) = 2;
+    else
+        Astar(i,i-1) = -1;
+        Astar(i,i) = 2;
+        Astar(i,i+1) = -1;
+    end
+end
+k = eig(Astar);
+
+lambda = sqrt(k) / dx;
+
+fprintf('lambda_1: %6.3f\n', lambda(1));
+fprintf('lambda_2: %6.3f\n\n', lambda(2));
+
+err = abs(lambda(1) - (pi/L)) / (pi/L);
+fprintf('Error in lambda_1: %5.2f%%\n', 100*err);
+
+
+
+
+
lambda_1:  1.047
+lambda_2:  2.091
+
+Error in lambda_1:  0.05%
+
+
+
+
+
+
+

4.3.4. Example: mass-spring system

+

Let’s analyze the motion of masses connected by springs in a system:

+
+
+ mass-spring system +
Figure: System with two masses connected by springs
+
+
+First, we need to write the equations of motion, based on doing a free-body diagram on each mass: +\begin{align} +m_1 \frac{d^2 x_1}{dt^2} &= -k x_1 + k(x_2 - x_1) \\ +m_2 \frac{d^2 x_2}{dt^2} &= -k (x_2 - x_1) - k x_2 +\end{align} +We can condense these equations a bit: +\begin{align} +x_1^{\prime\prime} - \frac{k}{m_1} \left( -2 x_1 + x_2 \right) &= 0 \\ +x_2^{\prime\prime} - \frac{k}{m_2} \left( x_1 - 2 x_2 \right) &= 0 +\end{align} +

To proceed, we can assume that the masses will move in a sinusoidal fashion, with a shared frequency but separate amplitude:

+
+(4.69)\[\begin{align} +x_i &= A_i \sin (\omega t) \\ +x_i^{\prime\prime} &= -A_i \omega^2 \sin (\omega t) +\end{align}\]
+

We can plug these into the ODEs:

+
+(4.70)\[\begin{align} +\sin (\omega t) \left[ \left( \frac{2k}{m_1} - \omega^2 \right) A_1 - \frac{k}{m_1} A_2 \right] &= 0 \\ +\sin (\omega t) \left[ -\frac{k}{m_2} A_1 + \left( \frac{2k}{m_2} - \omega^2 \right) A_2 \right] &= 0 +\end{align}\]
+

or

+
+(4.71)\[\begin{align} +\left( \frac{2k}{m_1} - \omega^2 \right) A_1 - \frac{k}{m_1} A_2 &= 0 \\ +-\frac{k}{m_2} A_1 + \left( \frac{2k}{m_2} - \omega^2 \right) A_2 &= 0 +\end{align}\]
+

Let’s put some numbers in, and try to solve for the eigenvalues: \(\omega^2\). +Let \(m_1 = m_2 = 40 \) kg and \(k = 200\) N/m.

+

Now, the equations become

+
+(4.72)\[\begin{align} +\left( 10 - \omega^2 \right) A_1 - 5 A_2 &= 0 \\ +-5 A_1 + \left( 10 - \omega^2 \right) A_2 &= 0 +\end{align}\]
+

or \(A \mathbf{y} = \mathbf{0}\), which we can represent as

+
+(4.73)\[\begin{equation} +\begin{bmatrix} 10-\omega^2 & -5 \\ -5 & 10-\omega^2 \end{bmatrix} +\begin{bmatrix} A_1 \\ A_2 \end{bmatrix} = +\begin{bmatrix} 0 \\ 0 \end{bmatrix} +\end{equation}\]
+

Here, \(\omega^2\) are the eigenvalues, and we can find them with \(\det(A) = 0\):

+
+(4.74)\[\begin{align} +\det(B) &= 0 \\ +\det (B^* - \omega^2 I) &= 0 +\end{align}\]
+
+
+
clear all; clc
+
+Bstar = [10 -5; -5 10];
+omega_squared = eig(Bstar);
+omega = sqrt(omega_squared);
+
+fprintf('omega_1 = %5.2f rad/s\n', omega(1));
+fprintf('omega_2 = %5.2f rad/s\n', omega(2));
+
+
+
+
+
omega_1 =  2.24 rad/s
+omega_2 =  3.87 rad/s
+
+
+
+
+

We find there are two modes of oscillation, each associated with a different natural frequency. Unfortunately, we cannot calculate independent and unique values for the amplitudes, but if we insert the values of \(\omega\) into the above equations, we can find relations connecting the amplitudes:

+
+(4.75)\[\begin{align} +\omega_1: \quad A_1 &= A_2 \\ +\omega_2: \quad A_1 &= -A_2 +\end{align}\]
+

So, for the first mode, we have the two masses moving in sync with the same amplitude. In the second mode, they move with opposite (but equal) amplitude. With the two different frequencies, they also have two different periods:

+
+
+
t = linspace(0, 3);
+subplot(1,5,1)
+plot(sin(omega(1)*t), t); hold on
+plot(0,0, 's');
+set (gca, 'ydir', 'reverse' )
+box off; set(gca,'Visible','off')
+
+subplot(1,5,2)
+plot(sin(omega(1)*t), t); hold on
+plot(0,0, 's');
+set (gca, 'ydir', 'reverse' )
+text(-2.5,-0.2, 'First mode')
+box off; set(gca,'Visible','off')
+
+subplot(1,5,4)
+plot(-sin(omega(2)*t), t); hold on
+plot(0,0, 's');
+set (gca, 'ydir', 'reverse' )
+box off; set(gca,'Visible','off')
+
+subplot(1,5,5)
+plot(sin(omega(2)*t), t); hold on
+plot(0,0, 's');
+set (gca, 'ydir', 'reverse' )
+box off; set(gca,'Visible','off')
+text(-2.7,-0.2, 'Second mode')
+
+
+
+
+../../_images/eigenvalue_11_0.png +
+
+

We can confirm that the system would actually behave in this way by setting up the system of ODEs and integrating based on initial conditions matching the amplitudes of the two modes.

+

For example, let’s use \(x_1 (t=0) = x_2(t=0) = 1\) for the first mode, and \(x_1(t=0) = 1\) and \(x_2(t=0) = -1\) for the second mode. We’ll use zero initial velocity for both cases.

+

Then, we can solve by converting the system of two 2nd-order ODEs into a system of four 1st-order ODEs:

+
+
+
%%file masses.m
+function dxdt = masses(t, x)
+% this is a function file to calculate the derivatives associated with the system
+
+m1 = 40;
+m2 = 40;
+k = 200;
+
+dxdt = zeros(4,1);
+
+dxdt(1) = x(2);
+dxdt(2) = (k/m1)*(-2*x(1) + x(3));
+dxdt(3) = x(4);
+dxdt(4) = (k/m2)*(x(1) - 2*x(3));
+
+
+
+
+
Created file '/Users/niemeyek/projects/ME373-book/content/bvps/masses.m'.
+
+
+
+
+
+
+
clear all; clc
+
+% this is the integration for the system in the first mode
+[t, X] = ode45('masses', [0 3], [1.0 0.0 1.0 0.0]);
+subplot(1,5,1)
+plot(X(:,1), t); 
+ylabel('displacement (m)'); xlabel('time (s)')
+set (gca, 'ydir', 'reverse' )
+%box off; set(gca,'Visible','off')
+
+subplot(1,5,2)
+plot(X(:,3), t); xlabel('time (s)')
+set (gca, 'ydir', 'reverse' )
+text(-4,-0.2, 'First mode')
+
+% this is the integration for the system in the second mode
+[t, X] = ode45('masses', [0 3], [1.0 0.0 -1.0 0.0]);
+subplot(1,5,4)
+plot(X(:,1), t);
+ylabel('displacement (m)'); xlabel('time (s)')
+set (gca, 'ydir', 'reverse' )
+%box off; set(gca,'Visible','off')
+
+subplot(1,5,5)
+plot(X(:,3), t); xlabel('time (s)')
+set (gca, 'ydir', 'reverse' )
+text(-4,-0.2, 'Second mode')
+
+
+
+
+../../_images/eigenvalue_14_0.png +
+
+

This shows that we get either of the pure modes of motion with the appropriate initial conditions.

+

What about if the initial conditions don’t match either set of amplitude patterns?

+
+
+
[t, X] = ode45('masses', [0 3], [0.25 0.0 0.75 0.0]);
+subplot(1,5,1)
+plot(X(:,1), t);
+%plot(0,0, 's');
+set (gca, 'ydir', 'reverse' )
+%box off; set(gca,'Visible','off')
+
+subplot(1,5,2)
+plot(X(:,3), t);
+set (gca, 'ydir', 'reverse' )
+
+
+
+
+../../_images/eigenvalue_16_0.png +
+
+

In this case, the resulting motion will be a complicated superposition of the two modes.

+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/bvps/finite-difference.html b/docs/content/bvps/finite-difference.html new file mode 100644 index 0000000..c49b37b --- /dev/null +++ b/docs/content/bvps/finite-difference.html @@ -0,0 +1,1028 @@ + + + + + + + + 4.2. Finite difference method — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ + +
+
+ +
+ +
+

4.2. Finite difference method

+
+

4.2.1. Finite differences

+

Another method of solving boundary-value problems (and also partial differential equations, as we’ll see later) involves finite differences, which are numerical approximations to exact derivatives.

+

Recall that the exact derivative of a function \(f(x)\) at some point \(x\) is defined as:

+
+(4.13)\[\begin{equation} +f^{\prime}(x) = \frac{df}{dx}(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} +\end{equation}\]
+

So, we can approximate this derivative using a finite difference (rather than an infinitesimal difference as in the exact derivative):

+
+(4.14)\[\begin{equation} +f^{\prime}(x) \approx \frac{f(x+\Delta x) - f(x)}{\Delta x} +\end{equation}\]
+

which involves some error. This is a forward difference for approximating the first derivative. +We can also approximate the first derivative using a backward difference:

+
+(4.15)\[\begin{equation} +f^{\prime}(x) \approx \frac{f(x) - f(x - \Delta x)}{\Delta x} +\end{equation}\]
+

To understand the error involved in these differences, we can use Taylor’s theorem to obtain Taylor series expansions:

+
+(4.16)\[\begin{align} +f(x + \Delta x) &= f(x) + \Delta x \, f^{\prime}(x) + \Delta x^2 \frac{1}{2!} f^{\prime\prime}(x) + \cdots \\ +\rightarrow \frac{f(x + \Delta x) - f(x)}{\Delta x} &= f^{\prime}(x) + \mathcal{O}\left( \Delta x \right) \\ +f(x - \Delta x) &= f(x) - \Delta x \, f^{\prime}(x) + \Delta x^2 \frac{1}{2!} f^{\prime\prime}(x) + \cdots \\ +\rightarrow \frac{f(x) - f(x - \Delta x)}{\Delta x} &= f^{\prime}(x) + \mathcal{O}\left( \Delta x \right) \\ +\end{align}\]
+

where the \(\mathcal{O}()\) notation stands for “order of magnitude of”. So, we can see that each of these approximations is first-order accurate.

+
+
+

4.2.2. Second-order finite differences

+

We can obtain higher-order approximations for the first derivative, and an approximations for the second derivative, by combining these Taylor series expansions:

+
+(4.17)\[\begin{align} +f(x + \Delta x) &= f(x) + \Delta x \, f^{\prime}(x) + \Delta x^2 \frac{1}{2!} f^{\prime\prime}(x) + \mathcal{O}\left( \Delta x^3 \right) \\ +f(x - \Delta x) &= f(x) - \Delta x \, f^{\prime}(x) + \Delta x^2 \frac{1}{2!} f^{\prime\prime}(x) + \mathcal{O}\left( \Delta x^3 \right) +\end{align}\]
+

Subtracting the Taylor series for \(f(x+\Delta x)\) by that for \(f(x-\Delta x)\) gives:

+
+(4.18)\[\begin{align} +f(x + \Delta x) - f(x - \Delta x) &= 2 \Delta x \, f^{\prime}(x) + \mathcal{O}\left( \Delta x^3 \right) \\ +f^{\prime}(x) &= \frac{f(x + \Delta x) - f(x - \Delta x)}{2 \Delta x} + \mathcal{O}\left( \Delta x^2 \right) +\end{align}\]
+

which is a second-order accurate approximation for the first derivative.

+

Adding the Taylor series for \(f(x+\Delta x)\) to that for \(f(x-\Delta x)\) gives:

+
+(4.19)\[\begin{align} +f(x + \Delta x) + f(x - \Delta x) &= 2 f(x) + \Delta x^2 f^{\prime\prime}(x) + \mathcal{O}\left( \Delta x^3 \right) \\ +f^{\prime\prime}(x) &= \frac{f(x + \Delta x) - 2 f(x) + f(x - \Delta x)}{\Delta x^2} + \mathcal{O}\left( \Delta x^2 \right) +\end{align}\]
+

which is a second-order accurate approximation for the second derivative.

+
+
+

4.2.3. Solving ODEs with finite differences

+

We can use finite differences to solve ODEs by substituting them for exact derivatives, and then applying the equation at discrete locations in the domain. This gives us a system of simultaneous equations to solve.

+

For example, let’s consider the ODE

+
+(4.20)\[\begin{equation} +y^{\prime\prime} + x y^{\prime} - x y = 2 x \;, +\end{equation}\]
+

with the boundary conditions \(y(0) = 1\) and \(y(2) = 8\).

+

First, we discretize the continuous domain: divide it into a number of discrete segments. For now, let’s choose \(\Delta x = 0.5\), which creates four segments and thus five points: \(x_1 = 0, x_2 = 0.5, x_3 = 1.0, x_4 = 1.5, x_5 = 2.0\).

+

Our goal is then to find approximate values of \(y(x)\) at these points: \(y_1\) through \(y_5\). So, we have five unknowns, and need five equations to solve for them. We can use the ODE to provide these equations, by replacing the derivatives with finite differences, and applying the equation at particular discrete locations.

+

Recall that \(y(x)\) is a function just like \(f(x)\), and so we can apply the above finite difference equations to \(y(x)\) and \(y(x+\Delta x)\). Now that we have points, or nodes, at locations separated by \(\Delta x\), we can consider a point \(x_i\) where \(y(x_i) = y_i\), \(y(x_i + \Delta x) = y(x_{i+1}) = y_{i+1}\), and \(y(x_i - \Delta x) = y(x_{i-1}) = y_{i-1}\).

+

To do this, we’ll follow a few steps:

+

1.) Replace exact derivatives in the original ODE with finite differences, and apply the equation at a particular location \((x_i, y_i)\).

+

For our example, this gives:

+
+(4.21)\[\begin{equation} +\frac{y_{i+1} - 2y_i + y_{i-1}}{\Delta x^2} + x_i \left( \frac{y_{i+1} - y_{i-1}}{2 \Delta x}\right) - x_i y_i = 2 x_i +\end{equation}\]
+

which applies at location \((x_i, y_i)\).

+

2.) Next, rearrange the equation into a recursion formula:

+
+(4.22)\[\begin{equation} +y_{i-1} \left(1 - x_i \frac{\Delta x}{2}\right) + y_i \left( -2 -\Delta x^2 x_i \right) + y_{i+1} \left(1 + x_i \frac{\Delta x}{2}\right) = 2 x_i \Delta x^2 +\end{equation}\]
+

We can use this equation to get an equation for each of the interior points in the domain.

+

For the first and last points—the boundary points—we already have equations, given by the boundary conditions.

+

3.) Set up system of linear equations

+

Applying the recursion formula to the interior points, and the boundary conditions for the boundary points, we can get a system of simultaneous linear equations:

+
+(4.23)\[\begin{align} +y_1 &= 1 \\ +y_1 (0.875) + y_2 (-2.125) + y_3 (1.125) &= 0.25 \\ +y_2 (0.75) + y_3 (-2.25) + y_4 (1.25) &= 0.5 \\ +y_3 (0.625) + y_4 (-2.375) + y_5 (1.375) &= 0.75 \\ +y_5 &= 8 +\end{align}\]
+

This is a system of five equations and five unknowns, which we can solve! But, solving using substitution would be painful, so let’s represent this system of equations using a matrix and vectors:

+
+(4.24)\[\begin{equation} +\begin{bmatrix} +1 & 0 & 0 & 0 & 0 \\ +0.875 & -2.125 & 1.125 & 0 & 0 \\ +0 & 0.75 & -2.25 & 1.25 & 0 \\ +0 & 0 & 0.625 & -2.375 & 1.375 \\ +0 & 0 & 0 & 0 & 1 +\end{bmatrix} +\begin{bmatrix} y_1 \\ y_2 \\ y_3 \\ y_4 \\ y_5 \end{bmatrix} = +\begin{bmatrix} 1 \\ 0.25 \\ 0.5 \\ 0.75 \\ 8 \end{bmatrix} +\end{equation}\]
+

or, more compactly, \(A \mathbf{y} = \mathbf{b}\).

+

4.) Solve the linear system of equations

+

The final step is just to solve. We can do this in Matlab with y = A \ b. (This is equivalent to y = inv(A)*b, but faster.)

+
+
+
A = [1.0 0 0 0 0;
+     0.875 -2.125 1.125 0 0;
+         0 0.75 -2.25 1.25 0;
+         0 0 0.625 -2.375 1.375;
+         0 0 0 0 1];
+
+b = [1.0; 0.25; 0.5; 0.75; 8.0];
+
+x = [0 : 0.5 : 2];
+y = A \ b;
+plot(x, y, 'o-');
+
+
+
+
+../../_images/finite-difference_3_0.png +
+
+
+

4.2.3.1. Matlab implementation

+

Of course, all of this will be easier if we implement in Matlab in a general way. We’ll use a for loop to populate the coefficient matrix \(A\) and right-hand-side vector \(\mathbf{b}\):

+
+
+
clear all; clc
+
+dx = 0.5;
+x = [0 : dx : 2];
+n = length(x);
+A = zeros(n,n); b = zeros(n,1);
+
+for i = 1 : n
+    if i == 1
+        A(1,1) = 1;
+        b(1) = 1;
+    elseif i == n
+        A(n,n) = 1;
+        b(n) = 8;
+    else
+        A(i, i-1) = 1 - x(i)*dx/2;
+        A(i, i) = -2 - x(i)*dx^2;
+        A(i, i+1) = 1 + x(i)*dx/2;
+        b(i) = 2*x(i)*dx^2;
+    end
+end
+y = A \ b;
+plot(x, y, 'o-')
+
+
+
+
+../../_images/finite-difference_5_0.png +
+
+

This looks good, but we can get a more-accurate solution by reducing our step size \(\Delta x\):

+
+
+
clear all; clc
+
+dx = 0.001;
+x = [0 : dx : 2];
+n = length(x);
+A = zeros(n,n); b = zeros(n,1);
+
+for i = 1 : n
+    if i == 1
+        A(1,1) = 1;
+        b(1) = 1;
+    elseif i == n
+        A(n,n) = 1;
+        b(n) = 8;
+    else
+        A(i, i-1) = 1 - x(i)*dx/2;
+        A(i, i) = -2 - x(i)*dx^2;
+        A(i, i+1) = 1 + x(i)*dx/2;
+        b(i) = 2*x(i)*dx^2;
+    end
+end
+y = A \ b;
+plot(x, y)
+
+
+
+
+../../_images/finite-difference_7_0.png +
+
+
+
+
+

4.2.4. Boundary conditions

+

We will encounter four main kinds of boundary conditions. Consider the ODE \(y^{\prime\prime} + y = 0\), on the domain \(0 \leq x \leq L\).

+
    +
  • First type, or Dirichlet, boundary conditions specify fixed values of \(y\) at the boundaries: \(y(0) = a\) and \(y(L) = b\).

  • +
  • Second type, or Neumann, boundary conditions specify values of the derivative at the boundaries: \(y^{\prime}(0) = a\) and \(y^{\prime}(L) = b\).

  • +
  • Third type, or Robin, boundary conditions specify a linear combination of the function value and its derivative at the boundaries: \(a \, y(0) + b \, y^{\prime}(0) = g(0)\) and \(a \, y(L) + b \, y^{\prime}(L) = g(L)\), where \(g(x)\) is some function.

  • +
  • Mixed boundary conditions, which combine any of these three at the different boundaries. For example, we could have \(y(0) = a\) and \(y^{\prime}(L) = b\).

  • +
+

Whichever type of boundary condition we are dealing with, the goal will be to construct an equation representing the boundary condition to incorporate in our system of equations.

+

If we have a fixed value boundary condition, such as \(y(0) = a\), then this equation is straightforward:

+
+(4.25)\[\begin{equation} +y_1 = a +\end{equation}\]
+

where \(y_1\) is the first point in the grid of points, corresponding to \(x_1 = 0\). (We saw this in the example above.) In Matlab, we can implement this equation with

+
A(1,1) = 1;
+b(1) = a;
+
+
+

If we have a fixed derivative boundary condition, such as \(y^{\prime}(0) = 0\), then we need to use a finite difference to represent the derivative. When the boundary condition is at the starting location, \(x=0\), the easiest way to do this is with a forward difference:

+
+(4.26)\[\begin{align} +y^{\prime}(0) \approx \frac{y_2 - y_1}{\Delta x} &= 0 \\ +-y_1 + y_2 &= 0 +\end{align}\]
+

We can implement this in Matlab with

+
A(1,1) = -1;
+A(1,2) = 1;
+b(1) = 0;
+
+
+

When we have this sort of derivative boundary condition at the right side of the domain, at \(x=L\), then we can use a backward difference to represent the derivative:

+
+(4.27)\[\begin{align} +y^{\prime}(L) \approx \frac{y_n - y_{n_1}}{\Delta x} &= 0 \\ +-y_{n-1} + y_n &= 0 +\end{align}\]
+

where \(y_n\) is the final point (\(x_n = L\)) and \(y_{n-1}\) is the second-to-last point (\(x_{n-1} = L - \Delta x\)). We can implement this in Matlab with

+
A(n,n-1) = -1;
+A(n,n) = 1;
+b(n) = 0;
+
+
+

If we have a linear combination of a fixed value and fixed derivative, like \(a \, y(0) + b \, y^{\prime}(0) = c\), then we can combine the above approaches using a forward difference:

+
+(4.28)\[\begin{align} +a y(0) + b y^{\prime}(0) \approx a y_1 + b \frac{y_2 - y_1}{\Delta x} &= c \\ +(a \Delta x - b) y_1 + b y_2 &= c \Delta x +\end{align}\]
+

and in Matlab:

+
A(1,1) = a*dx - b;
+A(1,2) = b;
+b(1) = c * dx;
+
+
+
+

4.2.4.1. Using central differences for derivative BCs

+

When a boundary condition involves a derivative, we can use a central difference to approximate the first derivative; this is more accurate than a forward or backward difference.

+

Consider the formula for a central difference at \(x=0\), applied for the boundary condition \(y^{\prime}(0) = 0\):

+
+(4.29)\[\begin{align} +y^{\prime}(0) \approx \frac{y_2 - y_0}{2 \Delta x} &= 0 \\ +y_0 &= y_2 +\end{align}\]
+

where \(y_0\) is an imaginary, or ghost, node outside the domain. We can’t actually keep this point in our implementation, because it isn’t a real point.

+

We still need an equation for the point at the boundary, \(y_1\). To get this, we’ll apply the regular recursion formula, normally used at interior points:

+
+(4.30)\[\begin{align} +a y_{i-1} + b y_i + c y_{i+1} = f(x_i) \\ +a y_0 + b y_1 + c y_2 = f(x_1) \;, +\end{align}\]
+

where \(a\), \(b\), \(c\), and \(f(x)\) depend on the problem. Normally we wouldn’t use this at the boundary node, \(y_1\), because it references a point outside the domain to the left—but we have an equation for that! From above, based on the boundary condition, we have \(y_0 = y_2\). If we incorporate that into the recursion formula, we can eliminate the ghost node \(y_0\):

+
+(4.31)\[\begin{align} +a y_2 + b y_1 + c y_2 &= f(x_1) \\ +b y_1 + (a + c) y_2 &= f(x_1) \, +\end{align}\]
+

which is the equation we can actually use at the boundary point. +In Matlab, this looks like

+
A(1,1) = b;
+A(1,2) = a + c;
+b(1) = f(x(1));
+
+
+
+
+
+

4.2.5. Example: nonlinear BVP

+

So far we’ve seen how to handle a linear boundary value problem, but what if we have a nonlinear BVP? This is going to be trickier, because our work so far relies on using linear algebra to solve the system of (linear) equations.

+

For example, consider the 2nd-order ODE

+
+(4.32)\[\begin{equation} +y^{\prime\prime} = 3y + x^2 + 100 y^2 +\end{equation}\]
+

with the boundary conditions \(y(0) = y(1) = 0\). This is nonlinear, due to the \(y^3\) term on the right-hand side.

+

To solve this, let’s first convert it into a discrete form, by replacing the second derivative with a finite difference and any \(x\)/\(y\) present with \(x_i\) and \(y_i\). We’ll also move any constants (i.e., terms that don’t contain \(y_i\)) and the nonlinear term to the right-hand side:

+
+(4.33)\[\begin{equation} +\frac{y_{i-1} - 2y_i + y_{i+1}}{\Delta x^2} - 3y_i = x_i^2 + 100 y_i^2 +\end{equation}\]
+

where the boundary conditions are now \(y_1 = 0\) and \(y_n = 0\), with \(n\) as the number of grid points. We can rearrange and simplify into our recursion formula:

+
+(4.34)\[\begin{equation} +y_{i-1} + y_i \left( -2 - 3 \Delta x^2 \right) + y_{i+1} = x_i^2 \Delta x^2 + 100 \Delta x^2 y_i^2 +\end{equation}\]
+

The question is: how do we solve this now? The nonlinear term involving \(y_i^3\) on the right-hand side complicates things, but we know how to set up and solve this without the nonlinear term. We can use an approach known as successive iteration:

+
    +
  1. Solve the ODE without the nonlinear term to get an initial “guess” to the solution for \(y\).

  2. +
  3. Then, incorporate that guess solution in the nonlinear term on the right-hand side, treating it as a constant. We can call this \(y_{\text{old}}\). Then, solve the full system for a new \(y\) solution.

  4. +
  5. Check whether the new \(y\) matches \(y_{\text{old}}\) with some tolerance. For example, check whether \(\max\left(\left| y - y_{\text{old}} \right| \right) < \) some tolerance, such as \(10^{-6}\). If this is true, then we can consider the solution converged. If it is not true, then set \(y_{\text{old}} = y\), and repeat the process starting at step 2.

  6. +
+

Let’s implement that process in Matlab:

+
+
+
%% Initial setup
+clear all; clc
+
+dx = 0.01;
+x = 0 : dx : 1;
+n = length(x);
+
+A = zeros(n, n);
+b = zeros(n, 1);
+
+%% First, solve the problem without the nonlinear term:
+for i = 1 : n
+    if i == 1  % x = 0 boundary condition
+        A(1,1) = 1;
+        b(1) = 0;
+    elseif i == n % x = L boundary condition
+        A(n,n) = 1;
+        b(n) = 0;
+    else  % interior nodes, use recursion formula
+        A(i, i-1) = 1;
+        A(i, i) = -2 - 3*dx^2;
+        A(i, i+1) = 1;
+        b(i) = x(i)^2 * dx^2;
+    end
+end
+% get solution without nonlinear term
+y = A \ b;
+
+plot(x, y, '--'); hold on
+
+%% Now, set up iterative process to solve while incorporating nonlinear terms
+iter = 1;
+y_old = zeros(n, 1);
+while max(abs(y - y_old)) > 1e-6
+    y_old = y;
+    % A matrix is not changed, but the b vector does
+    for i = 2 : n - 1
+        b(i) = x(i)^2 * dx^2 + 100*(dx^2)*(y_old(i)^2);
+    end
+    
+    y = A \ b;
+    iter = iter + 1;
+end
+
+fprintf('Number of iterations: %d\n', iter);
+plot(x, y)
+legend('Initial solution', 'Final solution')
+
+
+
+
+
Number of iterations: 16
+
+
+../../_images/finite-difference_10_1.png +
+
+

Another option is just to set our “guess” for the \(y\) solution to be zero, rather than solve the problem in two steps:

+
+
+
%% Initial setup
+clear all; clc
+
+dx = 0.01;
+x = 0 : dx : 1;
+n = length(x);
+
+A = zeros(n, n);
+b = zeros(n, 1);
+
+%% Set up the coefficient matrix, which does not change
+for i = 1 : n
+    if i == 1  % x = 0 boundary condition
+        A(1,1) = 1;
+        b(1) = 0;
+    elseif i == n % x = L boundary condition
+        A(n,n) = 1;
+        b(n) = 0;
+    else  % interior nodes, use recursion formula
+        A(i, i-1) = 1;
+        A(i, i) = -2 - 3*dx^2;
+        A(i, i+1) = 1;
+        b(i) = x(i)^2 * dx^2;
+    end
+end
+
+% just use zeros as our initial guess for the solution
+y = zeros(n, 1);
+
+%% Successive iteration
+iter = 1;
+y_old = 100 * rand(n, 1); % setting this to some random values, just to enter the while loop
+while max(abs(y - y_old)) > 1e-6
+    y_old = y;
+    % A matrix is not changed, but the b vector does
+    for i = 2 : n - 1
+        b(i) = x(i)^2 * dx^2 + 100*(dx^2)*(y_old(i)^2);
+    end
+    
+    y = A \ b;
+    iter = iter + 1;
+end
+
+fprintf('Number of iterations: %d\n', iter);
+
+
+
+
+
Number of iterations: 17
+
+
+
+
+

This made our process take slightly more iterations, because the initial guess was slightly further away from the final solution. For other problems, having a bad initial guess could make the process take much longer, so coming up with a good initial guess may be important.

+
+
+

4.2.6. Example: heat transfer through a fin

+

Let’s now consider a more complicated example: heat transfer through an extended surface (a fin).

+
+
+ Heat transfer fin +
Figure: Geometry of a heat transfer fin
+
+
+

In this situation, we have the temperature of the body \(T_b\), the temperature of the ambient fluid \(T_{\infty}\); the length \(L\), width \(w\), and thickness \(t\) of the fin; the thermal conductivity of the fin material \(k\); and convection heat transfer coefficient \(h\).

+

The boundary conditions can be defined in different ways, but generally we can say that the temperature of the fin at the wall is the same as the body temperature, and that the fin is insulated at the tip. This gives us

+
+(4.35)\[\begin{align} +T(x=0) &= T_b \\ +q(x=L) = 0 \rightarrow \frac{dT}{dx} (x=0) &= 0 +\end{align}\]
+

Our goal is to solve for the temperature distribution \(T(x)\). To do this, we need to set up a governing differential equation. Let’s do a control volume analysis of heat transfer through the fin:

+
+
+ Control volume for heat transfer fin +
Figure: Control volume for heat transfer through the fin
+
+
+

Given a particular volumetric slice of the fin, we can define the heat transfer rates of conduction through the fin and convection from the fin to the air:

+
+(4.36)\[\begin{align} +q_{\text{conv}} &= h P \left( T - T_{\infty} \right) dx \\ +q_{\text{cond}, x} &= -k A_c \left(\frac{dT}{dx}\right)_{x} \\ +q_{\text{cond}, x+\Delta x} &= -k A_c \left(\frac{dT}{dx}\right)_{x+\Delta x} \;, +\end{align}\]
+

where \(P\) is the perimeter (so that \(P \, dx\) is the heat transfer area to the fluid) and \(A_c\) is the cross-sectional area.

+

Performing a balance through the control volume:

+
+(4.37)\[\begin{align} +q_{\text{cond}, x+\Delta x} &= q_{\text{cond}, x} - q_{\text{conv}} \\ +-k A_c \left(\frac{dT}{dx}\right)_{x+\Delta x} &= -k A_c \left(\frac{dT}{dx}\right)_{x} - h P \left( T - T_{\infty} \right) dx \\ +-k A_c \frac{\left.\frac{dT}{dx}\right|_{x+\Delta x} - \left.\frac{dT}{dx}\right|_{x}}{dx} &= -h P ( T - T_{\infty} ) \\ +\lim_{\Delta x \rightarrow 0} : -k A_c \left. \frac{d^2 T}{dx^2} \right|_x &= -h P (T - T_{\infty}) \\ +\frac{d^2 T}{dx^2} &= \frac{h P}{k A_c} (T - T_{\infty}) \\ +\frac{d^2 T}{dx^2} &= m^2 (T - T_{\infty}) +\end{align}\]
+

then we have as a governing equation

+
+(4.38)\[\begin{equation} +\frac{d^2 T}{dx^2} - m^2 (T - T_{\infty}) = 0 \;, +\end{equation}\]
+

where \(m^2 = (h P)/(k A_c)\).

+

We can obtain an exact solution for this ODE. For convenience, let’s define a new variable, \(\theta\), which is a normalized temperature:

+
+(4.39)\[\begin{equation} +\theta \equiv T - T_{\infty} +\end{equation}\]
+

where \(\theta^{\prime} = T^{\prime}\) and \(\theta^{\prime\prime} = T^{\prime\prime}\). +This gives us a new governing equation:

+
+(4.40)\[\begin{equation} +\theta^{\prime\prime} - m^2 \theta = 0 \;. +\end{equation}\]
+

This is a 2nd-order homogeneous ODE, which looks a lot like \(y^{\prime\prime} + a y = 0\). The exact solution is then

+
+(4.41)\[\begin{align} +\theta(x) &= c_1 e^{-m x} + c_2 e^{m x} \\ +T(x) &= T_{\infty} + c_1 e^{-m x} + c_2 e^{m x} +\end{align}\]
+

We’ll use this to look at the accuracy of a numerical solution, but we will not be able to find an exact solution for more complicated versions of this problem.

+

We can also solve this numerically using the finite difference method. Let’s replace the derivative with a finite difference:

+
+(4.42)\[\begin{align} +\frac{d^2 T}{dx^2} - m^2 (T - T_{\infty}) &= 0 \\ +\frac{T_{i-1} - 2T_i + T_{i+1}}{\Delta x^2} - m^2 \left( T_i - T_{\infty} \right) &= 0 +\end{align}\]
+

which we can rearrange into a recursion formula:

+
+(4.43)\[\begin{equation} +T_{i-1} + T_i \left( -2 - \Delta x^2 m^2 \right) + T_{i+1} = -m^2 \Delta x^2 \, T_{\infty} +\end{equation}\]
+

This gives us an equation for all the interior nodes; we can use the above boundary conditions to get equations for the boundary nodes. For the boundary condition at \(x=L\), \(T^{\prime}(x=L) = 0\), let’s use a backward difference:

+
+(4.44)\[\begin{align} +T_1 &= T_b \\ +\frac{T_n - T_{n-1}}{\Delta x} = 0 \rightarrow - T_{n-1} + T_n &= 0 +\end{align}\]
+

Combining all these equations, we can construct a linear system: \(A \mathbf{T} = \mathbf{b}\).

+
+

4.2.6.1. Heat transfer with radiation

+

Let’s now consider a more-complicated case, where we also have radiation heat transfer occuring along the length of the fin. Now, our governing ODE is

+
+(4.45)\[\begin{equation} +\frac{d^2 T}{dx^2} - \frac{h P}{k A_c} \left(T - T_{\infty}\right) - \frac{\sigma \epsilon P}{h A_c} \left(T^4 - T_{\infty}^4 \right) = 0 +\end{equation}\]
+

This is a bit trickier to solve because of the nonlinear term involving \(T^4\). But, we can handle it via the iterative solution method discussed above.

+
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/bvps/shooting-method.html b/docs/content/bvps/shooting-method.html new file mode 100644 index 0000000..c1a5d2a --- /dev/null +++ b/docs/content/bvps/shooting-method.html @@ -0,0 +1,801 @@ + + + + + + + + 4.1. Shooting Method — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+ +
+ + + + + + + + + + + + + + + + + + +
+ + + +
+
+
+
+ +
+ +
+

4.1. Shooting Method

+

Boundary-value problems are also ordinary differential equations—the difference is that our two constraints are at boundaries of the domain, rather than both being at the starting point.

+

For example, consider the ODE

+
+(4.1)\[\begin{equation} +y^{\prime\prime} + xy^{\prime} - xy = 2x +\end{equation}\]
+

with the boundary conditions \(y(0)=1\) and \(y(2)=8\).

+

The numerical methods we have already discussed (e.g., Forward Euler, Runge-Kutta) require values of \(y\) and \(y^{\prime}\) at the starting point, \(x=0\). So we can’t use these directly because we are missing \(y^{\prime}(0)\).

+

But, what if we could guess a value for the missing initial condition, then integrate towards the second boundary condition using one of our familiar numerical methods, and then adjust our guess if necessary and repeat? This concept is the shooting method.

+

The shooting method algorithm is:

+
    +
  1. Guess a value of the missing initial condition; in this case, that is \(y'(0)\).

  2. +
  3. Integrate the ODE like an initial-value problem, using our existing numerical methods, to get the given boundary condition(s); in this case, that is \(y(L)\).

  4. +
  5. Assuming your trial solution for \(y(L)\) does not match the given boundary condition, adjust your guess for \(y'(0)\) and repeat.

  6. +
+

Now, this algorithm will not work particularly well if all your guesses are random/uninformed. Fortunately, we can use linear interpolation to inform a third guess based on two initial attempts:

+
+(4.2)\[\begin{align} +\text{guess 3} &= \text{guess 2} + m \left( \text{target} - \text{solution 2} \right) \\ +m &= \frac{\text{guess 1} - \text{guess 2}}{\text{solution 1} - \text{solution 2}} +\end{align}\]
+

where “target” is the target boundary condition—in this case, \(y(L)\).

+
+

4.1.1. Example: linear ODE

+

Let’s try solving the given ODE using the shooting method:

+
+(4.3)\[\begin{equation} +y^{\prime\prime} + xy^{\prime} - xy = 2x +\end{equation}\]
+

with the boundary conditions \(y(0)=1\) and \(y(2)=8\).

+

First, we need to convert this 2nd-order ODE into a system of two 1st-order ODEs, where we can define \(u = y'\):

+
+(4.4)\[\begin{align} +y' &= u \\ +u' &= 2x + xy - xu +\end{align}\]
+
+
+
%%file shooting_rhs.m
+function dydx = shooting_rhs(x, y)
+
+dydx = zeros(2,1);
+dydx(1) = y(2);
+dydx(2) = 2*x - x*y(2) + x*y(1);
+
+
+
+
+
Created file '/Users/niemeyek/projects/ME373-book/content/bvps/shooting_rhs.m'.
+
+
+
+
+
+
+
clear all; clc
+
+% target boundary condition
+target = 8;
+
+% Pick a guess for y'(0) of 1
+guess1 = 1;
+[X, Y] = ode45('shooting_rhs', [0 2], [1 guess1]);
+solution1= Y(end,1);
+fprintf('Solution 1: %5.2f\n', solution1);
+
+% Pick a second guess for y'(0) of 4
+guess2 = 4;
+[X, Y] = ode45('shooting_rhs', [0 2], [1 guess2]);
+solution2 = Y(end,1);
+fprintf('Solution 2: %5.2f\n', solution2);
+
+% now use linear interpolation to find a new guess
+m = (guess1 - guess2)/(solution1 - solution2);
+guess3 = guess2 + m*(target-solution2);
+fprintf('Guess 3:    %5.2f\n', guess3);
+
+[X, Y] = ode45('shooting_rhs', [0 2], [1 guess3]);
+solution3 = Y(end,1);
+fprintf('Solution 3: %5.2f\n', solution3);
+fprintf('Target:     %5.2f\n', target);
+
+plot(X, Y(:,1)); axis([0 2 0 9])
+
+
+
+
+
Solution 1:  6.00
+Solution 2: 11.96
+Guess 3:     2.01
+Solution 3:  8.00
+Target:      8.00
+
+
+../../_images/shooting-method_3_1.png +
+
+

As you can see, using linear interpolation, we are able to find the correct guess for the missing initial condition \(y'(0)\) with in just three steps. This works so well because this is a linear ODE. If we had a nonlinear ODE, it would take more tries, as we’ll see shortly.

+
+
+

4.1.2. Example: nonlinear ODE

+

We can use the shooting method to solve a famous fluids problem: the Blasius boundary layer.

+
+
+ Laminar boundary layer, from https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg +
Figure: Laminar boundary layer, taken from https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg
+
+
+

To get to a solveable ODE, we start with the conservation of momentum equation (i.e., Navier–Stokes equation) in the \(x\)-direction:

+
+(4.5)\[\begin{equation} +u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2} +\end{equation}\]
+

and the conservation of mass equation:

+
+(4.6)\[\begin{equation} +\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \;, +\end{equation}\]
+

where \(u\) is the velocity component in the \(x\)-direction, \(v\) is the velocity component in the \(y\)-direction, and \(\nu\) is the fluid’s kinematic viscosity. The boundary conditions are that \(u = v = 0\) at \(y=0\), and that \(u = U_{\infty}\) as \(y \rightarrow \infty\), where \(U_{\infty}\) is the free-stream velocity.

+

Blasius solved this problem by converting the PDE into an ODE, by recognizing that the boundary layer thickness is given by \(\delta(x) \sim \sqrt{\frac{x \nu}{U_{\infty}}}\), and then nondimensionalizing the position coordinates using a similarity variable

+
+(4.7)\[\begin{equation} +\eta = y \sqrt{\frac{U_{\infty}}{2 \nu x}} +\end{equation}\]
+

By introducing the stream function, \(\psi (x,y)\), we can ensure the continuity equation is satisfied:

+
+(4.8)\[\begin{equation} +u = \frac{\partial \psi}{\partial y} \;, \quad v = -\frac{\partial \psi}{\partial x} +\end{equation}\]
+

Let’s check this, using SymPy:

+
+
+
%%python
+import sympy as sym
+sym.init_printing()
+x, y, u, v = sym.symbols('x y u v')
+
+# Streamfunction
+psi = sym.Function(r'psi')(x,y)
+
+# Define u and v based on the streamfunction
+u = psi.diff(y)
+v = -psi.diff(x)
+
+# Check the continuity equation:
+print(u.diff(x) + v.diff(y) == 0)
+
+
+
+
+
True
+
+
+
+
+

Using the boundary layer thickness and free-stream velocity, we can define the dimensionlesss stream function \(f(\eta)\):

+
+(4.9)\[\begin{equation} +f(\eta) = \frac{\psi}{U_{\infty}} \sqrt{\frac{U_{\infty}}{2 \nu x}} +\end{equation}\]
+

which relates directly to the velocity components:

+
+(4.10)\[\begin{align} +u &= \frac{\partial \psi}{\partial y} = \frac{\partial \psi}{\partial f} \frac{\partial f}{\partial \eta} \frac{\partial \eta}{\partial y} \\ + &= U_{\infty} \sqrt{\frac{2 \nu x}{U_{\infty}}} \cdot f^{\prime}(\eta) \cdot \sqrt{\frac{U_{\infty}}{2 \nu x}} \\ +u &= U_{\infty} f^{\prime} (\eta) \\ +v &= -\frac{\partial \psi}{\partial x} = -\left( \frac{\partial \psi}{\partial x} + \frac{\partial \psi}{\partial \eta} \frac{\partial \eta}{\partial x} \right) \\ + &= \sqrt{\frac{\nu U_{\infty}}{2x}} \left( \eta f^{\prime} - f \right) +\end{align}\]
+

We can insert these into the \(x\)-momentum equation, which leads to an ODE for the dimensionless stream function \(f(\eta)\):

+
+(4.11)\[\begin{equation} +f^{\prime\prime\prime} + f f^{\prime\prime} = 0 \;, +\end{equation}\]
+

with the boundary conditions \(f = f^{\prime} = 0\) at \(\eta = 0\), and \(f^{\prime} = 1\) as \(\eta \rightarrow \infty\).

+

This is a 3rd-order ODE, which we can solve by converting it into three 1st-order ODEs:

+
+(4.12)\[\begin{align} +y_1 &= f \quad y_1^{\prime} = y_2 \\ +y_2 &= f^{\prime} \quad y_2^{\prime} = y_3 \\ +y_3 &= f^{\prime\prime} \quad y_3^{\prime} = -y_1 y_3 +\end{align}\]
+

and we can use the shooting method to solve by recognizing that we have two initial conditions, \(y_1(0) = y_2(0) = 0\), and are missing \(y_3(0)\). We also have a target boundary condition: \(y_2(\infty) = 1\).

+

(Note: obviously we cannot truly integrate over \(0 \leq \eta < \infty\). Instead, we just need to choose a large enough number. In this case, using 10 is sufficient.)

+

Let’s create a function to evaluate the derivatives:

+
+
+
%%file blasius_rhs.m
+function dydx = blasius_rhs(eta, y)
+
+dydx = zeros(3,1);
+
+dydx(1) = y(2);
+dydx(2) = y(3);
+dydx(3) = -y(1) * y(3);
+
+
+
+
+
Created file '/Users/niemeyek/projects/ME373-book/content/bvps/blasius_rhs.m'.
+
+
+
+
+

First, let’s try the same three-step approach we used for the simpler example, taking two guesses and then using linear interpolation to find a third guess:

+
+
+
clear all; clc
+
+target = 1.0;
+
+guesses = zeros(3,1);
+solutions = zeros(3,1);
+
+guesses(1) = 1;
+[eta, F] = ode45('blasius_rhs', [0 10], [0 0 guesses(1)]);
+solutions(1) = F(end, 2);
+
+guesses(2) = 0.1;
+[eta, F] = ode45('blasius_rhs', [0 10], [0 0 guesses(2)]);
+solutions(2) = F(end, 2);
+
+m = (guesses(1) - guesses(2))/(solutions(1) - solutions(2));
+guesses(3) = guesses(2) + m*(target - solutions(2));
+
+[eta, F] = ode45('blasius_rhs', [0 10], [0 0 guesses(3)]);
+solutions(3) = F(end, 2);
+
+tries = [1; 2; 3];
+table(tries, guesses, solutions)
+fprintf('Target: %5.2f\n', target);
+
+
+
+
+
ans =
+
+  3x3 table
+
+    tries    guesses    solutions
+    _____    _______    _________
+
+      1            1     1.6553  
+      2          0.1     0.3566  
+      3      0.54587     1.1056  
+
+Target:  1.00
+
+
+
+
+

So, for this problem, using linear interpolation did not get us the correct solution on the third try. This is because the ODE is nonlinear. But, you can see that we are converging towards the correct solution—it will just take more tries.

+

Rather than manually take an unknown (and potentially large) number of guesses, let’s automate this with a while loop:

+
+
+
clear all; clc
+
+target = 1.0;
+
+% get these arrays of stored values started.
+% note: I'm only doing this to make it easier to show a table of values
+% at the end; otherwise, there's no need to store these values.
+tries = [1; 2; 3];
+guesses = zeros(3,1);
+solutions = zeros(3,1);
+
+guesses(1) = 1;
+[eta, F] = ode45('blasius_rhs', [0 10], [0 0 guesses(1)]);
+solutions(1) = F(end, 2);
+
+guesses(2) = 0.1;
+[eta, F] = ode45('blasius_rhs', [0 10], [0 0 guesses(2)]);
+solutions(2) = F(end, 2);
+
+num = 2;
+solutions(3) = -1000.; % doing this to kick off the while loop
+while abs(target - solutions(num)) > 1.e-9
+    num = num + 1;
+    m = (guesses(num-2) - guesses(num-1))/(solutions(num-2) - solutions(num-1));
+    guesses(num) = guesses(num-1) + m*(target - solutions(num-1));
+    [eta, F] = ode45('blasius_rhs', [0 1e3], [0 0 guesses(num)]);
+    solutions(num) = F(end, 2);
+    tries(num) = num;
+    
+    % we should probably set a maximum number of iterations, just to prevent
+    % an infinite while loop in case something goes wrong
+    if num >= 1e4
+        break
+    end
+end
+
+table(tries, guesses, solutions)
+fprintf('Number of iterations required: %d', num)
+
+
+
+
+
ans =
+
+  7x3 table
+
+    tries    guesses    solutions
+    _____    _______    _________
+
+      1            1      1.6553 
+      2          0.1      0.3566 
+      3      0.54587      1.1056 
+      4      0.48301       1.019 
+      5      0.46922     0.99951 
+      6      0.46957           1 
+      7      0.46957           1 
+
+Number of iterations required: 7
+
+
+
+
+
+
+
%plot -r 200
+plot(F(:, 2), eta); ylim([0 5])
+xlabel("f^{\prime}(\eta) = u/U_{\infty}")
+ylabel('\eta')
+
+
+
+
+../../_images/shooting-method_13_0.png +
+
+

We can see that this plot of \(\eta\), the \(y\) position normalized by the boundary-layer thickness, vs. nondimensional velocity matches the original figure.

+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/contributing.html b/docs/content/contributing.html new file mode 100644 index 0000000..9792293 --- /dev/null +++ b/docs/content/contributing.html @@ -0,0 +1,660 @@ + + + + + + + + Contributing to Jupyter Book — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ + +
+
+ +
+ +
+

Contributing to Jupyter Book

+

Welcome to the jupyter-book repository! We’re excited you’re here and want to contribute.

+

These guidelines are designed to make it as easy as possible to get involved. +If you have any questions that aren’t discussed below, please let us know by opening an issue!

+

Before you start you’ll need to set up a free GitHub account and sign in. +Here are some instructions.

+

Already know what you’re looking for in this guide? Use the TOC to the right +to navigate this page!

+
+

Joining the conversation

+

jupyter-book is a young project maintained by a growing group of enthusiastic developers— and we’re excited to have you join! +Most of our discussions will take place on open issues.

+

As a reminder, we expect all contributors to jupyter-book to adhere to the Jupyter Code of Conduct in these conversations.

+
+
+

Contributing through GitHub

+

git is a really useful tool for version control. +GitHub sits on top of git and supports collaborative and distributed working.

+

You’ll use Markdown to chat in issues and pull requests on GitHub. +You can think of Markdown as a few little symbols around your text that will allow GitHub +to render the text with formatting. +For example you could write words as bold (**bold**), or in italics (*italics*), +or as a link ([link](https://https://youtu.be/dQw4w9WgXcQ)) to another webpage.

+

GitHub has a helpful page on +getting started with writing and formatting Markdown on GitHub.

+
+
+

Understanding issues, milestones and project boards

+

Every project on GitHub uses issues slightly differently.

+

The following outlines how the jupyter-book developers think about these tools.

+

Issues are individual pieces of work that need to be completed to move the project forwards. +A general guideline: if you find yourself tempted to write a great big issue that +is difficult to describe as one unit of work, please consider splitting it into two or more issues.

+

Issues are assigned labels which explain how they relate to the overall project’s +goals and immediate next steps.

+
+

Issue labels

+

The current list of labels are here and include:

+
    +
  • Help Wanted These issues contain a task that a member of the team has determined we need additional help with.

    +

    If you feel that you can contribute to one of these issues, we especially encourage you to do so!

    +
  • +
  • Good First Issue These issues contain a task that a member of the team thinks could be a good entry point to the project.

    +

    If you’re new to the jupyter-book project, we think that this is a great place for your first contribution!

    +
  • +
  • Bugs These issues point to problems in the project.

    +

    If you find new a bug, please give as much detail as possible in your issue, including steps to recreate the error. +If you experience the same bug as one already listed, please add any additional information that you have as a comment.

    +
  • +
  • Enhancement These issues are asking for enhancements to be added to the project.

    +

    Please try to make sure that your enhancement is distinct from any others that have already been requested or implemented. +If you find one that’s similar but there are subtle differences please reference the other request in your issue.

    +
  • +
  • Question These are questions that users and contributors have asked.

    +

    Please check the issues (especially closed ones) to see if your question has been asked and answered before. +If you find one that’s similar but there are subtle differences please reference the other request in your issue.

    +
  • +
+
+
+
+

Repository Structure of Jupyter Book

+

This section covers the general structure of the +Jupyter Book repository, and +explains which pieces are where.

+

The Jupyter Book repository contains two main pieces:

+
+

The command-line tool and Python package

+

This is used to help create and build books. +It can be found at ./jupyter_book.

+
    +
  • The page module builds single pages. This module is meant to be self-contained for +converting single .ipynb/.md/etc pages into HTML. Jupyter Book uses this module when +building entire books, but the module can also be used on its own (it’s what jupyter-book page uses). +You can find the module at: jupyter_book/page.

  • +
  • The create.py and build.py create and build a book. They connect with the CLI and +are used to process multiple pages and stitch them together into a static website template.

  • +
+
+
+

The template SSG website

+

This is used when generating new books. This website defines the structure of +the site that is created when you run jupyter-book create. It contains the Javascript, CSS, and +HTML structure of a book. It can be found at +jupyter_book/book_template.

+
    +
  • The _includes/ +folder contains core HTML and javascript files for the site. For example, +_includes/head.html contains the HTML for the header of each page, which is where CSS and JS files are linked.

  • +
  • The assets/ +folder contains static CSS/JS files that don’t depend on site configuration.

  • +
  • The _sass/ +folder contains all of the book and page CSS rules. This is stitched together in a single CSS file +at build time (SCSS is a way to split up CSS rules among multiple files). Within this folder, the +_sass/page/ folder +has CSS files for a single page of content, while the other folders/files contain CSS rules for +the whole book.

  • +
  • The content/ +folder contains the content for the Jupyter Book documentation (e.g., the markdown for this page).

  • +
+
+
+

An example

+

Here are a few examples of how this code gets used to help you get started.

+
    +
  • when somebody runs jupyter-book create mybook/, the create.py module is used to generate an empty template using the template in jupyter_book/book_template/.

  • +
  • when somebody runs jupyter-book build mybook/, the build.py module to loop through your page content files, +and uses the page/ module to convert each one into HTML and places it in mybook/_build.

  • +
+

Hopefully this explanation gets you situated and helps you understand how the pieces all fit together. +If you have any questions, feel free to open an issue asking for help!

+
+
+
+

Making a change

+

We appreciate all contributions to jupyter-book, but those accepted fastest will follow a workflow similar to the following:

+

1. Comment on an existing issue or open a new issue referencing your addition.

+

This allows other members of the jupyter-book development team to confirm that you aren’t overlapping with work that’s currently underway and that everyone is on the same page with the goal of the work you’re going to carry out.

+

This blog is a nice explanation of why putting this work in up front is so useful to everyone involved.

+

2. Fork the jupyter-book repository to your profile.

+

This is now your own unique copy of jupyter-book. +Changes here won’t effect anyone else’s work, so it’s a safe space to explore edits to the code!

+

Make sure to keep your fork up to date with the master repository.

+

3. Make the changes you’ve discussed.

+

Try to keep the changes focused. +We’ve found that working on a new branch makes it easier to keep your changes targeted.

+

4. Submit a pull request.

+

A member of the development team will review your changes to confirm that they can be merged into the main code base. +When opening the pull request, we ask that you follow some specific conventions. +We outline these below.

+
+

Pull Requests

+

To improve understanding pull requests “at a glance”, we encourage the use of several standardized tags. +When opening a pull request, please use at least one of the following prefixes:

+
    +
  • [BRK] for changes which break existing builds or tests

  • +
  • [DOC] for new or updated documentation

  • +
  • [ENH] for enhancements

  • +
  • [FIX] for bug fixes

  • +
  • [REF] for refactoring existing code

  • +
  • [STY] for stylistic changes

  • +
  • [TST] for new or updated tests, and

  • +
+

You can also combine the tags above, for example if you are updating both a test and +the documentation: [TST, DOC].

+

Pull requests should be submitted early and often!

+

If your pull request is not yet ready to be merged, please open your pull request as a draft. +More information about doing this is available in GitHub’s documentation. +This tells the development team that your pull request is a “work-in-progress”, +and that you plan to continue working on it.

+

When your pull request is Ready for Review, you can select this option on the PR’s page, +and a project maintainer will review your proposed changes.

+
+
+
+

Recognizing contributors

+

We welcome and recognize all contributions from documentation to testing to code development. +You can see a list of current contributors in the contributors tab.

+
+
+

Thank you!

+

You’re awesome.

+
+

— Based on contributing guidelines from the STEMMRoleModels project.

+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/first-order.html b/docs/content/first-order.html new file mode 100644 index 0000000..bf6f12f --- /dev/null +++ b/docs/content/first-order.html @@ -0,0 +1,571 @@ + + + + + + + + 1. Solutions to 1st-order ODEs — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ + +
+
+ +
+ +
+

1. Solutions to 1st-order ODEs

+
+

1.1. 1. Solution by direct integration

+

When equations are of this form, we can directly integrate:

+
+(1.1)\[\begin{align} +\frac{dy}{dx} &= y^{\prime} = f(x) \\ +\int dy &= \int f(x) dx \\ +y(x) &= \int f(x) dx + C +\end{align}\]
+

For example:

+
+(1.2)\[\begin{align} +\frac{dy}{dx} &= x^2 \\ +y(x) &= \frac{1}{3} x^3 + C +\end{align}\]
+

While these problems look simple, there may not be an obvious closed-form solution to all:

+
+(1.3)\[\begin{align} +\frac{dy}{dx} &= e^{-x^2} \\ +y(x) &= \int e^{-x^2} dx + C +\end{align}\]
+

(You may recognize this as leading to the error function, \(\text{erf}\): +\(\frac{1}{2} \sqrt{\pi} \text{erf}(x) + C\), +so the exact solution to the integral over the range \([0,1]\) is 0.7468.)

+
+
+

1.2. 2. Solution by separation of variables

+

If the given derivative is a separate function of \(x\) and \(y\), then we can solve via separation of variables:

+
+(1.4)\[\begin{align} +\frac{dy}{dx} &= f(x) g(y) = \frac{h(x)}{j(y)} \\ +\int \frac{1}{g(y)} dy &= \int f(x) dx +\end{align}\]
+

For example, consider this problem:

+
+(1.5)\[\begin{equation} +y^{\prime} = \frac{dy}{dx} = 1 + y^2 \\ +\end{equation}\]
+

We can separate this into a problem that looks like \(f(y) dy = g(x) dx\), where \(dy = \frac{1}{1+y^2}\) and \(g(x) = 1\).

+
+(1.6)\[\begin{align} +\int \frac{dy}{1 + y^2} &= \int dx \\ +\arctan y &= x + c \\ +y(x) &= \tan(x+c) +\end{align}\]
+

Unfortunately, not every separable ODE can be integrated:

+
+(1.7)\[\begin{align} +\frac{dy}{dx} &= \frac{e^x / 2 + 5}{y^2 + \cos y} \\ +(y^2 + \cos y) dy &= (e^x / 2 + 5) dx +\end{align}\]
+
+
+

1.3. 3. General solution to linear 1st-order ODEs

+

Given a general linear 1st-order ODE of the form

+
+(1.8)\[\begin{equation} +\frac{dy}{dx} + p(x) y = q(x) +\end{equation}\]
+

we can solve by integration factor:

+
+(1.9)\[\begin{equation} +y(x) = e^{-\int p(x) dx} \left[ \int e^{\int p(x) dx} q(x) dx + C \right] +\end{equation}\]
+

For example, in this equation

+
+(1.10)\[\begin{equation} +y^{\prime} + xy - 5 e^x = 0 +\end{equation}\]
+

after rearranging to the standard form

+
+(1.11)\[\begin{equation} +y^{\prime} + xy = 5 e^x +\end{equation}\]
+

we see that \(p(x) = x\) and \(q(x) = 5e^x\).

+
+
+

1.4. 4. Solution to nonlinear 1st-order ODEs

+

Given a general nonlinear 1st-order ODE

+
+(1.12)\[\begin{equation} +\frac{dy}{dx} + p(x) y = q(x) y^a +\end{equation}\]
+

where \(a \neq 1\) and \(a\) is a constant. This is known as the Bernoulli equation.

+

We can solve by transforming to a linear equation, by changing the dependent variable from \(y\) to \(z\):

+
+(1.13)\[\begin{align} +\text{let} \quad z &= y^{1-a} \\ +\frac{dz}{dx} &= (1-a) y^{-a} \frac{dy}{dx} +\end{align}\]
+

Multiply the original equation by \((1-a) y^{-a}\):

+
+(1.14)\[\begin{align} +(1-a) y^{-a} \frac{dy}{dx} + (1-a) y^{-a} p(x) y &= (1-a) y^{-a} q(x) y^a \\ +\frac{dz}{dx} + p(x) (1-a) z &= q(x) (1-a) \;, +\end{align}\]
+

which is now a linear first-order ODE, that looks like

+
+(1.15)\[\begin{equation} +\frac{dz}{dx} + p(x)^{\prime} z = q(x)^{\prime} +\end{equation}\]
+

where \(p(x)^{\prime} = (1-a) p(x)\) and \(q(x)^{\prime} = (1-a)q(x)\).

+

We can solve this using the integrating-factor approach discussed above. Then, once we have \(z(x)\), we can find \(y(x)\):

+
+(1.16)\[\begin{align} +z &= y^{1-a} \\ +y &= z^{\frac{1}{1-a}} +\end{align}\]
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/installing-jupyter.html b/docs/content/installing-jupyter.html new file mode 100644 index 0000000..ba30acd --- /dev/null +++ b/docs/content/installing-jupyter.html @@ -0,0 +1,512 @@ + + + + + + + + Installing Jupyter for Matlab — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+ +
+ + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ Contents +
+ +
+
+
+
+
+ +
+ +
+

Installing Jupyter for Matlab

+

This guide assumes you have Matlab already installed. I’ll be assuming you have Matlab R2019_b installed.

+

To use Jupyter notebooks with Matlab, you need to install Jupyter (which relies on Python) and the Matlab engine for Python.

+
+

macOS/Linux

+
    +
  1. I recommend you install Anaconda to manage your Python environment—it makes installing and managing packages very easy. This comes with the Jupyter notebook.

  2. +
  3. Create and activate an environment called jmatlab for Jupyter with Python 3.7:

  4. +
+
$ conda create -vv -n jmatlab python=3.7 jupyter
+$ conda activate jmatlab
+
+
+

Before the second command, you may need to tell your shell (e.g., bash, zsh) about conda, by doing conda init zsh for example. If you get an error with the conda activate command, your terminal should tell you to do this.

+

You should activate this environment whenever you want to run Jupyter.

+
    +
  1. Install the Matlab kernel for Jupyter:

  2. +
+
$ pip install matlab_kernel
+
+
+
    +
  1. Install the Python engine for Matlab:

  2. +
+
$ cd /Applications/MATLAB_R2019b.app/extern/engines/python
+$ python setup.py install
+
+
+

If you are using a different version of Matlab, then that path will be different.

+
    +
  1. Run Jupyter notebook:

  2. +
+
$ jupyter notebook
+
+
+

and create a new Matlab notebook with “New” then “Matlab” under “Notebook:”.

+
+
+

Windows

+
    +
  1. Install Anaconda for Windows

  2. +
  3. Open the Anaconda Prompt

  4. +
+

(to be continued)

+
+
+

Try it out

+

Now try it out:

+
+
+
disp('This is running Matlab!')
+
+
+
+
+
This is running Matlab!
+
+
+
+
+
+
+ + + + +
+ + + + + + +
+ + +
+ +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/intro.html b/docs/content/intro.html new file mode 100644 index 0000000..fed0121 --- /dev/null +++ b/docs/content/intro.html @@ -0,0 +1,431 @@ + + + + + + + + Introduction — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+ +
+ + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
+
+ +
+ +
+

Introduction

+

This website is an interactive Jupyter Book for ME 373, Mechanical Engineering Methods, taught at Oregon State University.

+
+
+
+
+
+
+
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/numerical-methods/error.html b/docs/content/numerical-methods/error.html new file mode 100644 index 0000000..594fd00 --- /dev/null +++ b/docs/content/numerical-methods/error.html @@ -0,0 +1,474 @@ + + + + + + + + 2.2. Error — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+ +
+ + + + + + + + + + + + + + + + +
+ + +
+ +
+ Contents +
+ +
+
+
+
+
+ +
+ +
+

2.2. Error

+

Applying the trapezoidal rule and Simpson’s rule introduces the concept of error in numerical solutions.

+

In our work so far, we have come across two obvious kinds of error, that we’ll come back to later:

+
    +
  • local truncation error, which represents how “wrong” each interval/step is compared with the exact solution; and

  • +
  • global truncation error, which is the sum of the truncation errors over the entire method.

  • +
+

In any numerical solution, there are five main sources of error:

+
    +
  1. Error in input data: this comes from measurements, and can be systematic (for example, due to uncertainty in measurement devices) or random.

  2. +
  3. Rounding errors: loss of significant digits. This comes from the fact that computers cannot represent real numbers exactly, and instead use a floating-point representation.

  4. +
  5. Truncation error: due to an infinite process being broken off. For example, an infinite series or sum ending after a finite number of terms, or discretization error by using a finite step size to approximate a continuous function.

  6. +
  7. Error due to simplifications in a mathematical model: “All models are wrong, but some are useful” (George E.P. Box) All models make some idealizations, or simplifying assumptions, which introduce some error with respect to reality. For example, we may assume gases are continuous, that a spring has zero mass, or that a process is frictionless.

  8. +
  9. Human error and machine error: there are many potential sources of error in any code. These can come from typos, human programming errors, errors in input data, or (more rarely) a pure machine error. Even textbooks, tables, and formulas may have errors.

  10. +
+
+

2.2.1. Absolute and relative error

+

We can also differentiate between absolute and relative error in a quantity. If \(y\) is an exact value and \(\tilde{y}\) is an approximation to that value, then we have

+
    +
  • absolute error: \(\Delta y = | \tilde{y} - y |\)

  • +
  • relative error: \(\frac{\Delta y}{y} = \left| \frac{\tilde{y} - y}{y} \right|\)

  • +
+

If \(y\) is a vector, then we can define error using the maximum of the elements:

+
+(2.4)\[\begin{equation} +\max_i \frac{ |y_i - \tilde{y}_{i} |}{|y_i|} +\end{equation}\]
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/numerical-methods/initial-value-methods.html b/docs/content/numerical-methods/initial-value-methods.html new file mode 100644 index 0000000..b00db69 --- /dev/null +++ b/docs/content/numerical-methods/initial-value-methods.html @@ -0,0 +1,808 @@ + + + + + + + + 2.3. Numerical Solutions of 1st-order ODEs — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+ +
+ + + + + + + + + + + + + + + + + + +
+ + + +
+
+
+
+ +
+ +
+

2.3. Numerical Solutions of 1st-order ODEs

+

For numerically solving definite integrals (\(\int_a^b f(x) dx\)) we have methods like the trapezoidal rule and Simpson’s rule. When we need to solve 1st-order ODEs of the form

+
+(2.5)\[\begin{equation} +y^{\prime} = \frac{dy}{dx} = f(x, y) +\end{equation}\]
+

for \(y(x)\), we need other methods. All of them will work by starting at the initial conditions, and then using information provided by the ODE to march forward in the solution, based on an increment (i.e., step size) \(\Delta x\).

+

For example, let’s say we want to solve

+
+(2.6)\[\begin{equation} +\frac{dy}{dx} = 4 x - \frac{2 y}{x} \;, \quad y(1) = 1 +\end{equation}\]
+

This problem is fairly simple, and we can find the general and particular solutions to compare our numerical results against:

+
+(2.7)\[\begin{align} +\text{general: } y(x) &= x^2 + \frac{x}{x^2} \\ +\text{particular: } y(x) &= x^2 +\end{align}\]
+
+

2.3.1. Forward Euler method

+

Recall that the derivative, \(y^{\prime}\), is the same as the slope. At the starting point, \((x,y) = (1,1)\), where \(y^{\prime} = 2\), this looks like:

+
+
+
format compact
+%plot inline
+
+x = linspace(1, 3);
+y = x.^2;
+plot(x, y); hold on
+plot([1, 2], [1, 3], '--')
+legend(['Solution'], ['Slope at start'])
+hold off
+
+
+
+
+../../_images/initial-value-methods_1_0.png +
+
+

Remember that the slope, or derivative, is

+
+(2.8)\[\begin{equation} +\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} +\end{equation}\]
+

Let’s consider the initial condition—the starting point—as \((x_i, y_i)\), and the next point in our numerical solution is \((x_{i+1}, y_{i+1})\), where \(i\) represents an index starting at 1 and ending at the number of steps \(N\). Our step size is then \(\Delta x = x_{i+1} - x_i\).

+

Based on our (very simple) approximation to the first derivative based on slope, we can relate the derivative to our two points:

+
+(2.9)\[\begin{equation} +\left(\frac{dy}{dx}\right)_{i} = \frac{y_{i+1} - y_i}{x_{i+1} - x_i} = \frac{y_{i+1} - y_i}{\Delta x} +\end{equation}\]
+

Then, solve this for our unknown:

+
+(2.10)\[\begin{equation} +y_{i+1} = y_i + \left(\frac{dy}{dx}\right)_i \Delta x +\end{equation}\]
+

This is the Forward Euler method.

+

Based on a given step size \(\Delta x\), we’ll use this formula (called a recursion formula) to march forward and obtain the full solution over given \(x\) domain. That will look something like this:

+
+
+
clear x y
+x_exact = linspace(1, 3);
+y_exact = x_exact.^2;
+plot(x_exact, y_exact); hold on
+
+% our derivative function, dy/dx
+f = @(x,y) 4*x - (2*y)/x;
+
+dx = 0.5;
+x = 1 : dx : 3;
+y(1) = 1;
+for i = 1 : length(x)-1
+    y(i+1) = y(i) + f(x(i), y(i))*dx;
+end
+plot(x, y, 'o--', 'MarkerFaceColor', 'r')
+
+legend(['Exact solution'], ['Numerical solution'], 'Location','northwest')
+hold off
+
+
+
+
+../../_images/initial-value-methods_3_0.png +
+
+

Another way to obtain the recursion formula for the Forward Euler method is to use a Taylor series expansion. +Recall that for well-behaved functions, the Taylor series expansion says

+
+(2.11)\[\begin{equation} +y(x + \Delta x) = y(x) + \Delta x y^{\prime}(x) + \frac{1}{2} \Delta x^2 y^{\prime\prime}(x) + \frac{1}{3!} \Delta x^3 y^{\prime\prime\prime}(x) \dots \;. +\end{equation}\]
+

This is exact for an infinite series. We can apply this formula to our (unknown) solution \(y_i\) and cut off the terms of order \(\Delta x^2\) and higher; the derivative \(y^{\prime}\) is given by our original ODE. +This gives us the same recursion formula as above:

+
+(2.12)\[\begin{equation} +\therefore y_{i+1} \approx y_i + \left( \frac{dy}{dx}\right)_i \Delta x +\end{equation}\]
+

where we can now see that we are introducing some error on the order of \(\Delta x^2\) at each step. This is the local truncation error. The global error is the accumulation of error over all the steps, and is on the order of \(\Delta x\). Thus, the Forward Euler method is a first-order method, because its global error is on the order of the step size to the first power: error \(\sim \mathcal{O}(\Delta x)\).

+

Forward Euler is also an explicit method, because its recursion formula is explicity defined for \(y_{i+1}\). (You’ll see when that may not be the case soon.)

+

In general, for an \(n\)th-order method:

+
+(2.13)\[\begin{align} +\text{local error } &\sim \mathcal{O}(\Delta x^{n+1}) \\ +\text{global error } &\sim \mathcal{O}(\Delta x^{n}) +\end{align}\]
+

(This only applies for \(\Delta x < 1\); in cases where you have a \(\Delta x > 1\), you should nondimensionalize the problem based on the domain size such that \(0 \leq x \leq 1\).)

+

Applying the Forward Euler method then requires:

+
    +
  1. Have a given first-order ODE: \(\frac{dy}{dx} = y^{\prime} = f(x,y)\). Complex and/or nonlinear problems are fine!

  2. +
  3. Specify the step size \(\Delta x\) (or \(\Delta t\)).

  4. +
  5. Specify the domain over which to integrate: \(x_1 \leq x \leq x_n\)

  6. +
  7. Specify the initial condition: \(y(x=x_1) = y_1\)

  8. +
+

Let’s do another example:

+
+(2.14)\[\begin{equation} +y^{\prime} = 8 e^{-x}(1+x) - 2y +\end{equation}\]
+

with the initial condition \(y(0) = 1\), and the domain \(0 \leq x \leq 7\). This is a linear 1st-order ODE that we can find the analytical solution for comparison:

+
+(2.15)\[\begin{equation} +y(x) = e^{-2x} (8 x e^x + 1) +\end{equation}\]
+

To solve, we’ll create an anonymous function for the derivative and then incorporate that into our Forward Euler code. We’ll start with \(\Delta x = 0.2\).

+
+
+
clear
+
+f = @(x,y) 8*exp(-x)*(1 + x) - 2*y;
+
+dx = 0.2;
+x = 0 : dx : 7;
+n = length(x);
+y(1) = 1;
+
+% Forward Euler loop
+for i = 1 : n - 1
+    y(i+1) = y(i) + dx*f(x(i), y(i));
+end
+
+x_exact = linspace(0, 7);
+y_exact = exp(-2.*x_exact).*(8*x_exact.*exp(x_exact) + 1);
+plot(x_exact, y_exact); hold on
+plot(x, y, 'o--')
+legend('Exact solution', 'Forward Euler solution')
+
+
+
+
+../../_images/initial-value-methods_6_0.png +
+
+

Notice the visible error in that plot, which is between 0.2–0.25, or in other words \(\mathcal{O}(\Delta x)\).

+

How can we reduce the error? Just like with the trapezoidal rule, we have two main options:

+
    +
  • Reduce the step size \(\Delta x\)

  • +
  • Choose a higher-order (i.e., more accurate) method

  • +
+

The downside to reducing \(\Delta x\) is the increased number of steps we then have to take, which may make the solution too computationally expensive. A more-accurate method would have less error per step, which might allow us to use the same \(\Delta x\) but get a better solution. Let’s next consider some better methods.

+
+
+

2.3.2. Heun’s method

+

Heun’s method is a predictor-corrector method; these work by predicting a solution at some intermediate location and then using that information to get a better overall answer at the next location (correcting). Heun’s uses the Forward Euler method to predict the solution at \(x_{i+1}\), then uses the average of the slopes at \(y_i\) and the predicted \(y_{i+1}\) to get a better overall answer for \(y_{i+1}\).

+
+(2.16)\[\begin{align} +\text{predictor: } y_{i+1}^p &= y_i + \Delta x f(x_i, y_i) \\ +\text{corrector: } y_{i+1} &= y_i + \frac{\Delta x}{2} \left( f(x_i, y_i) + f(x_{i+1}, y_{i+1}^p) \right) +\end{align}\]
+

Heun’s method is second-order accurate, meaning the global error is \(\mathcal{O}(\Delta x^2)\) and explicit.

+

Let’s see this method in action:

+
+
+
clear
+
+f = @(x,y) 8*exp(-x)*(1 + x) - 2*y;
+
+dx = 0.2;
+x = 0 : dx : 7;
+n = length(x);
+y(1) = 1;
+
+% Heun's method loop
+for i = 1 : n - 1
+    y_p = y(i) + dx*f(x(i), y(i));
+    y(i+1) = y(i) + (dx/2)*(f(x(i), y(i)) + f(x(i+1), y_p));
+end
+
+x_exact = linspace(0, 7);
+y_exact = exp(-2.*x_exact).*(8*x_exact.*exp(x_exact) + 1);
+plot(x_exact, y_exact); hold on
+plot(x, y, 'o--')
+legend('Exact solution', "Heun's method solution")
+fprintf('Maximum error: %5.3f', abs(max(y_exact) - max(y)))
+
+
+
+
+
Maximum error: 0.055
+
+
+../../_images/initial-value-methods_8_1.png +
+
+

Notice how the error is visibly smaller than for the Forward Euler method–the maximum error is around 0.05, which is very close to \(\Delta x^2 = 0.04\).

+
+
+

2.3.3. Midpoint method

+

The midpoint method, also known as the modified Euler method, is another predictor-corrector method, that instead predicts the solution at the midpoint (\(x + \Delta x/2\)):

+
+(2.17)\[\begin{align} +y_{i + \frac{1}{2}} &= y_i + \frac{\Delta x}{2} f(x_i, y_i) \\ +y_{i+1} &= y_i + \Delta x f \left( x_{i+\frac{1}{2}} , y_{i + \frac{1}{2}} \right) +\end{align}\]
+

Like Heun’s method, the midpoint method is explicit and second-order accurate:

+
+
+
clear
+
+f = @(x,y) 8*exp(-x)*(1 + x) - 2*y;
+
+dx = 0.2;
+x = 0 : dx : 7;
+n = length(x);
+y(1) = 1;
+
+% midpoint method loop
+for i = 1 : n - 1
+    y_half = y(i) + (dx/2)*f(x(i), y(i));
+    y(i+1) = y(i) + dx * f(x(i) + dx/2, y_half);
+end
+
+x_exact = linspace(0, 7);
+y_exact = exp(-2.*x_exact).*(8*x_exact.*exp(x_exact) + 1);
+plot(x_exact, y_exact); hold on
+plot(x, y, 'o--')
+legend('Exact solution', "Midpoint method solution")
+
+fprintf('Maximum error: %5.3f', abs(max(y_exact) - max(y)))
+
+
+
+
+
Maximum error: 0.050
+
+
+../../_images/initial-value-methods_10_1.png +
+
+
+
+

2.3.4. Fourth-order Runge–Kutta method

+

Runge–Kutta methods are a family of methods that use one or more stages; the methods we have discussed so far (Forward Euler, Heun’s, and midpoint) actually all fall in this family. There is also a popular fourth-order method: the fourth-order Runge–Kutta method (RK4). This uses four stages to get a more accurate solution:

+
+(2.18)\[\begin{align} +y_{i+1} &= y_i + \frac{\Delta x}{6} (k_1 + 2 k_2 + 2 k_3 + k_4) \\ +k_1 &= f(x_i, y_i) \\ +k_2 &= f \left( x_i + \frac{\Delta x}{2}, y_i + \frac{\Delta x}{2} k_1 \right) \\ +k_3 &= f \left( x_i + \frac{\Delta x}{2}, y_i + \frac{\Delta x}{2} k_2 \right) \\ +k_4 &= f \left( x_i + \Delta x, y_i + \Delta x \, k_3 \right) +\end{align}\]
+

This method is explicit and fourth-order accurate: error \(\sim \mathcal{O}(\Delta x^4)\):

+
+
+
clear
+
+f = @(x,y) 8*exp(-x)*(1 + x) - 2*y;
+
+dx = 0.2;
+x = 0 : dx : 7;
+n = length(x);
+y(1) = 1;
+
+% 4th-order Runge-Kutta method loop
+for i = 1 : n - 1
+    k1 = f(x(i), y(i));
+    k2 = f(x(i) + dx/2, y(i) + dx*k1/2);
+    k3 = f(x(i) + dx/2, y(i) + dx*k2/2);
+    k4 = f(x(i) + dx, y(i) + dx*k3);
+    y(i+1) = y(i) + (dx/6) * (k1 + 2*k2 + 2*k3 + k4);
+end
+
+x_exact = linspace(0, 7);
+y_exact = @(x) exp(-2.*x).*(8*x.*exp(x) + 1);
+plot(x_exact, y_exact(x_exact)); hold on
+plot(x, y, 'o--')
+legend('Exact solution', "RK4 solution")
+
+fprintf('Maximum error: %6.4f', max(abs(y_exact(x) - y)))
+
+
+
+
+
Maximum error: 0.0004
+
+
+../../_images/initial-value-methods_12_1.png +
+
+

The maximum error (0.0004) is actually a bit smaller than \(\Delta x^4 = 0.0016\), but approximately the same order of magnitude.

+

Matlab also offers a built-in RK4 integrator: ode45. (It is actually slightly more complicated than the equations shown just now, because it automatically adjusts the step size \(\Delta x\) to control error.) You can call this function with the syntax:

+
[X, Y] = ode45(function_name, [x_start x_end], [IC]);
+
+
+

where function_name is the name of a function that provides the derivative (this can be a regular function given in a file, or an anonymous function); [x_start x_end] provides the domain of integration (\(x_{\text{start}} \leq x \leq x_{\text{end}}\)), and [IC] provides the initial condition \(y(x=x_{\text{start}})\).

+

For example, let’s use this and compare with our exact solution:

+
+
+
clear
+
+f = @(x,y) 8*exp(-x)*(1 + x) - 2*y;
+
+[X, Y] = ode45(f, [0 7], [1]);
+
+x_exact = linspace(0, 7);
+y_exact = @(x) exp(-2.*x).*(8*x.*exp(x) + 1);
+plot(x_exact, y_exact(x_exact)); hold on
+plot(X, Y, 'o--')
+legend('Exact solution', "ode45 solution")
+
+fprintf('Maximum error: %6.4f', max(abs(y_exact(X) - Y)))
+
+
+
+
+
Maximum error: 0.0007
+
+
+../../_images/initial-value-methods_14_1.png +
+
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/numerical-methods/integrals.html b/docs/content/numerical-methods/integrals.html new file mode 100644 index 0000000..d79300f --- /dev/null +++ b/docs/content/numerical-methods/integrals.html @@ -0,0 +1,655 @@ + + + + + + + + 2.1. Numerical integrals — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+ +
+ + + + + + + + + + + + + + + + + + +
+ + + +
+
+
+
+ +
+ +
+

2.1. Numerical integrals

+

What about when we cannot integrate a function analytically? In other words, when there is no (obvious) closed-form solution. In these cases, we can use numerical methods to solve the problem.

+

Let’s use this problem:

+
+(2.1)\[\begin{align} +\frac{dy}{dx} &= e^{-x^2} \\ +y(x) &= \int e^{-x^2} dx + C +\end{align}\]
+

(You may recognize this as leading to the error function, \(\text{erf}\): +\(\frac{1}{2} \sqrt{\pi} \text{erf}(x) + C\), +so the exact solution to the integral over the range \([0,1]\) is 0.7468.)

+
+
+
x = linspace(0, 1);
+f = @(x) exp(-x.^2);
+plot(x, f(x))
+axis([0 1 0 1])
+
+
+
+
+../../_images/integrals_1_0.png +
+
+
+

2.1.1. Numerical integration: Trapezoidal rule

+

In such cases, we can find the integral by using the trapezoidal rule, which finds the area under the curve by creating trapezoids and summing their areas:

+
+(2.2)\[\begin{equation} +\text{area under curve} = \sum \left( \frac{f(x_{i+1}) + f(x_i)}{2} \right) \Delta x +\end{equation}\]
+

Let’s see what this looks like with four trapezoids (\(\Delta x = 0.25\)):

+
+
+
hold off
+x = linspace(0, 1);
+plot(x, f(x)); hold on
+axis([0 1 0 1])
+
+x = 0 : 0.25 : 1;
+
+% plot the trapezoids
+for i = 1 : length(x)-1
+    xline = [x(i), x(i)];
+    yline = [0, f(x(i))];
+    line(xline, yline, 'Color','red','LineStyle','--')
+    xline = [x(i+1), x(i+1)];
+    yline = [0, f(x(i+1))];
+    line(xline, yline, 'Color','red','LineStyle','--')
+    xline = [x(i), x(i+1)];
+    yline = [f(x(i)), f(x(i+1))];
+    line(xline, yline, 'Color','red','LineStyle','--')
+end
+hold off
+
+
+
+
+../../_images/integrals_3_0.png +
+
+

Now, let’s integrate using the trapezoid formula given above:

+
+
+
dx = 0.1;
+x = 0.0 : dx : 1.0;
+
+area = 0.0;
+for i = 1 : length(x)-1
+    area = area + (dx/2)*(f(x(i)) + f(x(i+1)));
+end
+
+fprintf('Numerical integral: %f\n', area)
+exact = 0.5*sqrt(pi)*erf(1);
+fprintf('Exact integral: %f\n', exact)
+fprintf('Error: %f %%\n', 100.*abs(exact-area)/exact)
+
+
+
+
+
Numerical integral: 0.746211
+Exact integral: 0.746824
+Error: 0.082126 %
+
+
+
+
+

We can see that using the trapezoidal rule, a numerical integration method, with an internal size of \(\Delta x = 0.1\) leads to an approximation of the exact integral with an error of 0.08%.

+

You can make the trapezoidal rule more accurate by:

+
    +
  • using more segments (that is, a smaller value of \(\Delta x\), or

  • +
  • using higher-order polynomials (such as with Simpson’s rules) over the simpler trapezoids.

  • +
+

First, how does reducing the segment size (step size) by a factor of 10 affect the error?

+
+
+
dx = 0.01;
+x = 0.0 : dx : 1.0;
+
+area = 0.0;
+for i = 1 : length(x)-1
+    area = area + (dx/2)*(f(x(i)) + f(x(i+1)));
+end
+
+fprintf('Numerical integral: %f\n', area)
+exact = 0.5*sqrt(pi)*erf(1);
+fprintf('Exact integral: %f\n', exact)
+fprintf('Error: %f %%\n', 100.*abs(exact-area)/exact)
+
+
+
+
+
Numerical integral: 0.746818
+Exact integral: 0.746824
+Error: 0.000821 %
+
+
+
+
+

So, reducing our step size by a factor of 10 (using 100 segments instead of 10) reduced our error by a factor of 100!

+
+
+

2.1.2. Numerical integration: Simpson’s rule

+

We can increase the accuracy of our numerical integration approach by using a more sophisticated interpolation scheme with each segment. In other words, instead of using a straight line, we can use a polynomial. Simpson’s rule, also known as Simpson’s 1/3 rule, refers to using a quadratic polynomial to approximate the line in each segment.

+

Simpson’s rule defines the definite integral for our function \(f(x)\) from point \(a\) to point \(b\) as

+
+(2.3)\[\begin{equation} +\int_a^b f(x) \approx \frac{1}{6} \Delta x \left( f(a) + 4 f \left(\frac{a+b}{2}\right) + f(b) \right) +\end{equation}\]
+

where \(\Delta x = b - a\).

+

That equation comes from interpolating between points \(a\) and \(b\) with a third-degree polynomial, then integrating by parts.

+
+
+
hold off
+x = linspace(0, 1);
+plot(x, f(x)); hold on
+axis([-0.1 1.1 0.2 1.1])
+
+plot([0 1], [f(0) f(1)], 'Color','black','LineStyle',':');
+
+% quadratic polynomial
+a = 0; b = 1; m = (b-a)/2;
+p = @(z) (f(a).*(z-m).*(z-b)/((a-m)*(a-b))+f(m).*(z-a).*(z-b)/((m-a)*(m-b))+f(b).*(z-a).*(z-m)/((b-a)*(b-m)));
+plot(x, p(x), 'Color','red','LineStyle','--');
+
+xp = [0 0.5 1];
+yp = [f(0) f(m) f(1)];
+plot(xp, yp, 'ok')
+hold off
+legend('exact', 'trapezoid fit', 'polynomial fit', 'points used')
+
+
+
+
+../../_images/integrals_10_0.png +
+
+

We can see that the polynomial fit, used by Simpson’s rule, does a better job of of approximating the exact function, and as a result Simpson’s rule will be more accurate than the trapezoidal rule.

+

Next let’s apply Simpson’s rule to perform the same integration as above:

+
+
+
dx = 0.1;
+x = 0.0 : dx : 1.0;
+
+area = 0.0;
+for i = 1 : length(x)-1
+    area = area + (dx/6.)*(f(x(i)) + 4*f((x(i)+x(i+1))/2.) + f(x(i+1)));
+end
+
+fprintf('Simpson rule integral: %f\n', area)
+exact = 0.5*sqrt(pi)*erf(1);
+fprintf('Exact integral: %f\n', exact)
+fprintf('Error: %f %%\n', 100.*abs(exact-area)/exact)
+
+
+
+
+
Simpson rule integral: 0.746824
+Exact integral: 0.746824
+Error: 0.000007 %
+
+
+
+
+

Simpson’s rule is about three orders of magnitude (~1000x) more accurate than the trapezoidal rule.

+

In this case, using a more-accurate method allows us to significantly reduce the error while still using the same number of segments/steps.

+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/numerical-methods/numerical-methods.html b/docs/content/numerical-methods/numerical-methods.html new file mode 100644 index 0000000..56b26a0 --- /dev/null +++ b/docs/content/numerical-methods/numerical-methods.html @@ -0,0 +1,435 @@ + + + + + + + + 2. Numerical Methods — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+ +
+ + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
+
+ +
+ +
+

2. Numerical Methods

+

This chapter describes numerical methods used to solve integrals and first-order ordinary differential equations, along with concepts related to these such as error and stability.

+
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/numerical-methods/stability.html b/docs/content/numerical-methods/stability.html new file mode 100644 index 0000000..aa2a5b3 --- /dev/null +++ b/docs/content/numerical-methods/stability.html @@ -0,0 +1,785 @@ + + + + + + + + 2.4. Stability and Stiffness — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+ +
+ + + + + + + + + + + + + + + + + + +
+ + + +
+
+
+
+ +
+ +
+

2.4. Stability and Stiffness

+

In the past when you’ve talked about stability, it has likely been regarding the stability of a system. Stable systems are those will well-behaved exact solutions, meaning they do not grow unbounded. +In engineering we mostly focus (or want!) stable systems, although there are some interesting unstable systems such as those involving resonance, nonlinear dynamics, or chaos—generally we want to know when that happens so we can prevent it.

+

We can also define the stability of a numerical scheme, which is when the numerical solution exhibits unphysical behavior. In other words, it blows up.

+

For example, let’s consider the relatively simple 1st-order ODE

+
+(2.19)\[\begin{equation} +\frac{dy}{dt} = -3 y +\end{equation}\]
+

with the initial condition \(y(0) = 1\). As we will see, this ODE can cause explicit numerical schemes to become unstable, and thus it is a stiff ODE. (Note that we can easily obtain the exact solution for this problem, which is \(y(t) = e^{-3 t}\).)

+

Let’s try solving this with the Forward Euler method, integrating over \(0 \leq t \leq 10\), for a range of time-step size values: \(\Delta t = 0.1, 0.25, 0.5, 0.75\):

+
+
+
clear
+f = @(t,y) -3*y;
+
+dt = 0.1;
+t = 0 : dt : 20;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) + dt*f(t(i), y(i));
+end
+subplot(4,1,1);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+dt = 0.25;
+t = 0 : dt : 20;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) + dt*f(t(i), y(i));
+end
+subplot(4,1,2);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+dt = 0.5;
+t = 0 : dt : 20;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) + dt*f(t(i), y(i));
+end
+subplot(4,1,3);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+dt = 0.75;
+t = 0 : dt : 20;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) + dt*f(t(i), y(i));
+end
+subplot(4,1,4);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+
+
+
+../../_images/stability_1_0.png +
+
+

At the smaller step sizes, \(\Delta t = 0.1\) and \(\Delta t = 0.25\), we see that the solution is well-behaved. But, when we increase \(\Delta t\) to 0.5, we see some instability that goes away with time. Then, when we increase \(\Delta t\) to 0.75, the solution eventually blows up, leading to error much larger than what we should expect based on the method’s order of accuracy (first) and the step size value.

+

Compare this behavior to that for the ODE

+
+(2.20)\[\begin{equation} +\frac{dy}{dt} = e^{-t} +\end{equation}\]
+

which is non-stiff:

+
+
+
clear
+f = @(t,y) exp(-t);
+
+dt = 0.1;
+t = 0 : dt : 10;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) + dt*f(t(i), y(i));
+end
+subplot(4,1,1);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+dt = 0.25;
+t = 0 : dt : 10;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) + dt*f(t(i), y(i));
+end
+subplot(4,1,2);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+dt = 0.5;
+t = 0 : dt : 10;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) + dt*f(t(i), y(i));
+end
+subplot(4,1,3);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+dt = 0.75;
+t = 0 : dt : 10;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) + dt*f(t(i), y(i));
+end
+subplot(4,1,4);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+
+
+
+../../_images/stability_3_0.png +
+
+

In this case, we see that the solution remains well-behaved even for larger time-step sizes, and the error matches the expected order based on the method and step-size value.

+

In general numerical schemes can be:

+
    +
  • unstable: the scheme blows up for any choice of parameters

  • +
  • conditionally stable: the scheme is stable for a particular choice of parameters (for example, \(\Delta t\) is less than some threshold

  • +
  • unconditionally stable: the scheme is always stable

  • +
+

Schemes may be stable for some problem/system and not for another, and vice versa.

+

Stability is related to robustness of a method, which is generally a tradeoff between complexity and computational cost. The choice of method and solution strategy depends on what you want, and how long you can wait for it. In general, we almost always want to use the largest \(\Delta t\) allowable.

+

Rather than reducing \(\Delta t\) to avoid stability problems, we can also use a method that is unconditionally stable, such as the Backward Euler method.

+
+

2.4.1. Backward Euler method

+

The Backward Euler method is very similar to the Forward Euler method, except in one way: it uses the slope at the next time step:

+
+(2.21)\[\begin{equation} + \left(\frac{dy}{dx}\right)_{i+1} \approx \frac{y_{i+1} - y_i}{\Delta x} +\end{equation}\]
+

Then, the resulting recursion formula is

+
+(2.22)\[\begin{equation} +y_{i+1} = y_i + \Delta x \left(\frac{dy}{dx}\right)_{i+1}, \text{ or} \\ +y_{i+1} = y_i + \Delta x \, f(x_{i+1}, y_{i+1}) +\end{equation}\]
+

where \(f(x,y) = dy/dx\).

+

Notice that this recursion formula cannot be directly solved, because \(y_{i+1}\) shows up on both sides. This is an implicit method, where all the other methods we have covered (Forward Euler, Heun’s, Midpoint, and 4th-order Runge-Kutta) are explicit. Implicit methods require more work to actually implement.

+
+

2.4.1.1. Backward Euler example

+

For example, consider the problem

+
+(2.23)\[\begin{equation} +\frac{dy}{dx} = f(x,y) = 8 e^{-x} (1+x) - 2y +\end{equation}\]
+

To actually solve this problem with the Backward Euler method, we need to incorporate the derivative function \(f(x,y)\) into the recursion formula and solve for \(y_{i+1}\):

+
+(2.24)\[\begin{align} +y_{i+1} &= y_i + \Delta x \, f(x_{i+1}, y_{i+1}) \\ +y_{i+1} &= y_i + \Delta x \left[ 8 e^{-x_{i+1}} (1 + x_{i+1}) - 2 y_{i+1} \right] \\ +y_{i+1} &= y_i + 8 e^{-x_{i+1}} (1 + x_{i+1}) \Delta x - 2 \Delta x \, y_{i+1} \\ +y_{i+1} + 2 \Delta x \, y_{i+1} &= y_i + 8 e^{-x_{i+1}} (1 + x_{i+1}) \Delta x \\ +y_{i+1} &= \frac{ y_i + 8 e^{-x_{i+1}} (1 + x_{i+1}) \Delta x }{ 1 + 2 \Delta x } +\end{align}\]
+

Now we have a useable recursion formula that we can use to solve this problem. Let’s use the initial condition \(y(0) = 1\), the domain \(0 \leq x \leq 7\), and \(\Delta x = 0.2\).

+
+
+
clear
+
+dx = 0.2;
+x = 0 : dx : 7;
+y = zeros(length(x), 1);
+y(1) = 1;
+
+% Backward Euler loop
+for i = 1 : length(x) - 1
+    y(i+1) = (y(i) + 8*exp(-x(i+1))*(1 + x(i+1))*dx) / (1 + 2*dx);
+end
+
+x_exact = linspace(0, 7);
+y_exact = exp(-2.*x_exact).*(8*x_exact.*exp(x_exact) + 1);
+plot(x_exact, y_exact); hold on
+plot(x, y, 'o--')
+legend('Exact solution', 'Backward Euler solution')
+
+
+
+
+../../_images/stability_6_0.png +
+
+

This matches nearly what we saw with the Forward Euler method before—Backward Euler is also a first-order method, so the global error should be proportional to \(\Delta x\).

+

Let’s now return to the stiff ODE \(y^{\prime} = -3 y\), and see how the Backward Euler method does. First, we need to obtain our useable recursion formula:

+
+(2.25)\[\begin{align} +y_{i+1} &= y_i + \Delta t \, f(t_{i+1}, y_{i+1}) \\ +y_{i+1} &= y_i + \Delta t \, \left( -3 y_{i+1} \right) \\ +y_{i+1} + 3 y_{i+1} \Delta t &= y_i \\ +y_{i+1} &= \frac{y_i}{1 + 3 \Delta t} +\end{align}\]
+
+
+
clear
+
+dt = 0.1;
+t = 0 : dt : 10;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) / (1 + 3*dt);
+end
+subplot(4,1,1);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+dt = 0.25;
+t = 0 : dt : 10;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) / (1 + 3*dt);
+end
+subplot(4,1,2);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+dt = 0.5;
+t = 0 : dt : 10;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) / (1 + 3*dt);
+end
+subplot(4,1,3);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+dt = 0.75;
+t = 0 : dt : 10;
+y = zeros(length(t), 1);
+y(1) = 1;
+for i = 1 : length(t) - 1
+    y(i+1) = y(i) / (1 + 3*dt);
+end
+subplot(4,1,4);
+plot(t, y); title(sprintf('dt = %4.2f', dt));
+
+
+
+
+../../_images/stability_8_0.png +
+
+

In this case, we see that the solution remains well-behaved for all the step sizes, not showing any of the instability we saw with the Forward Euler method. This is because the Backward Euler method is unconditionally stable.

+
+
+
+

2.4.2. Stability analysis

+
+

2.4.2.1. Stability analysis of Forward Euler

+

We can perform a stability analysis of the stiff problem to identify when the Forward Euler method becomes unstable. Let’s apply the method to the ODE at hand:

+
+(2.26)\[\begin{align} +\frac{dy}{dt} &= -3 y \\ +y_{i+1} &= y_i + \Delta t f(t_i, y_i) \\ +y_{i+1} &= y_i + \Delta t (-3 y_i) \\ + &= y_i (1 - 3 \Delta t) \\ +\frac{y_{i+1}}{y_i} &= \sigma = 1 - 3 \Delta t +\end{align}\]
+

where \(\sigma\) is the amplification factor. This defines whether the solution grows or decays each step—for a stable physical system, we expect the solution to get smaller or remain contant with each step.

+

Therefore, for the method to remain stable, we must have \(\sigma | \leq 1\). We can use this stability criterion to find conditions on \(\Delta t\) for stability:

+
+(2.27)\[\begin{gather} +| \sigma | = | 1 - 3 \Delta t | \leq 1 \\ +-1 \leq 1 - 3 \Delta t \leq 1 \\ +-1 \leq 1 - 3 \Delta t \quad \text{or} \quad 1 - 3 \Delta t \leq 1 \\ +\frac{-2}{3} \leq -\Delta t \quad \quad -\Delta t \leq 0 \\ +\rightarrow \Delta t \leq \frac{2}{3} \quad \text{and} \quad \Delta t \geq 0 \\ +\therefore 0 \leq \Delta t \leq \frac{2}{3} +\end{gather}\]
+

for stability. (For safety, we might use \(\Delta t < 1/2\) for safety, to stay away from the absolute stability limit.)

+

The Forward Euler method is then conditionally stable.

+

As a general rule of thumb, all explicit methods are conditionally stable; these are methods where the recursion formula for \(y_{i+1}\) can be written and calculated explicitly in terms of known quantities.

+
+
+

2.4.2.2. Stability analysis of Backward Euler

+

We can also perform a stability analysis on the Backward Euler method to show that its stability does not depend on the step size:

+
+(2.28)\[\begin{align} +\frac{dy}{dt} &= -3 y \\ +y_{i+1} &= y_i + \Delta t f(t_{i+1}, y_{i+1}) \\ +y_{i+1} &= y_i + \Delta t (-3 y_{i+1}) \\ +y_{i+1} (1 + 3 \Delta t) &= y_i \\ +\sigma &= \frac{y_{i+1}}{y_i} = \frac{1}{1 + 3 \Delta t} +\end{align}\]
+

For stability, we need \(| \sigma | \leq 1\):

+
+(2.29)\[\begin{align} +| \sigma | &= \left| \frac{1}{1 + 3 \Delta t} \right| \leq 1 \\ +\rightarrow \Delta t &> 0 +\end{align}\]
+

Therefore the Backward Euler method is unconditionally stable.

+
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/pdes/elliptic.html b/docs/content/pdes/elliptic.html new file mode 100644 index 0000000..385c9f9 --- /dev/null +++ b/docs/content/pdes/elliptic.html @@ -0,0 +1,1345 @@ + + + + + + + + 5.1. Elliptic PDEs — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ + +
+
+ +
+ +
+

5.1. Elliptic PDEs

+

The classic example of an elliptic PDE is Laplace’s equation (yep, the same Laplace that gave us the Laplace transform), which in two dimensions for a variable \(u(x,y)\) is

+
+(5.2)\[\begin{equation} +\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \nabla^2 u = 0 \;, +\end{equation}\]
+

where \(\nabla\) is del, or nabla, and represents the gradient operator: \(\nabla = \frac{\partial}{\partial x} + \frac{\partial}{\partial y}\).

+

Laplace’s equation shows up in a number of physical problems, including heat transfer, fluid dynamics, and electrostatics. For example, the heat equation for conduction in two dimensions is

+
+(5.3)\[\begin{equation} +\frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) \;, +\end{equation}\]
+

where \(u(x,y,t)\) is temperature and \(\alpha\) is thermal diffusivity. Steady-state heat transfer (meaning after any initial transient period) is then described by Laplace’s equation.

+

A related elliptic PDE is Poisson’s equation:

+
+(5.4)\[\begin{equation} +\nabla^2 u = f(x,y) \;, +\end{equation}\]
+

which also appears in multiple physical problems—most notably, when solving for pressure in the Navier–Stokes equations.

+

To numerically solve these equations, and any elliptic PDE, we can use finite differences, where we replace the continuous \(x,y\) domain with a discrete grid of points. This is similar to what we did with boundary-value problems in one dimension—but now we have two dimensions.

+

To approximate the second derivatives in Laplace’s equation, we can use central differences in both the \(x\) and \(y\) directions, applied around the \(u_{i,j}\) point:

+
+(5.5)\[\begin{align} +\frac{\partial^2 u}{\partial x^2} &\approx \frac{u_{i-1,j} - 2u_{i,j} + u_{i+1,j}}{\Delta x^2} \\ +\frac{\partial^2 u}{\partial y^2} &\approx \frac{u_{i,j-1} - 2u_{i,j} + u_{i,j+1}}{\Delta y^2} +\end{align}\]
+

where \(i\) is the index used in the \(x\) direction, \(j\) is the index in the \(y\) direction, and \(\Delta x\) and \(\Delta y\) are the step sizes in the \(x\) and \(y\) directions. +In other words, \(x_i = (i-1) \Delta x\) and \(y_j = (j-1) \Delta y\).

+

The following figure shows the points necessary to approximate the partial derivatives in the PDE at a location \((x_i, y_j)\), for a general 2D region. This is known as a five-point stencil:

+
+five-point stencil +

Fig. 5.1 Five-point finite difference stencil

+
+

Applying these finite differences gives us an approximation for Laplace’s equation:

+
+(5.6)\[\begin{equation} +\frac{u_{i-1,j} - 2u_{i,j} + u_{i+1,j}}{\Delta x^2} + \frac{u_{i,j-1} - 2u_{i,j} + u_{i,j+1}}{\Delta y^2} = 0 \;. +\end{equation}\]
+

If we use a uniform grid where \(\Delta x = \Delta y = h\), then we can simplify to

+
+(5.7)\[\begin{equation} +u_{i+1,j} + u_{i,j+1} + u_{i-1,j} + u_{i,j-1} - 4 u_{i,j} = 0 \;. +\end{equation}\]
+
+

5.1.1. Example: heat transfer in a square plate

+

As an example, let’s consider the problem of steady-state heat transfer in a square solid object. If \(u(x,y)\) is temperature, then this is described by Laplace’s equation:

+
+(5.8)\[\begin{equation} +\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \nabla^2 u = 0 \;, +\end{equation}\]
+

and we can solve this using finite differences. Using a uniform grid where \(\Delta x = \Delta y = h\), Laplace’s equation gives us a recursion formula that relates the values at neighboring points:

+
+(5.9)\[\begin{equation} +u_{i+1,j} + u_{i,j+1} + u_{i-1,j} + u_{i,j-1} - 4 u_{i,j} = 0 \;. +\end{equation}\]
+

Consider a case where the square has sides of length \(L\), and the boundary conditions are that the temperature is fixed at 100 on the left, right, and bottom sides, and fixed at 0 on the top. +For now, we’ll use two segments to discretize the domain in each directions, giving us nine total points in the grid. +The following figures show the example problem, and the grid of points we’ll use.

+
+Heat transfer in a square +

Fig. 5.2 Heat transfer in a square object

+
+
+3x3 grid of points +

Fig. 5.3 Simple 3x3 grid of points

+
+

Using the above recursion formula, we can write an equation for each of the nine unknown points (in the interior, not the boundary points):

+
+(5.10)\[\begin{align} +u_{1,1} &= 100 \\ +u_{2,1} &= 100 \\ +u_{3,1} &= 100 \\ +u_{1,2} &= 100 \\ +\text{for } u_{2,2}: \quad u_{3,2} + u_{2,3} + u_{1,2} + u_{2,1} - 4u_{2,2} &= 0 \\ +u_{3,2} &= 100 \\ +u_{1,3} &= 100 \\ +u_{2,3} &= 0 \\ +u_{3,3} &= 100 +\end{align}\]
+

where \(u_{i,j}\) are the unknowns. Note that in this we used the side boundary condition values for the corner points \(u_{1,3}\) and \(u_{3,3}\), rather than the top value. (In reality this would represent a discontinuity in temperature, so these aren’t very realistic boundary conditions.)

+

This is a system of linear equations, that we can represent as a matrix-vector product:

+
+(5.11)\[\begin{align} +\begin{bmatrix} +1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ +0 & 1 & 0 & 1 & -4 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix} +\begin{bmatrix} u_{1,1} \\ u_{2,1} \\ u_{3,1} \\ u_{1,2} \\ u_{2,2} \\ u_{3,2} \\ u_{1,3} \\ u_{2,3} \\ u_{3,3} \end{bmatrix} &= +\begin{bmatrix} 100 \\ 100 \\ 100 \\ +100 \\ 0 \\ 100 \\ +100 \\ 0 \\ 100 \end{bmatrix} \\ +\text{or} \quad A \mathbf{u} &= \mathbf{b} +\end{align}\]
+

where \(A\) is a \(9\times 9\) coefficient matrix, \(\mathbf{u}\) is a nine-element vector of unknown variables, and \(\mathbf{b}\) is a nine-element right-hand side vector. +For \(\mathbf{u}\), we had to take variables that physically represent points in a two-dimensional space and combine them in some order to form a one-dimensional column vector. Here, we used a row-major mapping, where we started with the point in the first row and first column, then added the remaining points in that row, before moving to the next row and repeating. We’ll discuss this a bit more later.

+

If we set this up in Matlab, we can solve with u = A \ b:

+
+
+
clear all; clc
+
+A = [
+1 0 0 0  0 0 0 0 0;
+0 1 0 0  0 0 0 0 0;
+0 0 1 0  0 0 0 0 0;
+0 0 0 1  0 0 0 0 0;
+0 1 0 1 -4 1 0 1 0;
+0 0 0 0  0 1 0 0 0;
+0 0 0 0  0 0 1 0 0;
+0 0 0 0  0 0 0 1 0;
+0 0 0 0  0 0 0 0 1];
+b = [100; 100; 100; 100; 0; 100; 100; 0; 100];
+
+% solve system of linear equations
+u = A \ b;
+
+disp(u)
+
+
+
+
+
   100
+   100
+   100
+   100
+    75
+   100
+   100
+     0
+   100
+
+
+
+
+

This gives us the values for temperature at each of the nine points. In this example, we really only have one unknown temperature: \(u_{2,2}\), located in the middle. Does the value given make sense? We can check by rearranging the recursion formula for Laplace’s equation:

+
+(5.12)\[\begin{equation} +u_{i,j} = \frac{u_{i+1,j} + u_{i,j+1} + u_{i-1,j} + u_{i,j-1}}{4} \;, +\end{equation}\]
+

which shows that in such problems the value of the middle point should be the average of the four surrounding points. This matches the value of 75 found above.

+

We can use a contour plot to visualize the results, though we’ll need to convert the one-dimensional solution array into a two-dimensional matrix to plot. The Matlab reshape() function can help us here: it reshapes an array into a matrix, by specifying the target number of desired columns and rows:

+
+
+
% Example of using the reshape function, with a simple array going from 1 to 10
+
+% We want to convert it into a matrix with 5 columns and 2 rows.
+% The expected output is:
+% [1 2 3 4 5; 
+%  6 7 8 9 10]
+
+b = (1 : 10)';
+A = reshape(b, [5, 2]);
+disp('b array:')
+disp(b)
+disp('A matrix:')
+disp(A)
+
+
+
+
+
b array:
+     1
+     2
+     3
+     4
+     5
+     6
+     7
+     8
+     9
+    10
+
+A matrix:
+     1     6
+     2     7
+     3     8
+     4     9
+     5    10
+
+
+
+
+

This behavior may be a bit unexpected, because reshape() uses a column-major mapping. We can fix this by taking the transpose of the resulting matrix:

+
+
+
disp('transpose of output matrix:')
+disp(A')
+
+
+
+
+
transpose of output matrix:
+     1     2     3     4     5
+     6     7     8     9    10
+
+
+
+
+
+
+
% We can use the reshape function to convert the calculated temperatures
+% into a 3x3 matrix:
+
+n = 3; m = 3;
+u_square = reshape(u, [n, m]);
+
+contourf(u_square')
+colorbar
+
+
+
+
+../../_images/elliptic_8_0.png +
+
+

Overall that looks correct: the boundary conditions are right, and we see that the center is the average of the boundaries.

+

But, clearly only using nine points (with eight of those being boundary conditions) doesn’t give us a very good solution. To make this more accurate, we’ll need to use more points, which also means we need to automate the construction of the system of equations.

+
+
+

5.1.2. Row-major mapping

+

For a two-dimensional elliptic PDE like Laplace’s equation, we can generate a general recursion formula, but we need a way to take a grid of points where location is defined by row and column index and map these into a one-dimensional column vector, which has its own index.

+

The following figure shows a general 2D grid of points, with \(n\) number of columns in the \(x\) direction (using index \(i\)) and \(m\) number of rows in the \(y\) direction (using index \(j\)):

+
+2D grid of points +

Fig. 5.4 2D grid of points with n columns and m columns.

+
+

We want to convert the rows and columns of \(u_{i,j}\) points defined by column and row index into a single column array using a different index, \(k\) (this choice is arbitrary):

+
+(5.13)\[\begin{equation} +\begin{bmatrix} u_{1,1} \\ u_{2,1} \\ u_{3,1} \\ \vdots \\ u_{n,1} \\ +u_{1,2} \\ u_{2,2} \\ u_{3,2} \\ \vdots \\ u_{n, 2} \\ u_{1,3} \\ \vdots \\ +u_{1,m} \\ u_{2,m} \\ \vdots \\ u_{n,m} +\end{bmatrix} +\end{equation}\]
+

where \(k\) refers to the index used in that array.

+

To do this mapping, we can use this formula:

+
+(5.14)\[\begin{equation} +k_{i,j} = (j-1)n + i +\end{equation}\]
+

where \(k_{i,j}\) refers to the 1D index \(k\) mapped from the 2D indices \(i\) and \(j\).

+
+3x3 grid of points +

Fig. 5.5 Simple 3x3 grid of points

+
+

For example, in this \(3\times 3\) grid, where \(n=3\) and \(m=3\), consider the point where \(i=2\) and \(j=2\) (the point right in the center). Using our formula,

+
+(5.15)\[\begin{equation} +k_{2,2} = (2-1)3 + 2 = 5 +\end{equation}\]
+

which matches what we can visually confirm.

+

Using that mapping, we can also identify the 1D indices associated with the points surrounding location \((i,j)\):

+
+(5.16)\[\begin{align} +k_{i-1,j} &= (j-1)n + i - 1 \\ +k_{i+1,j} &= (j-1)n + i + 1 \\ +k_{i,j-1} &= (j-2)n + i \\ +k_{i,j+1} &= j n + i +\end{align}\]
+

which we can use to determine the appropriate locations to place values in the coefficient and right-hand side matrices.

+
+
+

5.1.3. Example: heat transfer in a square plate (redux)

+

Let’s return to the example of steady-state heat transfer in a square plate—but this time we’ll set the solution up more generally so we can vary the step size \(h = \Delta x = \Delta y\).

+
+
+
clear; clc; close all
+
+h = 0.1;
+x = [0 : h : 1]; n = length(x);
+y = [0 : h : 1]; m = length(y);
+
+% The coefficient matrix A is now m*n by m*n, since that is the total number of points.
+% The right-hand side vector b is m*n by 1.
+A = zeros(m*n, m*n);
+b = zeros(m*n, 1);
+
+u_left = 100;
+u_right = 100;
+u_bottom = 100;
+u_top = 0;
+
+for j = 1 : m
+    for i = 1 : n
+        % for convenience we calculate all the indices once
+        kij = (j-1)*n + i;
+        kim1j = (j-1)*n + i - 1;
+        kip1j = (j-1)*n + i + 1;
+        kijm1 = (j-2)*n + i;
+        kijp1 = j*n + i;
+        
+        if i == 1 
+            % this is the left boundary
+            A(kij, kij) = 1;
+            b(kij) = u_left;
+        elseif i == n 
+            % right boundary
+            A(kij, kij) = 1;
+            b(kij) = u_right;
+        elseif j == 1 
+            % bottom boundary
+            A(kij, kij) = 1;
+            b(kij) = u_bottom;
+        elseif j == m 
+            % top boundary
+            A(kij, kij) = 1;
+            b(kij) = u_top;
+        else
+            % these are the coefficients for the interior points,
+            % based on the recursion formula
+            A(kij, kim1j) = 1;
+            A(kij, kip1j) = 1;
+            A(kij, kijm1) = 1;
+            A(kij, kijp1) = 1;
+            A(kij, kij) = -4;
+        end
+    end
+end
+u = A \ b;
+
+u_square = reshape(u, [n, m]);
+contourf(x, y, u_square')
+c = colorbar;
+c.Label.String = 'Temperature';
+
+
+
+
+../../_images/elliptic_12_0.png +
+
+
+
+

5.1.4. Neumann (derivative) boundary conditions

+

So far, we have only discussed cases where we have Dirichlet boundary conditions; in other words, when we have all fixed values at the boundary. Frequently we also encounter Neumann-style boundary conditions, where we have the derivative specified at the boundary.

+

We can handle this in the same way we do for one-dimensional boundary value problems: either with a forward or backward difference (both of which are first-order accurate), or with a central difference using an imaginary point/ghost node (which is second-order accurate). Let’s focus on using the central difference, since it is more accurate.

+
+ghost node at boundary +

Fig. 5.6 Ghost/imaginary node beyond an upper boundary

+
+

For example, let’s say that at the upper boundary, the derivative of temperature is zero:

+
+(5.17)\[\begin{equation} +\left. \frac{\partial u}{\partial y} \right|_{\text{boundary}} = 0 +\end{equation}\]
+

Let’s consider this boundary condition applied at the point shown, \(u_{2,3}\). +We can approximate this derivative using a central difference:

+
+(5.18)\[\begin{align} +\frac{u_{2,3}}{\partial y} \approx \frac{u_{2,4} - u_{2,2}}{\Delta x} &= 0 \\ +u_{2,4} &= u_{2,2} +\end{align}\]
+

This tells us the value of the point above the boundary, \(u_{2,4}\); however, this point is a “ghost” or imaginary point located outside the boundary, so we don’t really care about its value. Instead, we can use this relationship to give us a usable equation for the boundary point, by incorporating it into the normal recursion formula for Laplace’s equation:

+
+(5.19)\[\begin{align} +u_{1,3} + u_{3,3} + u_{2,4} + u_{2,2} - 4u_{2,3} &= 0 \\ +u_{1,3} + u_{3,3} + u_{2,2} + u_{2,2} - 4u_{2,3} &= 0 \\ +\rightarrow u_{1,3} + u_{3,3} + 2 u_{2,2} - 4u_{2,3} &= 0 +\end{align}\]
+

The recursion formula for points along the upper boundary would then become

+
+(5.20)\[\begin{equation} +u_{i+1,j} + u_{i-1,j} + 2 u_{i,j-1} - 4 u_{i,j} = 0 \;. +\end{equation}\]
+

Now let’s try solving the above example, but with \(\frac{\partial u}{\partial y} = 0\) at the top boundary and \(u = 0\) at the bottom boundary:

+
+
+
clear; clc; close all
+
+h = 0.1;
+x = [0 : h : 1]; n = length(x);
+y = [0 : h : 1]; m = length(y);
+
+% The coefficient matrix A is now m*n by m*n, since that is the total number of points.
+% The right-hand side vector b is m*n by 1.
+A = zeros(m*n, m*n);
+b = zeros(m*n, 1);
+
+u_left = 100;
+u_right = 100;
+u_bottom = 0;
+
+for j = 1 : m
+    for i = 1 : n
+        % for convenience we calculate all the indices once
+        kij = (j-1)*n + i;
+        kim1j = (j-1)*n + i - 1;
+        kip1j = (j-1)*n + i + 1;
+        kijm1 = (j-2)*n + i;
+        kijp1 = j*n + i;
+        
+        if i == 1 
+            % this is the left boundary
+            A(kij, kij) = 1;
+            b(kij) = u_left;
+        elseif i == n 
+            % right boundary
+            A(kij, kij) = 1;
+            b(kij) = u_right;
+        elseif j == 1 
+            % bottom boundary
+            A(kij, kij) = 1;
+            b(kij) = u_bottom;
+        elseif j == m 
+            % top boundary, using the ghost node + recursion formula
+            A(kij, kim1j) = 1;
+            A(kij, kip1j) = 1;
+            A(kij, kijm1) = 2;
+            A(kij, kij) = -4;
+        else
+            % these are the coefficients for the interior points,
+            % based on the recursion formula
+            A(kij, kim1j) = 1;
+            A(kij, kip1j) = 1;
+            A(kij, kijm1) = 1;
+            A(kij, kijp1) = 1;
+            A(kij, kij) = -4;
+        end
+    end
+end
+u = A \ b;
+
+u_square = reshape(u, [n, m]);
+% the "20" indicates the number of levels for the contour plot
+contourf(x, y, u_square', 20);
+c = colorbar;
+c.Label.String = 'Temperature';
+
+
+
+
+../../_images/elliptic_14_0.png +
+
+
+
+

5.1.5. Iterative solutions for (very) large problems

+

So far, we’ve been able to solve our systems of linear equations in Matlab by using y = A \ b, which directly finds the solution to the equation \(A \mathbf{y} = \mathbf{b}\).

+

However, this approach will become very slow as the grid resolution (\(h = \Delta x = \Delta y\)) becomes smaller, and eventually unfeasable due to the associated computational requirements. First, let’s create a function that takes as input the segment size \(h\), then returns the time ittakes to solve the problem for different sizes.

+
+
+
%%file heat_equation.m
+function [time, num] = heat_equation(h)
+
+x = [0 : h : 1]; n = length(x);
+y = [0 : h : 1]; m = length(y);
+
+% The coefficient matrix A is now m*n by m*n, since that is the total number of points.
+% The right-hand side vector b is m*n by 1.
+A = zeros(m*n, m*n);
+b = zeros(m*n, 1);
+
+num = m*n; % number of points
+
+tic;
+
+u_left = 100;
+u_right = 100;
+u_bottom = 100;
+u_top = 0;
+
+for j = 1 : m
+    for i = 1 : n
+        % for convenience we calculate all the indices once
+        kij = (j-1)*n + i;
+        kim1j = (j-1)*n + i - 1;
+        kip1j = (j-1)*n + i + 1;
+        kijm1 = (j-2)*n + i;
+        kijp1 = j*n + i;
+        
+        if i == 1 
+            % this is the left boundary
+            A(kij, kij) = 1;
+            b(kij) = u_left;
+        elseif i == n 
+            % right boundary
+            A(kij, kij) = 1;
+            b(kij) = u_right;
+        elseif j == 1 
+            % bottom boundary
+            A(kij, kij) = 1;
+            b(kij) = u_bottom;
+        elseif j == m 
+            % top boundary
+            A(kij, kij) = 1;
+            b(kij) = u_top;
+        else
+            % these are the coefficients for the interior points,
+            % based on the recursion formula
+            A(kij, kim1j) = 1;
+            A(kij, kip1j) = 1;
+            A(kij, kijm1) = 1;
+            A(kij, kijp1) = 1;
+            A(kij, kij) = -4;
+        end
+    end
+end
+u = A \ b;
+
+u_square = reshape(u, [n, m]);
+% the "20" indicates the number of levels for the contour plot
+%contourf(x, y, u_square', 20);
+%c = colorbar;
+%c.Label.String = 'Temperature';
+
+time = toc;
+
+
+
+
+
Created file '/Users/kyle/projects/ME373-book/content/pdes/heat_equation.m'.
+
+
+
+
+

Now, we can see how long it takes to solve as we increase the resolution, and get an idea about the relationship between time-to-solution and number of unknowns.

+
+
+
clear all; clc;
+
+step_sizes = [0.1, 0.05, 0.025, 0.02, 0.0125, 0.01];
+
+n = length(step_sizes);
+nums = zeros(n,1); times = zeros(n,1);
+
+for i = 1 : n
+    [times(i), nums(i)] = heat_equation(step_sizes(i));
+end
+
+loglog(nums, times, '-o')
+xlabel('Number of unknowns'); 
+ylabel('Time for direct solution (sec)')
+hold on
+
+x = nums(3:end);
+n2 = x.^2 * (times(3) / x(1)^2);
+n3 = x.^3 * (times(3) / x(1)^3);
+plot(x, n2, '-x');
+plot(x, n3, '-s');
+legend('Actual cost', 'Quadratic cost', 'Cubic cost', 'Location', 'northwest')
+
+
+
+
+../../_images/elliptic_18_0.png +
+
+

Interestingly, we see that the slope in this log-log plot that after about 400 unknowns (so a coefficient matrix of about 160,000), the cost begins to increase exponentially, somewhere between quadratic (\(\mathcal{O}(n^2)\)) and cubic (\(\mathcal{O}(n^3)\)).

+

If we try to reduce the step size further, for example to 0.005, we’ll see that we cannot get a solution in a reasonable amount of time. But, clearly we want to get solutions for large numbers of unknowns, so what can we do?

+

We can solve larger systems of linear equations using iterative methods. There are a number of these, and we’ll focus on two:

+
    +
  • Jacobi method

  • +
  • Gauss-Seidel method

  • +
+
+

5.1.5.1. Jacobi method

+

The Jacobi method essentially works by starting with an initial guess to the solution, then using the recursion formula to solve for values at each point, then repeating this until the values converge (i.e., stop changing).

+

An algorithm we can use to solve Laplace’s equation:

+
    +
  1. Set some initial guess for all unknowns: \(u_{i,j}^{\text{old}}\)

  2. +
  3. Set the boundary values

  4. +
  5. For each point in the interior, use the recursion formula to solve for new values based on old values at the surrounding points: \(u_{i,j} = \left( u_{i+1,j}^{\text{old}} + u_{i-1,j}^{\text{old}} + u_{i,j+1}^{\text{old}} + u_{i,j-1}^{\text{old}} \right)/4\).

  6. +
  7. Check for convergence: is \(\epsilon\) less than some tolerance, such as \(10^{-6}\)? Where \(\epsilon = \max \left| u_{i,j} - u_{i,j}^{\text{old}} \right|\). If no, then return to step 2 and repeat.

  8. +
+

More formally, if we have a system \(A \mathbf{x} = \mathbf{b}\), where

+
+(5.21)\[\begin{equation} +A = \begin{bmatrix} +a_{11} & a_{12} & \cdots & a_{1n} \\ +a_{21} & a_{22} & \cdots & a_{2n} \\ +\vdots & \vdots & \ddots & \vdots \\ +a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} +\quad \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} +\quad \mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix} +\end{equation}\]
+

then we can solve iterative for \(\mathbf{x}\) using

+
+(5.22)\[\begin{equation} +x_i^{(k+1)} = \frac{1}{a_{ii}} \left( b_i - \sum_{j \neq i} a_{ij} x_j^{(k)} \right) , \quad i = 1,2,\ldots, n +\end{equation}\]
+

where \(x_i^{(k)}\) is a value of the solution at iteration \(k\) and \(x_i^{(k+1)}\) is at the next iteration.

+
+
+
%%file heat_equation_jacobi.m
+function [time, num_point, num_iter] = heat_equation_jacobi(h)
+
+x = [0 : h : 1]; n = length(x);
+y = [0 : h : 1]; m = length(y);
+
+% The coefficient matrix A is now m*n by m*n, since that is the total number of points.
+% The right-hand side vector b is m*n by 1.
+A = zeros(m*n, m*n);
+b = zeros(m*n, 1);
+num_point = m*n;
+
+tic;
+
+u_left = 100;
+u_right = 100;
+u_bottom = 100;
+u_top = 0;
+
+% initial guess
+u = 100*ones(m*n, 1);
+
+% dummy value for residual variable
+epsilon = 1.0; 
+
+num_iter = 0;
+while epsilon > 1e-6
+    u_old = u;
+    
+    epsilon = 0;
+    for j = 1 : m
+        for i = 1 : n
+            kij = (j-1)*n + i;
+            kim1j = (j-1)*n + i - 1;
+            kip1j = (j-1)*n + i + 1;
+            kijm1 = (j-2)*n + i;
+            kijp1 = j*n + i;
+
+            if i == 1 
+                % this is the left boundary
+                u(kij) = u_left;
+            elseif i == n 
+                % right boundary
+                u(kij) = u_right;
+            elseif j == 1 
+                % bottom boundary
+                u(kij) = u_bottom;
+            elseif j == m 
+                % top boundary
+                u(kij) = u_top;
+            else
+                % interior points
+                u(kij) = (u_old(kip1j) + u_old(kim1j) + u_old(kijm1) + u_old(kijp1))/4.0;
+            end
+        end
+    end
+    
+    epsilon = max(abs(u - u_old));
+    num_iter = num_iter + 1;
+end
+
+u_square = reshape(u, [n, m]);
+% the "20" indicates the number of levels for the contour plot
+contourf(x, y, u_square', 20);
+c = colorbar;
+c.Label.String = 'Temperature';
+
+time = toc;
+fprintf('Number of iterations: %d\n', num_iter)
+
+
+
+
+
Created file '/Users/kyle/projects/ME373-book/content/pdes/heat_equation_jacobi.m'.
+
+
+
+
+
+
+
step_sizes = [0.1, 0.05, 0.025, 0.02, 0.0125, 0.01, 0.005];
+n = length(step_sizes);
+
+nums_jac = zeros(n,1); times_jac = zeros(n,1); num_iter_jac = zeros(n,1);
+
+for i = 1 : n
+    [times_jac(i), nums_jac(i), num_iter_jac(i)] = heat_equation_jacobi(step_sizes(i));
+end
+
+
+
+
+
Number of iterations: 291
+Number of iterations: 1061
+Number of iterations: 3803
+Number of iterations: 5717
+Number of iterations: 13421
+Number of iterations: 20067
+Number of iterations: 69037
+
+
+../../_images/elliptic_22_1.png +
+
+
+
+

5.1.5.2. Gauss-Seidel method

+

The Gauss-Seidel method is very similar to the Jacobi method, but with one important difference: rather than using all old values to calculate the new values, incorporate updated values as they are available. Because the method incorporates newer information more quickly, it tends to converge faster (meaning, with fewer iterations) than the Jacobi method.

+

Formally, if we have a system \(A \mathbf{x} = \mathbf{b}\), where

+
+(5.23)\[\begin{equation} +A = \begin{bmatrix} +a_{11} & a_{12} & \cdots & a_{1n} \\ +a_{21} & a_{22} & \cdots & a_{2n} \\ +\vdots & \vdots & \ddots & \vdots \\ +a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} +\quad \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} +\quad \mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix} +\end{equation}\]
+

then we can solve iterative for \(\mathbf{x}\) using

+
+(5.24)\[\begin{equation} +x_i^{(k+1)} = \frac{1}{a_{ii}} \left( b_i - \sum_{j=1}^{i-1} a_{ij} x_j^{(k+1)} - \sum_{j =i+1}^n a_{ij} x_j^{(k)} \right) , \quad i = 1,2,\ldots, n +\end{equation}\]
+

where \(x_i^{(k)}\) is a value of the solution at iteration \(k\) and \(x_i^{(k+1)}\) is at the next iteration.

+
+
+
%%file heat_equation_gaussseidel.m
+function [time, num_point, num_iter] = heat_equation_gaussseidel(h)
+
+x = [0 : h : 1]; n = length(x);
+y = [0 : h : 1]; m = length(y);
+
+% The coefficient matrix A is now m*n by m*n, since that is the total number of points.
+% The right-hand side vector b is m*n by 1.
+A = zeros(m*n, m*n);
+b = zeros(m*n, 1);
+num_point = m*n;
+
+tic;
+
+u_left = 100;
+u_right = 100;
+u_bottom = 100;
+u_top = 0;
+
+% initial guess
+u = 100*ones(m*n, 1);
+
+% dummy value for residual variable
+epsilon = 1.0; 
+
+num_iter = 0;
+while epsilon > 1e-6
+    u_old = u;
+    
+    epsilon = 0;
+    for j = 1 : m
+        for i = 1 : n
+            kij = (j-1)*n + i;
+            kim1j = (j-1)*n + i - 1;
+            kip1j = (j-1)*n + i + 1;
+            kijm1 = (j-2)*n + i;
+            kijp1 = j*n + i;
+
+            if i == 1 
+                % this is the left boundary
+                u(kij) = u_left;
+            elseif i == n 
+                % right boundary
+                u(kij) = u_right;
+            elseif j == 1 
+                % bottom boundary
+                u(kij) = u_bottom;
+            elseif j == m 
+                % top boundary
+                u(kij) = u_top;
+            else
+                % interior points
+                u(kij) = (u(kip1j) + u(kim1j) + u(kijm1) + u(kijp1))/4.0;
+            end
+        end
+    end
+    
+    epsilon = max(abs(u - u_old));
+    num_iter = num_iter + 1;
+end
+
+u_square = reshape(u, [n, m]);
+%% the "20" indicates the number of levels for the contour plot
+contourf(x, y, u_square', 20);
+c = colorbar;
+c.Label.String = 'Temperature';
+
+time = toc;
+fprintf('Number of iterations: %d\n', num_iter)
+
+
+
+
+
Created file '/Users/kyle/projects/ME373-book/content/pdes/heat_equation_gaussseidel.m'.
+
+
+
+
+
+
+
step_sizes = [0.1, 0.05, 0.025, 0.02, 0.0125, 0.01, 0.005];
+n = length(step_sizes);
+
+nums_gs = zeros(n,1); times_gs = zeros(n,1); num_iter_gs = zeros(n,1);
+
+for i = 1 : n
+    [times_gs(i), nums_gs(i), num_iter_gs(i)] = heat_equation_gaussseidel(step_sizes(i));
+end
+
+loglog(nums, times, '-o'); hold on
+loglog(nums_jac, times_jac, '-^')
+loglog(nums_gs, times_gs, '-x')
+xlabel('Number of unknowns'); 
+ylabel('Time for solution (sec)')
+legend('Direct solution', 'Jacobi solution', 'Gauss-Seidel solution', 'Location', 'northwest')
+
+
+
+
+
Number of iterations: 156
+Number of iterations: 564
+Number of iterations: 2025
+Number of iterations: 3048
+Number of iterations: 7181
+Number of iterations: 10762
+Number of iterations: 37378
+
+
+../../_images/elliptic_25_1.png +
+
+
+
+
loglog(nums_jac, num_iter_jac, '-^')
+hold on;
+loglog(nums_gs, num_iter_gs, '-x')
+xlabel('Number of unknowns')
+ylabel('Number of iterations required')
+legend('Jacobi method', 'Gauss-Seidel method', 'Location', 'northwest')
+
+
+
+
+../../_images/elliptic_26_0.png +
+
+

These results show us a few things:

+
    +
  • For very small problems, the direct solution method is faster.

  • +
  • For the heat equation, once we get to around 1000 unknowns, the methods perform similarly. Beyond this, the direct solution method becomes unreasonably slow, and even fails to solve in a reasonable time for a step size of 0.005.

  • +
  • The Gauss-Seidel method converges with around half the number of iterations than the Jacobi method.

  • +
+

For larger, more-realistic problems, iterative solution methods like Jacobi and Gauss-Seidel are essential.

+
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/pdes/parabolic.html b/docs/content/pdes/parabolic.html new file mode 100644 index 0000000..f157376 --- /dev/null +++ b/docs/content/pdes/parabolic.html @@ -0,0 +1,733 @@ + + + + + + + + 5.2. Parabolic PDEs — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+ +
+ + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ Contents +
+ +
+
+
+
+
+ +
+ +
+

5.2. Parabolic PDEs

+

A classic example of a parabolic partial differential equation (PDE) is the one-dimensional unsteady heat equation:

+
+(5.25)\[\begin{equation} +\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial t^2} +\end{equation}\]
+

where \(T(x,t)\) is the temperature varying in space and time, and \(\alpha\) is the thermal diffusivity: \(\alpha = k / (\rho c_p)\), which is a constant.

+

We can solve this using finite differences to represent the spatial derivatives and time derivatives separately. +First, let’s rearrange the PDE slightly:

+
+(5.26)\[\begin{equation} +\frac{\partial^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t} +\end{equation}\]
+
+

5.2.1. Explicit scheme

+

Let’s use a central difference for the spatial derivative with a spacing of \(\Delta x\), and a forward difference for the time derivative with a time-step size of \(\Delta t\). With these choices, we can obtain an approximation to the PDE that applies at time \(t^k\) and spatial location \(x_i\):

+
+(5.27)\[\begin{equation} +\frac{T_{i-1}^k - 2 T_i^k + T_{i+1}^k}{\Delta x^2} = \frac{1}{\alpha} \left( \frac{T_i^{k+1} - T_i^k}{\Delta t} \right) +\end{equation}\]
+

where \(T_i^k\) is the temperature at time \(t^k\) and spatial location \(x_i\). The following figure shows the stencil of points involved in the PDE, for a domain with five points in the \(x\)-direction.

+
+stencil for explicit parabolic solution +

Fig. 5.7 Stencil for explicit solution to heat equation

+
+

To solve the heat equation for a one-dimensional domain over \(0 \leq x \leq L\), we will need both initial conditions at \(t = 0\) and boundary conditions at \(x=0\) and \(x=L\) (for all time). In terms of our nodal values, this means we need \(T_i^{k=1}\) for \(i = 1 \ldots n\), where \(n\) is the number of points, as well as information about \(T_1^k\) and \(T_n^k\) for all times \(k\).

+

We can rearrange the above equation to obtain our recursion formula:

+
+(5.28)\[\begin{equation} +T_i^{k+1} = \left( T_{i+1}^k + T_{i-1}^k \right) \frac{\alpha \Delta t}{\Delta x^2} + T_i^k \left( 1 - 2 \frac{\alpha \Delta t}{\Delta x^2} \right) \;. +\end{equation}\]
+

This is an explicit scheme in time, similar to the Forward Euler method we used for ordinary differential equations, and like that method it may have stability issues. The combination of terms we see repeated is also known as the Fourier number: \(\text{Fo} = \frac{\alpha \Delta t}{\Delta x^2}\), and governs the stability. +We can rewrite the recursion formula using this:

+
+(5.29)\[\begin{equation} +T_i^{k+1} = \left( T_{i+1}^k + T_{i-1}^k \right) \text{Fo} + T_i^k \left( 1 - 2 \text{Fo} \right) \;. +\end{equation}\]
+

The term in parentheses there must be greater than or equal to zero for stability (\(1 - 2 \text{Fo} \geq 0\)); if not, the solution may become unstable and blow up. This gives us some conditions on our choice of step sizes:

+
+(5.30)\[\begin{align} +1 - 2 \text{Fo} &\geq 0 \\ +1 & \geq 2 \text{Fo} \\ +\text{Fo} &\leq \frac{1}{2} \\ +\frac{\alpha \Delta t}{\Delta x^2} &\leq \frac{1}{2} +\end{align}\]
+

This is the stability criterion for the explicit method: the Fourier number must be smaller than 0.5. +For a given thermal diffusivity and chosen spatial step size, this also gives us the limit on the time-step size: \(\Delta t \leq \Delta x^2 / (2 \alpha)\).

+

Let’s look at an example where the initial temperature is 200, the temperature at the boundaries are 50, the thermal diffusivity is \(\alpha = 2.3 \times 10^{-1}\) m\(^2 /\) s, and \(L = 1\). +In other words,

+
+(5.31)\[\begin{align} +T(x, t=0) &= 200 \\ +T(x=0, t) &= 50 \\ +T(x=L, t) &= 50 +\end{align}\]
+

We’ll integrate out to \(t = 1\), using a Fourier number of 0.25 to be comfortably below the stability limit (Fo = 0.25):

+
+
+
clear all
+
+dx = 0.1;
+alpha = 2.3e-1;
+
+% for stability, set the Fourier number at 0.25 (half the stability limit of 0.5)
+Fo = 0.25;
+% then choose the time step based on the Fourier number
+dt = Fo * dx^2 / alpha;
+
+x = [0 : dx : 1]; n = length(x);
+t = [0 : dt : 1]; m = length(t);
+
+T = zeros(m, n);
+
+% initial conditions
+T(1,:) = 200;
+
+plot(x, T(1,:))
+axis([0 1 50 200]);
+xlabel('Distance'); ylabel('Temperature');
+F(1) = getframe(gcf);
+
+for k = 1 : m - 1
+    for i = 1 : n
+        if i == 1
+            T(k+1, 1) = 50;
+        elseif i == n
+            T(k+1, n) = 50;
+        else
+            T(k+1, i) = (T(k,i+1) + T(k,i-1))*Fo + T(k,i)*(1 - 2*Fo);
+        end
+    end
+    plot(x, T(k+1,:))
+    axis([0 1 50 200]);
+    xlabel('Distance'); ylabel('Temperature');
+    F(k+1) = getframe(gcf);
+end
+close
+
+%% If you are working interactively, you can use this to make a movie in Matlab
+%fig = figure;
+%movie(fig, F, 2)
+
+%% This generates a GIF of the results (for use in Jupyter Notebook)
+filename = 'parabolic_animated.gif';
+for i = 1 : length(F)
+    im = frame2im(F(i)); 
+    [imind,cm] = rgb2ind(im,256); 
+    % Write to GIF
+    if i == 1 
+        imwrite(imind,cm,filename,'gif', 'Loopcount',inf, 'DelayTime',1e-3); 
+    else 
+        imwrite(imind,cm,filename,'gif','WriteMode','append', 'DelayTime',1e-3); 
+    end
+end
+
+
+
+
+
+movie of parabolic PDE solution +

Fig. 5.8 Animated solution to 1D transient heat transfer PDE

+
+

This shows the temperature decaying exponentially from the initial conditions, constrained by the boundary conditions.

+

What happens if we tried to use a Fourier number larger than 0.5, or arbitrarily chose a time-step size that was too large (and resulted in \(\text{Fo} > 0.5\))?

+
+
+
clear all
+
+dx = 0.1;
+alpha = 2.3e-1;
+
+%% Purposely choose a Fourier number that is past the stability limit:
+Fo = 0.75;
+dt = Fo * dx^2 / alpha;
+
+x = [0 : dx : 1]; n = length(x);
+t = [0 : dt : 1]; m = length(t);
+
+T = zeros(m, n);
+
+% initial conditions
+T(1,:) = 200;
+
+plot(x, T(1,:))
+axis([0 1 50 200]);
+xlabel('Distance'); ylabel('Temperature');
+F(1) = getframe(gcf);
+
+for k = 1 : m - 1
+    for i = 1 : n
+        if i == 1
+            T(k+1, 1) = 50;
+        elseif i == n
+            T(k+1, n) = 50;
+        else
+            T(k+1, i) = (T(k,i+1) + T(k,i-1))*Fo + T(k,i)*(1 - 2*Fo);
+        end
+    end
+    plot(x, T(k+1,:))
+    xlabel('Distance'); ylabel('Temperature');
+    F(k+1) = getframe(gcf);
+end
+close
+
+%% If you are working interactively, you can use this to make a movie in Matlab
+%fig = figure;
+%movie(fig, F, 2)
+
+%% This generates a GIF of the results (for use in Jupyter Notebook)
+filename = 'parabolic_unstable_animated.gif';
+for i = 1 : length(F)
+    im = frame2im(F(i)); 
+    [imind,cm] = rgb2ind(im,256); 
+    % Write to GIF
+    if i == 1 
+        imwrite(imind,cm,filename,'gif', 'Loopcount',2, 'DelayTime',1e-3); 
+    else 
+        imwrite(imind,cm,filename,'gif','WriteMode','append', 'DelayTime',1e-3); 
+    end
+end
+
+
+
+
+
+movie of unstable parabolic PDE solution +

Fig. 5.9 Animated unstable solution

+
+

In this case, the solution becomes unstable and blows up, leading to unphysical results.

+

For this explicit scheme, the choice of \(\Delta t\) is limited by the stability criterion. This means that we may be stuck using a small time-step size.

+

Rather than being forced to use a very small time-step size, we can also explore implicit schemes that are unconditionally stable.

+
+
+

5.2.2. Implicit scheme

+
+
+
clear all
+
+alpha = 2.3e-1;
+dx = 0.1;
+x = [0 : dx : 1]; n = length(x);
+
+% choose a Fourier number that is deliberately past the explicit method stability limit
+Fo = 0.75;
+dt = Fo * dx^2 / alpha;
+
+t = [0 : dt : 1]; m = length(t);
+
+T = zeros(m, n);
+
+% Initial conditions
+T(1,:) = 200;
+
+plot(x, T(1,:))
+axis([0 1 50 200]);
+xlabel('Distance'); ylabel('Temperature');
+F(1) = getframe(gcf);
+
+for k = 1 : m - 1
+    A = zeros(n,n);
+    b = zeros(n,1);
+    for i = 1 : n
+        if i == 1
+            A(1,1) = 1;
+            b(1) = 50;
+        elseif i == n
+            A(n,n) = 1;
+            b(n) = 50;
+        else
+            A(i,i-1) = Fo;
+            A(i,i) = -2*Fo - 1;
+            A(i,i+1) = Fo;
+            b(i) = -T(k,i);
+        end
+    end
+    
+    T(k+1, :) = A \ b;
+    plot(x, T(k+1,:))
+    axis([0 1 50 200]);
+    xlabel('Distance'); ylabel('Temperature');
+    F(k+1) = getframe(gcf);
+end
+close
+
+%% This generates a GIF of the results (for use in Jupyter Notebook)
+filename = 'parabolic_implicit_animated.gif';
+for i = 1 : length(F)
+    im = frame2im(F(i)); 
+    [imind,cm] = rgb2ind(im,256); 
+    % Write to GIF
+    if i == 1 
+        imwrite(imind,cm,filename,'gif', 'Loopcount',inf, 'DelayTime',1e-3); 
+    else 
+        imwrite(imind,cm,filename,'gif','WriteMode','append', 'DelayTime',1e-3); 
+    end
+end
+
+
+
+
+
+movie of implicit parabolic PDE solution +

Fig. 5.10 Solution to 1D heat equation with implicit method. Fo = 0.75

+
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/pdes/partial-differential-equations.html b/docs/content/pdes/partial-differential-equations.html new file mode 100644 index 0000000..28d5211 --- /dev/null +++ b/docs/content/pdes/partial-differential-equations.html @@ -0,0 +1,449 @@ + + + + + + + + 5. Partial Differential Equations — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
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+ +
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+ +
+ + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
+
+ +
+ +
+

5. Partial Differential Equations

+

This chapter focuses on numerical methods for solving partial differential equations (PDEs), which involve derivatives in multiple dimensions.

+

We can write a general, linear 2nd-order PDE for a variable \(u(x,y)\) as

+
+(5.1)\[\begin{equation} +A \frac{\partial^2 u}{\partial x^2} + 2 B \frac{\partial^2 u}{\partial x \, \partial y} + C \frac{\partial^2 u}{\partial y^2} = F \left( x, y, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y} \right) +\end{equation}\]
+

where \(A\), \(B\), and \(C\) are constants. Depending on their value, we can categorize a PDE into one of three categories:

+
    +
  • \(B^2 - AC < 0\): elliptic

  • +
  • \(B^2 - AC = 0\): parabolic

  • +
  • \(B^2 - AC > 0\): hyperbolic

  • +
+

The different PDE types will exhibit different characteristics and will also require slightly different solution approaches.

+
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/quizzes/quiz2-IVPs.html b/docs/content/quizzes/quiz2-IVPs.html new file mode 100644 index 0000000..474583d --- /dev/null +++ b/docs/content/quizzes/quiz2-IVPs.html @@ -0,0 +1,685 @@ + + + + + + + + Sample Quiz 2 problems: IVPs — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+ +
+ + + + + + + + + + + + + + + + +
+ + + +
+
+
+
+ +
+ +
+

Sample Quiz 2 problems: IVPs

+
+

Problem 1: Stability analysis

+

Given the first-order ODE

+
+(1)\[\begin{equation} +\frac{dy}{dt} = -8y \;, +\end{equation}\]
+

a.) Perform a linear stability analysis to find the amplification factor (\(\sigma\)) for the midpoint method (also known as the modified Euler method) applied to this ODE.

+

b.) Using your result from part (a), show whether the numerical solution would be stable or unstable for time-step sizes of \(\Delta t = 0.2\) and \(\Delta t = 0.4\). (Show for each value)

+

c.) Based on your results for parts (a) and (b), is the midpoint method unstable, conditionally stable, or unconditionally stable for this ODE?

+

d.) What is the order of accuracy for the midpoint method? Based on this, what (approximate) global errors would you expect in the solution when using time-step sizes of \(\Delta t = 0.2\) and 0.4?

+

e.) What are your two options for reducing error in the solution?

+
+

Solution

+

a.)

+
+(2)\[\begin{align} +y_{i+1/2} &= y_i + \frac{\Delta t}{2} \, f(t_i, y_i) \\ +y_{i+1} &= y_i + \Delta t \, f(t_i + \frac{\Delta t}{2}, y_{i+1/2}) +\end{align}\]
+

For this ODE:

+
+(3)\[\begin{align} +y_{i+1/2} &= y_i + \frac{\Delta t}{2} \, (-8 y_i) = y_i - 4 \Delta t \, y_i \\ +y_{i+1} &= y_i + \Delta t \, \left[ -8 \left( y_i - 4 \Delta t \, y_i \right) \right] \\ +&= y_i - 8 \Delta t \, y_i + 32 \Delta t^2 \, y_i \\ +&= y_i (1 - 8 \Delta t + 32 \Delta t^2) +\end{align}\]
+

So,

+
+(4)\[\begin{equation} +\sigma = \frac{y_{i+1}}{y_i} = 1 - 8 \Delta t + 32 \Delta t^2 +\end{equation}\]
+

b.) For stability, \(|\sigma| \leq 1\).

+

\(\Delta t = 0.2\):

+
+(5)\[\begin{equation} +\sigma = 1 - 8(0.2) + 32 (0.2)^2 = 0.68 +\end{equation}\]
+

so stable.

+

\(\Delta t = 0.4\):

+
+(6)\[\begin{equation} +\sigma = 1 - 8(0.4) + 32 (0.4)^2 = 2.92 +\end{equation}\]
+

so unstable.

+

c.) Conditionally stable; the method is stable for some values of \(\Delta t\) and unstable for other values.

+

d.) The midpoint method is 2nd-order accurate. So, for \(\Delta t = 0.2\), we should expect global errors on the order of 0.04, and for \(\Delta t = 0.4\) we should expect errors on the order of 0.16.

+

e.)

+
    +
  • reduce the step size

  • +
  • choose a higher order method, such as 4th-order Runge-Kutta

  • +
+
+
+
+

Problem 2: Second-order Backward Euler

+

Given the second-order ODE

+
+(7)\[\begin{equation} +2 y^{\prime\prime} + y^{\prime} + 4y = 3x +\end{equation}\]
+

a.) Find the recursion formulas (i.e., \(y_{i+1} = \ldots\)) for numerically solving this ODE using the backward Euler method. Clearly define/state any variable or function you use.

+

b.) What is the order of accuracy for the backward Euler method? Given a step size \(\Delta x = 0.15\), approximately what local error and what global error would you expect in your solution? What is the difference between these two errors?

+

c.) Why would you want to use this method to solve an ODE over a simpler method like forward Euler?

+
+

Solution

+

a.) System of Backward Euler recursion formulas, for \(y(x)\) and \(u(x) = y^{\prime}\), where \(f(x,y,u) = dy/dx\) and \(g(x,y,u) = du/dx = y^{\prime\prime}\):

+
+(8)\[\begin{align} +y_{i+1} &= y_i + \Delta x \, f(x_{i+1}, y_{i+1}, u_{i+1}) = y_i + \Delta x \, u_{i+1} \\ +u_{i+1} &= u_i + \Delta x \, g(x_{i+1}, y_{i+1}, u_{i+1}) = u_i + \Delta x \, \left( \frac{3}{2}x_{i+1} - \frac{1}{2} u_{i+1} - 2 y_{i+1} \right) +\end{align}\]
+

This form is implicit and cannot be used directly, so we need to solve the system of equations.

+

There are two ways to approach this, that give an equivalent solution

+

Option 1: Use substitution and elimination to solve:

+
+(9)\[\begin{align} +u_{i+1} &= u_i + \frac{3}{2} \Delta x \, x_{i+1} - \frac{1}{2} \Delta x \, u_{i+1} - 2 \Delta x \left( y_i + \Delta x \, u_{i+1} \right) \\ +&= u_i + \frac{3}{2} \Delta x \, x_{i+1} - \frac{1}{2} \Delta x \, u_{i+1} - 2 \Delta x \, y_i - 2 \Delta x^2 \, u_{i+1} \\ +u_{i+1} \left( 1 + 2 \Delta x^2 + \frac{1}{2} \Delta x \right) &= u_i + \frac{3}{2} \Delta x \, x_{i+1} - 2 \Delta x \, y_i +\end{align}\]
+

Thus,

+
+(10)\[\begin{align} +u_{i+1} &= \frac{u_i + \frac{3}{2} \Delta x \, x_{i+1} - 2 \Delta x \, y_i}{1 + 2 \Delta x^2 + \frac{1}{2} \Delta x} \\ +y_{i+1} &= y_i + \Delta x \frac{u_i + \frac{3}{2} \Delta x \, x_{i+1} - 2 \Delta x \, y_i}{1 + 2 \Delta x^2 + \frac{1}{2} \Delta x} +\end{align}\]
+

Option 2: Or, use Cramer’s rule:

+
+(11)\[\begin{align} +y_{i+1} - \Delta x u_{i+1} &= y_i \\ +2 \Delta x \, y_{i+1} + \left( 1 + \frac{1}{2} \Delta x \right) u_{i+1} &= u_i + \frac{3}{2} \Delta x \, x_{i+1} \\ +\rightarrow \begin{bmatrix} 1 & -\Delta x \\ 2 \Delta x & \left( 1 + \frac{1}{2} \Delta x \right)\end{bmatrix} \begin{bmatrix} y_{i+1} \\ u_{i+1} \end{bmatrix} &= +\begin{bmatrix} y_i \\ u_i + \frac{3}{2} \Delta x \, x_{i+1} \end{bmatrix} +\end{align}\]
+

Then,

+
+(12)\[\begin{align} +y_{i+1} &= \frac{y_i \left(1 + \frac{\Delta x}{2} \right) + \Delta x \left( u_i + \frac{3}{2} \Delta x \, x_{i+1} \right)}{1 + \frac{\Delta x}{2} + 2 \Delta x^2} \\ +u_{i+1} &= \frac{u_i + \frac{3}{2} \Delta x \, x_{i+1} - 2 \Delta x \, y_i}{1 + \frac{\Delta x}{2} + 2 \Delta x^2} +\end{align}\]
+

b.) Backward Euler is 1st-order accurate, so the global error is on the order of the step size (\(\Delta x\)).

+

The global error should then be on the order of 0.15, and the local error on the order of \(0.15^2 = 0.0225\).

+

The difference: the local (or truncation) error is the error at each step of the solution, while the global error is the overall error that accumulates over the whole solution.

+

c.) Backward Euler is unconditionally stable, while Forward Euler is conditionally stable.

+
+
+
+

Problem 3: Fourier series

+

Given the input waveform \(R(t)\) shown here,

+

Increasing square wave form

+

a.) What is the period and fundamental frequency of the input forcing function?

+

b.) Is the periodic function \(R(t)\) odd, even, or neither?

+

c.) Find the coefficients \(a_0\) and \(a_n\) of the Fourier series representation of \(R(t)\). (For purposes of time, you do not need to find \(b_n\)):

+
+(13)\[\begin{equation} +R(t) = a_0 + \sum_{n=1}^{\infty} a_n \cos(n \omega t) + \sum_{n=1}^{\infty} b_n \sin(n \omega t) +\end{equation}\]
+
+

Solution

+

a.)

+
+(14)\[\begin{equation} +T = 4\pi \quad \omega = \frac{2\pi}{T} = \frac{1}{2} +\end{equation}\]
+

b.) Neither.

+

c.)

+
+(15)\[\begin{align} +a_0 &= \frac{1}{T} \int_0^T f(t) dt = \frac{1}{4\pi} \left[ \int_0^{\pi} 0 dt + \int_{\pi}^{2\pi} 3 dt + \int_{2\pi}^{3\pi} 2 dt + \int_{3\pi}^{4\pi} 1 dt \right] \\ +&= \frac{1}{4\pi} \left[ 3 \pi + 2 \pi + 1 \pi \right] \\ +a_0 &= \frac{3}{2} +\end{align}\]
+
+(16)\[\begin{align} +a_n &= \frac{2}{T} \int_0^T f(t) \cos(n \omega t) dt \\ +&= \frac{2}{4\pi} \left[ \int_0^{\pi} 0 \cos \left(\frac{n t}{2}\right) dt + \int_{\pi}^{2\pi} 3 \cos \left(\frac{n t}{2}\right) dt + \int_{2\pi}^{3\pi} 2 \cos \left(\frac{n t}{2}\right) dt + \int_{3\pi}^{4\pi} 1 \cos \left(\frac{n t}{2}\right) dt \right] \\ +&\cdots \\ +a_n &= \frac{1}{n\pi} \left[ \sin\left(\frac{3 n \pi}{2}\right) - 3 \sin\left(\frac{n \pi}{2}\right) \right] +\end{align}\]
+
+
+
+

Problem 4: Second-order analytical

+

The displacement \(y(t)\) of a harmonically forced mass-spring system is given by:

+
+(17)\[\begin{equation} +y^{\prime\prime} + 8y^{\prime} + 16y = 6 e^{-4t} +\end{equation}\]
+

a.) For initial conditions \(y(0)=0\) and \(y^{\prime}(0) = 2\), find the response of the system \(y(t)\).

+

b.) Given a specified time increment \(\Delta t\) and a domain, write the recursion formulas for solving this equation with the forward Euler method. Clearly define any variables or functions used. You do not need to write Matlab code.

+
+

Solution

+

a.) First, get the homogeneous solution:

+
+(18)\[\begin{align} +y_H^{\prime\prime} + 8y_H^{\prime} + 16y_H &= 0 \\ +\rightarrow \lambda^2 + 8 \lambda + 16 &= 0 = (\lambda + 4)^2 \\ +\lambda &= 4 \text{ (repeated)} \\ +\text{so } y_H &= c_1 e^{-4t} + c_2 t e^{-4t} +\end{align}\]
+

Next, use the method of undetermined coefficients to get the inhomogeneous solution:

+
+(19)\[\begin{align} +y_{IH} &= K t^2 e^{-4t} \\ +y_{IH}^{\prime} &= K e^{-4t} \left( 2t - 4t^2 \right) \\ +y_{IH}^{\prime\prime} &= K e^{-4t} \left( -8t + 2 - 8t + 16t^2 \right) \\ +\rightarrow K &= 3 +\end{align}\]
+

Then, the general solution is

+
+(20)\[\begin{equation} +y(t) = c_1 e^{-4t} + c_2 t e^{-4t} + 3 t^2 e^{-4t} +\end{equation}\]
+

Applying the initial conditions:

+
+(21)\[\begin{equation} +y(t) = 2 t e^{-4t} + 3 t^2 e^{-4t} +\end{equation}\]
+

b.) If \(z_1 = y\) and \(z_2 = y^{\prime}\), then

+
+(22)\[\begin{align} +z_1^{\prime} &= z_2 \\ +z_2^{\prime} &= y^{\prime\prime} = 6e^{-4t} - 8z_2 - 16z_1 = f(t, z_1, z_2) +\end{align}\]
+

Then, the recursion formulas are:

+
+(23)\[\begin{align} +z_{1, i+1} &= z_{1,i} + \Delta t \, z_{2,i} \\ +z_{2, i+1} &= z_{2, i} + \Delta t \left( 6 e^{-4t} - 8 z_{2,i} - 16 z_{1,i} \right) +\end{align}\]
+

where \(z_{1,1} = 0\) and \(z_{2,1} = 4\).

+
+
+
+ + + + +
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+
+ +
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Sample Quiz 3 problems: BVPs

+
+

Problem 1: Finite difference method

+

The temperature distribution \(T(r)\) in an annular fin of inner radius \(r_1\) and outer radius \(r_2\) is described by the equation

+
+(24)\[\begin{equation} +r \frac{d^2 T}{dr^2} + \frac{dT}{dr} - rm^2 (T - T_{\infty}) = 0 \;, +\end{equation}\]
+

where \(r\) is the radial distance from the centerline (the independent +variable) and \(m^2\) is a constant that depends on the heat transfer coefficient, thermal conductivity, and thickness of the annulus. Assuming we choose a spatial step size \(\Delta r\),

+
+
+ Annular fin +
Figure: Annular fin
+
+
+

a.) Write the finite-difference representation of the ODE (that applies at a location \(r_i\)), using central differences.

+

b.) Based on the last part, write the recursion formula.

+

c.) The boundary condition at the outer radius \(r = r_2\) is described by convection heat transfer:

+
+(25)\[\begin{equation} +-k \left. \frac{dT}{dr} \right|_{r=r_2} = h \left[ T(r=r_2) - T_{\infty} \right] \;. +\end{equation}\]
+

Write the boundary condition at \(r = r_2\) in recursion form (i.e., the equation you would implement into your system of equations to solve for temperature).

+
+

Solution

+

a.) Replace the derivatives in the given ODE with finite differences, and replace any locations with the \(i\) location:

+
+(26)\[\begin{equation} +r_i \frac{T_{i-1} - 2T_i + T_{i+1}}{\Delta r^2} + \frac{T_{i+1} - T_{i-1}}{2\Delta r} - r_i m^2 (T_i - T_{\infty}) = 0 +\end{equation}\]
+

or

+
+(27)\[\begin{equation} +r_i (T_{i-1} - 2T_i + T_{i+1}) + \frac{\Delta r}{2} (T_{i+1} - T_{i-1}) - r_i m^2 \Delta r^2 (T_i - T_{\infty}) = 0 +\end{equation}\]
+

b.) Rearrange and combine terms:

+
+(28)\[\begin{align} +r_i (T_{i-1} - 2T_i + T_{i+1}) + \frac{\Delta r}{2} (T_{i+1} - T_{i-1}) - r_i m^2 \Delta r^2 (T_i - T_{\infty}) &= 0 \\ +r_i (T_{i-1} - 2T_i + T_{i+1}) + \frac{\Delta r}{2} (T_{i+1} - T_{i-1}) - r_i m^2 \Delta r^2 T_i &= -r_i m^2 \Delta r^2 T_{\infty} \\ +\left(r_i - \frac{\Delta r}{2}\right) T_{i-1} + \left( -2 r_i - r_i m^2 \Delta r^2 \right) T_i + \left( r_i + \frac{\Delta r}{2} \right) T_{i+1} &= -r_i m^2 \Delta r^2 T_{\infty} +\end{align}\]
+

c.) We can use a backward difference to approximate the \(dT/dr\) term. \(T_n\) represents the temperature at node \(n\) where \(r_n = r_2\):

+
+(29)\[\begin{align} +-k \frac{T_n - T_{n-1}}{\Delta r} &= h (T_n - T_{\infty}) \\ +-k (T_n - T_{n-1}) &= h \Delta r (T_n - T_{\infty}) \\ +k T_{n-1} - (k + h\Delta r) T_n &= -h \Delta r T_{\infty} +\end{align}\]
+
+
+
+

Problem 2: eigenvalue

+

Given the equation \(y^{\prime\prime} + 9 \lambda^2 y = 0\) with \(y(0) = 0\) and \(y(2) = 0\),

+

a.) Find the expression that gives all eigenvalues (\(\lambda\)). What is the eigenfunction?

+

b.) Calculate the principal eigenvalue.

+
+

Solution

+

a.)

+
+(30)\[\begin{gather} +y(x) = A \sin (3 \lambda x) + B \cos (3 \lambda x) \\ +\text{Apply BCs: } y(x=0) = 0 = A \sin(0) + B \cos(0) = B \\ +\therefore B = 0 \\ +y(x) = A \sin (3 \lambda x) \\ +y(x=2) = 0 = A \sin (3 \lambda 2) \\ +A \neq 0 \text{ so } \sin(3 \lambda 2) = \sin(6 \lambda) = 0 \therefore 6 \lambda = n \pi \quad n=1,2,3,\ldots \\ +\lambda = \frac{n \pi}{6} \quad n=1,2,3,\ldots,\infty +\end{gather}\]
+

The eigenfunction is then the solution function associated with an eigenvalue:

+
+(31)\[\begin{equation} +y_n = A_n \sin \left( \frac{n \pi x}{2} \right) \quad n = 1, 2, 3, \ldots, \infty +\end{equation}\]
+

b.) The principal eigenvalue is just that associated with \(n = 1\):

+
+(32)\[\begin{equation} +\lambda_p = \lambda_1 = \frac{\pi}{6} +\end{equation}\]
+
+
+
+

Problem 3: shooting method

+

Use the shooting method to solve the boundary value problem

+
+(33)\[\begin{equation} +y^{\prime\prime} - 4y = 0 +\end{equation}\]
+

where \(y(0) = 0\) and \(y(1) = 3\). Find the initial value of \(y'\) (meaning, \(y'(0)\)) that satisfies the given boundary conditions. Use the forward Euler method with a step size of \(\Delta x = 0.5\).

+
+

Solution

+

First decompose into two 1st-order ODEs:

+
+(34)\[\begin{align} +z_1' &= y' = z_2 \\ +z_2' &= y'' = 4 z_1 +\end{align}\]
+

with BCs \(z_1 (x=0) = z_{1,1} = 0\) and \(z_1(x=1) = z_{1,3} = 3\), we do not know \(y'(0) = z_2(x=0) = z_{2,1} = ?\)

+

Try some guess #1: \(y' (0) = 0 = z_2 (0)\), with the forward Euler method:

+
+(35)\[\begin{align} +z_{1,2} = z_1 (0.5) &= z_1 (0) + z_2(0) 0.5 = 0 \\ +z_{2,2} = z_2 (0.5) &= z_2 (0) + \left( 4z_1(0) \right) 0.5 = 0 \\ +z_{1,3} = z_1 (1.0) &= z_1 (0.5) + z_2(0.5) 0.5 = 0 \leftarrow \text{solution 1} \\ +z_{2,3} = z_2 (1.0) &= z_2 (0.5) + \left( 4z_1(0.5) \right) 0.5 = 0 +\end{align}\]
+

so for solution 1: \(y(1) = 0 \neq 3\).

+

For guess #2: \(y' (0) = 2 = z_2 (0)\), with the forward Euler method:

+
+(36)\[\begin{align} +z_1 (0.5) &= z_1 (0) + z_2(0) 0.5 = 1.0 \\ +z_2 (0.5) &= z_2 (0) + \left( 4z_1(0) \right) 0.5 = 2.0 \\ +z_1 (1.0) &= z_1 (0.5) + z_2(0.5) 0.5 = 2.0 \leftarrow \text{solution 2} \\ +z_2 (1.0) &= z_2 (0.5) + \left( 4z_1(0.5) \right) 0.5 = 4.0 +\end{align}\]
+

so for solution 1: \(y(1) = 2 \neq 3\).

+

For guess #3, we can interpolate:

+
+(37)\[\begin{align} +m &= \frac{\text{guess 1} - \text{guess 2}}{\text{solution 1} - \text{solution 2}} = \frac{0 - 2}{0 - 2} = 1 \\ +\text{guess 3} &= \text{guess 2} + m (\text{target} - \text{solution 2}) = 2 + 1(3-2) = 3 +\end{align}\]
+

then, use this guess:

+
+(38)\[\begin{align} +z_1 (0.5) &= z_1 (0) + z_2(0) 0.5 = 1.5 \\ +z_2 (0.5) &= z_2 (0) + \left( 4z_1(0) \right) 0.5 = 3.0 \\ +z_1 (1.0) &= z_1 (0.5) + z_2(0.5) 0.5 = 3.0 \leftarrow \text{solution 3} \\ +z_2 (1.0) &= z_2 (0.5) + \left( 4z_1(0.5) \right) 0.5 = 6.0 +\end{align}\]
+

so for solution 3: \(y(1) = 3\) which is the target.

+

So our answer is \(y'(0) = 3\).

+

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+ + + + + + \ No newline at end of file diff --git a/docs/content/second-order/analytical.html b/docs/content/second-order/analytical.html new file mode 100644 index 0000000..4ea25d6 --- /dev/null +++ b/docs/content/second-order/analytical.html @@ -0,0 +1,743 @@ + + + + + + + + 3.1. Analytical Solutions to 2nd-order ODEs — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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3.1. Analytical Solutions to 2nd-order ODEs

+

Let’s first focus on simple analytical solutions for 2nd-order ODEs. As before, let’s categorize problems based on their solution approach.

+
+

3.1.1. Solution by direct integration

+

If you have a 2nd-order ODE of this form:

+
+(3.1)\[\begin{equation} +\frac{d^2 y}{dx^2} = f(x) +\end{equation}\]
+

then you can solve by direct integration.

+

For example, let’s say we are trying to solve for the deflection of a cantilever beam \(y(x)\) with a force \(P\) at the end, where \(E\) is the modulus and \(I\) is the moment of intertia, and the initial conditions are \(y(0)=0\) and \(y^{\prime}(0) = 0\):

+

Cantilever beam with force at end

+
+(3.2)\[\begin{align} +\frac{d^2 y}{dx^2} &= \frac{-P (L-x)}{EI} \\ +\frac{d}{dx} \left(\frac{dy}{dx}\right) &= \frac{-P}{EI} (L-x) \\ +\int d \left(\frac{dy}{dx}\right) &= \frac{-P}{EI} \int (L-x) dx \\ +y^{\prime} = \frac{dy}{dx} &= \frac{-P}{EI} \left(Lx - \frac{x^2}{2}\right) + C_1 \\ +\int dy &= \int \left( \frac{-P}{EI} \left(Lx - \frac{x^2}{2}\right) + C_1 \right) dx \\ +y(x) &= \frac{-P}{EI} \left(\frac{L}{2} x^2 - \frac{1}{6} x^3\right) + C_1 x + C_2 +\end{align}\]
+

That is our general solution; we can obtain the specific solution by applying our two initial conditions:

+
+(3.3)\[\begin{align} +y(0) &= 0 = C_2 \\ +y^{\prime}(0) &= 0 = C_1 \\ +\therefore y(x) &= \frac{P}{EI} \left( \frac{x^3}{6} - \frac{L x^2}{2} \right) +\end{align}\]
+
+
+

3.1.2. Solution by substitution

+

If we have a 2nd-order ODE of this form:

+
+(3.4)\[\begin{equation} +\frac{d^2 y}{dx^2} = f(x, y^{\prime}) +\end{equation}\]
+

then we can solve by substitution, meaning by substituting a new variable for \(y^{\prime}\). (Notice that \(y\) itself does not show up in the ODE.)

+

Let’s substitute \(u\) for \(y^{\prime}\) in the above ODE:

+
+(3.5)\[\begin{align} +u &= y^{\prime} \\ +u^{\prime} &= y^{\prime\prime} \\ +\rightarrow u^{\prime} &= f(f, u) +\end{align}\]
+

Now we have a 1st-order ODE! Then, we can apply the methods previously discussed to solve this; once we find \(u(x)\), we can integrate that once more to get \(y(x)\).

+
+

3.1.2.1. Example: falling object

+

For example, consider a falling mass where we are solving for the downward distance as a function of time, \(y(t)\), that is experiencing the force of gravity downward and a drag force upward. It starts at some reference point so \(y(0) = 0\), and has a zero initial downward velocity: \(y^{\prime}(0) = 0\). The governing equation is:

+
+(3.6)\[\begin{equation} +m \frac{d^2 y}{dt^2} = mg - c \left( \frac{dy}{dt} \right)^2 +\end{equation}\]
+

where \(m\) is the mass, \(g\) is acceleration due to gravity, and \(c\) is the drag proportionality constant. +We can substitute \(V\) for \(y^{\prime}\), which gives us a first-order ODE:

+
+(3.7)\[\begin{align} +\text{let} \quad \frac{dy}{dt} = V \\ +\rightarrow m \frac{dV}{dt} &= mg - c V^2 \\ +\end{align}\]
+

Then, we can solve this for \(V(t)\) using our initial condition for velocity \(V(0) = 0\). Once we have that, we can integrate once more:

+
+(3.8)\[\begin{equation} +y(t) = \int V(t) dt +\end{equation}\]
+

and apply our initial condition for position, \(y(0) = 0\), to obtain \(y(t)\).

+

Here is the full process:

+
+(3.9)\[\begin{align} +\frac{dV}{dt} &= g - \frac{c}{m} V^2 \\ +\frac{dV}{g - \frac{c}{m} V^2} &= dt \\ +\frac{m}{c} \int \frac{dV}{a^2 - V^2} &= \int dt = t + \bar{c}, \quad \text{where} \quad a = \sqrt{\frac{mg}{c}} \\ +\frac{m}{c} \frac{1}{a} \tanh^{-1} \left(\frac{V}{a}\right) &= t + c_1 \\ +V &= a \tanh \left( \frac{a c}{m} t + c_1 \right) \\ +\therefore V(t) &= \sqrt{\frac{mg}{c}} \tanh \left(\sqrt{\frac{gc}{m}} t + c_1\right) +\end{align}\]
+

Applying the initial condition for velocity, \(V(0) = 0\):

+
+(3.10)\[\begin{align} +V(0) &= 0 = \sqrt{\frac{mg}{c}} \tanh \left(0 + c_1\right) \\ +\therefore c_1 &= 0 \\ +V(t) &= \sqrt{\frac{mg}{c}} \tanh \left(\sqrt{\frac{gc}{m}} t\right) +\end{align}\]
+

Then, to get \(y(t)\), we just need to integrate once more:

+
+(3.11)\[\begin{align} +\frac{dy}{dt} = V(t) &= \sqrt{\frac{mg}{c}} \tanh \left(\sqrt{\frac{gc}{m}} t\right) \\ +\int dy &= \sqrt{\frac{mg}{c}} \int \tanh \left(\sqrt{\frac{gc}{m}} t\right) dt \\ +y(t) &= \sqrt{\frac{mg}{c}} \sqrt{\frac{m}{gc}} \log\left(\cosh\left(\sqrt{\frac{gc}{m}} t\right)\right) + c_2 \\ +\rightarrow y(t) &= \frac{m}{c} \log\left(\cosh\left(\sqrt{\frac{gc}{m}} t\right)\right) + c_2 +\end{align}\]
+

Finally, we can apply the initial condition for position, \(y(0) = 0\), to get our solution:

+
+(3.12)\[\begin{align} +y(0) &= 0 = \frac{m}{c} \log\left(\cosh\left(0\right)\right) + c_2 = c_2 \\ +\rightarrow c_2 &= 0 \\ +y(t) &= \frac{m}{c} \log\left(\cosh\left(\sqrt{\frac{gc}{m}} t\right)\right) +\end{align}\]
+
+
+

3.1.2.2. Example: catenary problem

+

The catenary problem describes the shape of a hanging chain or rope fixed between two points. (It was also a favorite of one of my professors, Joe Prahl, and I like to teach it as an example in his honor.) The downward displacement of the hanging string/chain/rope as a function of horizontal position, \(y(x)\), is governed by the equation

+
+(3.13)\[\begin{equation} +y^{\prime\prime} = \sqrt{1 + (y^{\prime})^2} +\end{equation}\]
+

Catenary problem (hanging rope/chain)

+

This is actually a boundary value problem, with the boundary conditions for the displacement at one side \(y(0) = 0\) and that the slope is zero in the middle: \(\frac{dy}{dx}\left(\frac{L}{2}\right) = 0\). (Please note that I have skipped the derivation of the governing equation, and left some details out.)

+

We can solve this via substitution, by letting a new variable \(u = y^{\prime}\); then, \(u^{\prime} = \frac{du}{dx} = y^{\prime\prime}\). This gives is a first-order ODE, which we can integrate:

+
+(3.14)\[\begin{align} +\frac{du}{dx} &= \sqrt{1 + u^2} \\ +\int \frac{du}{\sqrt{1 + u^2}} &= \int dx \\ +\sinh^{-1}(u) &= x + c_1, \quad \text{where } \sinh(x) = \frac{e^x - e^{-x}}{2} \\ +u(x) &= \sinh(x + c_1) +\end{align}\]
+

Then, we can integrate once again to get \(y(x)\):

+
+(3.15)\[\begin{align} +\frac{dy}{dx} &= u(x) = \sinh(x + c_1) \\ +\int dy &= \int \sinh(x + c_1) dx = \int \left(\sinh(x)\cosh(c_1) + \cosh(x)\sinh(c_1)\right)dx \\ +y(x) &= \cosh(x)\cosh(c_1) + \sinh(x)\sinh(c_1) + c_2 \\ +\rightarrow y(x) &= \cosh(x + c_1) + c_2 +\end{align}\]
+

This is the general solution to the catenary problem, and applies to any boundary conditions.

+

For our specific case, we can apply the boundary conditions and find the particular solution, though it involves some algebra…:

+
+(3.16)\[\begin{align} +y(0) &= 0 = \cosh(c_1) + c_2 \\ +\frac{dy}{dx}\left(\frac{L}{2}\right) &= u(0) = \sinh \left(\frac{L}{2} + c_1\right) \\ +\rightarrow c_1 &= -\frac{L}{2} \\ +0 &= \cosh \left( -\frac{L}{2} \right) + c_2 \\ +\rightarrow c_2 &= -\cosh\left( -\frac{L}{2} \right) = -\cosh\left(\frac{L}{2} \right) +\end{align}\]
+

So, the overall solution for the catenary problem with the given boundary conditions is

+
+(3.17)\[\begin{equation} +y(x) = \cosh \left(x - \frac{L}{2}\right) - \cosh\left( \frac{L}{2} \right) +\end{equation}\]
+

Let’s see what this looks like:

+
+
+
L = 1.0;
+x = linspace(0, 1);
+y = cosh(x - (L/2.)) - cosh(L/2.);
+plot(x, y)
+
+
+
+
+../../_images/analytical_4_0.png +
+
+

Please note that I’ve made some simplifications in the above work, and skipped the details of how the ODE is derived. In general, the solution for the shape is

+
+(3.18)\[\begin{equation} +y(x) = C \cosh \frac{x + c_1}{C} + c_2 +\end{equation}\]
+

where you would solve for the constants \(C\), \(c_1\), and \(c_2\) using the constraints:

+
+(3.19)\[\begin{align} +\int_{x_a}^{x_b} \sqrt{1 + (y^{\prime})^2} dx &= L \\ +y(x_a) &= y_a \\ +y(x_b) &= y_b \;, +\end{align}\]
+

where \(L\) is the length of the rope/chain.

+

You can read more about the catenary problem here (for example): http://euclid.trentu.ca/aejm/V4N1/Chatterjee.V4N1.pdf

+
+
+
+

3.1.3. Homogeneous 2nd-order ODEs

+

An important category of 2nd-order ODEs are those that look like

+
+(3.20)\[\begin{equation} +y^{\prime\prime} + p(x) y^{\prime} + q(x) y = 0 +\end{equation}\]
+

“Homogeneous” means that the ODE is unforced; that is, the right-hand side is zero.

+

Depending on what \(p(x)\) and \(q(x)\) look like, we have a few different solution approaches:

+
    +
  • constant coefficients: \(y^{\prime\prime} + a y^{\prime} + by = 0\)

  • +
  • Euler-Cauchy equations: \(x^2 y^{\prime\prime} + axy^{\prime} + by = 0\)

  • +
  • Series solutions

  • +
+

First, let’s talk about the characteristics of linear, homogeneous 2nd-order ODEs:

+
+

3.1.3.1. Solutions have two parts:

+

Solutions have two parts: \(y(x) = c_1 y_1 + c_2 y_2\), where \(y_1\) and \(y_2\) are each a basis of the solution.

+
+
+

3.1.3.2. Linearly independent:

+

The two parts of the solution \(y_1\) and \(y_2\) are linearly independent.

+

One way of defining this is that \(a_1 y_1 + a_2 y_2 = 0\) only has the trivial solution \(a_1=0\) and \(a_2=0\).

+

Another way of thinking about this is that \(y_1\) and \(y_2\) are linearly dependent if one is a multiple of the other, like \(y_1 = x\) and \(y_2 = 5x\). This cannot be solutions to a linear, homogeneous ODE.

+
+
+

3.1.3.3. Both parts satisfy the ODE:

+

\(y_1\) and \(y_2\) each satisfy the ODE. Meaning, you can plug each of them into the ODE for \(y\) and obtain 0.

+

However, we need both parts together to fully solve the ODE.

+
+
+

3.1.3.4. Reduction of order:

+

If \(y_1\) is known, we can get \(y_2\) by reduction of order. Let \(y_2 = u y_1\), where \(u\) is some unknown function of \(x\). Then, put \(y_2\) into the ODE \(y^{\prime}{\prime} + p(x) y^{\prime} + q(x) y = 0\):

+
+(3.21)\[\begin{align} +y_2 &= u y_1 \\ +y_2^{\prime} &= u y_1^{\prime} + u^{\prime} y_1 \\ +y_2^{\prime\prime} &= 2 u^{\prime} y_1^{\prime} + u^{\prime\prime} y_1 + u y_1^{\prime\prime} \\ +\rightarrow u^{\prime\prime} &= - \left[ p(x) + \left(\frac{2 y_1^{\prime}}{y_1}\right) \right] u^{\prime} +\text{or, } u^{\prime\prime} &= - \left( g(x) \right) u^{\prime} +\end{align}\]
+

Now, we have an ODE with only \(u^{\prime\prime}\), \(u^{\prime}\), and some function \(g(x)\)—so we can solve by substitution! Let \(u^{\prime} = v\), and then we have \(v^{\prime} = -g(x) v\):

+
+(3.22)\[\begin{align} +\frac{dv}{dx} &= - \left( p(x) + \frac{2 y_1^{\prime}}{y_1} \right) v \\ +\int \frac{dv}{v} &= - \int \left(p(x) + \frac{2 y_1^{\prime}}{y_1} \right) dx \\ +\text{Recall } 2 \frac{d}{dx} \left( \ln y_1 \right) &= 2 \frac{y_1^{\prime}}{y_1} \\ +\therefore \int \frac{dv}{v} &= - \int \left(p(x) + 2 \frac{d}{dx} \left( \ln y_1 \right) \right) dx \\ +\ln v &= -\int p(x) dx - 2 \ln y_1 \\ +\rightarrow v &= \frac{\exp\left( -\int p(x)dx \right)}{y_1^2} +\end{align}\]
+

So, the actual solution procedure is then:

+
    +
  1. Solve for \(v\): \(v = \frac{\exp\left( -\int p(x)dx \right)}{y_1^2}\)

  2. +
  3. Solve for \(u\): \(u = \int v dx\)

  4. +
  5. Get \(y_2\): \(y_2 = u y_1\)

  6. +
+

Here’s an example, where we know one part of the solution \(y_1 = e^{-x}\):

+
+(3.23)\[\begin{align} +y^{\prime\prime} + 2 y^{\prime} + y &= 0 \\ +\text{Step 1:} \quad v = \frac{\exp \left( -\int 2dx \right)}{ \left(e^{-x}\right)^2} = \frac{e^{-2x}}{e^{-2x}} &= 1 \\ +\text{Step 2:} \quad u = \int v dx = \int 1 dx &= x \\ +\text{Step 3:} \quad y_2 &= x e^{-x} +\end{align}\]
+

Then, the general solution to the ODE is \(y(x) = c_1 e^{-x} + c_2 x e^{-x}\).

+
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+ + + + + + \ No newline at end of file diff --git a/docs/content/second-order/fourier-series.html b/docs/content/second-order/fourier-series.html new file mode 100644 index 0000000..bfde201 --- /dev/null +++ b/docs/content/second-order/fourier-series.html @@ -0,0 +1,838 @@ + + + + + + + + 3.4. Fourier Series — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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3.4. Fourier Series

+

Fourier series are a method we can use to solve inhomogeneous 2nd-order ODEs of the form

+
+(3.72)\[\begin{equation} +y^{\prime\prime} + p(t) y^{\prime} + q(t) y = r(t) \;, +\end{equation}\]
+

where the forcing function \(r(t)\) is periodic. This means looking like one of these examples:

+
+
+
% periodic square wave
+subplot(3,1,1);
+squareWave = repmat([1,1,1,1,0,0,0,0], [1, 10]);
+t = linspace(0, 1, length(squareWave));
+plot(t, squareWave); ylim([-0.5, 1.5]);
+
+% sin wave
+subplot(3,1,2);
+t = linspace(0, 1, 100);
+plot(t, sin(t*10*pi)); ylim([-1.5, 1.5]);
+
+% sawtooth
+subplot(3,1,3);
+t = linspace(0, 1, 100);
+y = ((mod(t,2*pi/40)/(pi*2/40))*2)-1;
+plot(t, y); ylim([-1.5, 1.5]);
+
+
+
+
+../../_images/fourier-series_1_0.png +
+
+

(Actually, as we’ll see later, we can use a Fourier series to represent generic forcing function!)

+

Fourier series have been around a while, ever since in 1790 Jean-Baptiste Joseph Fourier found that generic periodic functions could be represented by a sum of series of sin() and cos() functions, harmonically related by frequency.

+

In general, a Fourier series represents a function \(f(t)\) by

+
+(3.73)\[\begin{equation} +f(t) = a_0 + \sum_{n=1}^{\infty} a_n \cos (n \omega t) + \sum_{n=1}^{\infty} b_n \sin (n \omega t) +\end{equation}\]
+

where \(a_0\), \(a_n\), and \(b_n\) are the Fourier coefficients, \(\omega = \frac{2\pi}{T}\) is the frequency of the function \(f(t)\), and \(T\) is the period. \(n\) is an integer used as an index.

+
+

3.4.1. Properties of Fourier Series

+

Considering that \(n\) is an integer, and the sine and cosine components of a Fourier series share the same fundamental frequency \(\omega\), Fourier series have some useful properties:

+
    +
  1. The integral of each component trigonometric function over the period is zero:

  2. +
+
+(3.74)\[\begin{equation} +\int_0^T \sin (n \omega t) dt = 0 = \int_0^T cos (n \omega t) dt +\end{equation}\]
+
    +
  1. The component trigonometric functions are orthogonal over their period:

  2. +
+
+(3.75)\[\begin{equation} +\int_0^T \cos(n \omega t) \sin (m \omega t) dt = 0 +\end{equation}\]
+

for all values of \(n, m = 1, 2, \ldots, \infty\).

+
    +
  1. The component trigonometric functions are also orthogonal with themselves over their period:

  2. +
+
+(3.76)\[\begin{align} +\int_0^T \cos (n \omega t) \cos (m \omega t) dt &= \begin{cases}0 \quad m \neq n \\ \frac{T}{2} \quad m = n \end{cases} \\ +% +\int_0^T \sin (n \omega t) \sin (m \omega t) dt &= \begin{cases}0 \quad m \neq n \\ \frac{T}{2} \quad m = n \end{cases} +\end{align}\]
+

for all values of \(n, m = 1, 2, \ldots, \infty\).

+
+
+

3.4.2. Fourier coefficients

+

We can use the above properties to calculate the Fourier coefficients, given a periodic function \(f(t)\). +First, recall

+
+(3.77)\[\begin{equation} +f(t) = a_0 + \sum_{n=1}^{\infty} a_n \cos (n \omega t) + \sum_{n=1}^{\infty} b_n \sin (n \omega t) +\end{equation}\]
+
    +
  1. To calculate \(a_0\), integrate both sides of the equation over the period:

  2. +
+
+(3.78)\[\begin{equation} +\int_0^T f(t) dt = \int_0^T a_0 dt + \int_0^T \left( \sum_{n=1}^{\infty} a_n \cos (n \omega t) \right)dt + \int_0^T \left( \sum_{n=1}^{\infty} b_n \sin (n \omega t) \right) dt +\end{equation}\]
+

For the integrals of the infinite sums, recall that the integral of the sum of some functions is the same as the sum of the integrals of the functions: \(\int (a + b + c) = \int a + \int b + \int c\). That means that

+
+(3.79)\[\begin{align} +\int_0^T \left( \sum_{n=1}^{\infty} a_n \cos (n \omega t) \right)dt = \int_0^T a_1 \cos (\omega t) dt + \int_0^T a_2 \cos (2 \omega t) dt + \ldots &= 0 \;, \text{ and} \\ +\int_0^T \left( \sum_{n=1}^{\infty} b_n \sin (n \omega t) \right)dt = \int_0^T b_1 \sin (\omega t) dt + \int_0^T b_2 \sin (2 \omega t) dt + \ldots &= 0 \;, +\end{align}\]
+

since the integrals of the trigonometric functions are all zero over the period. Thus,

+
+(3.80)\[\begin{equation} +a_0 = \frac{1}{T} \int_0^T f(t) dt +\end{equation}\]
+
    +
  1. To calculate \(a_n\), multiply both sides of the equation by \(\cos(m \omega t)\) and integrate over the period:

  2. +
+
+(3.81)\[\begin{equation} +\int_0^T f(t) \cos(m \omega t) dt = a_0 \int_0^T \cos(m \omega t) dt + \int_0^T \left( \sum_{n=1}^{\infty} a_n \cos (n \omega t) \cos(m \omega t) \right)dt + \int_0^T \left( \sum_{n=1}^{\infty} b_n \sin (n \omega t) \cos(m \omega t) \right) dt +\end{equation}\]
+

Let’s take a look at each of the three integrals on the right-hand side. First,

+
+(3.82)\[\begin{equation} +a_0 \int_0^T \cos(m \omega t) dt = 0 +\end{equation}\]
+

because it just integrates cosine over the period. +Skipping to the last term,

+
+(3.83)\[\begin{equation} +\int_0^T \left( \sum_{n=1}^{\infty} b_n \sin (n \omega t) \cos(m \omega t) \right) dt = b_1 \int_0^T sin(\omega t) \cos (m \omega t) dt + b_2 \int_0^T \sin (2 \omega t) \cos (m \omega t) dt + \ldots = 0 +\end{equation}\]
+

due to orthogonality. We are just left with the middle integral; let’s expand a few terms to see what that looks like:

+
+(3.84)\[\begin{equation} +\int_0^T \left( \sum_{n=1}^{\infty} a_n \cos (n \omega t) \cos(m \omega t) \right)dt = a_1 \int_0^T \cos (\omega t) \cos (m \omega t) dt + a_2 \int_0^T \cos (2 \omega t) \cos (m \omega t) dt + \ldots +\end{equation}\]
+

Again, due to orthogonality, all of the terms of this infinite sum of integrals will be zero, except for the term where \(n = m\). As a result, we are left with

+
+(3.85)\[\begin{align} +\int_0^T f(t) \cos(m \omega t) dt &= \int_0^T a_m \cos^2 (m \omega t) dt = a_m \frac{T}{2} \\ +a_n = a_m &= \frac{2}{T} \int_0^T f(t) \cos (n \omega t) dt +\end{align}\]
+
    +
  1. We can find \(b_n\) in the same way, multiplying the equation by \(\sin (m \omega t)\) and integrating over the period. The work is the same, so let’s skip that:

  2. +
+
+(3.86)\[\begin{equation} +b_n = \frac{2}{T} \int_0^T f(t) \sin (n \omega t) dt +\end{equation}\]
+
+
+

3.4.3. Example: periodic rectangular wave

+

Let’s find the Fourier series for representing this periodic function \(f(t)\):

+
+
+
x = [0 1 1 2 2 3 3 4]; y = [2 2 1 1 2 2 1 1];
+plot(x,y); ylim([0 2.5]); ylabel('f(t)'); xlabel('t')
+
+
+
+
+../../_images/fourier-series_6_0.png +
+
+

First, we need to identify the fundamental period and frequency: \(T = 2\) and then \(\omega = \frac{2\pi}{T} = \pi\). Our work is then to calculate the Fourier coefficients. Since our periodic function \(f(t)\) is not easily expressed with a function–hence the need for a Fourier series—we’ll use piecewise integration.

+

First, calculate \(a_0\):

+
+(3.87)\[\begin{align} +a_0 =& \frac{1}{T} \int_0^T f(t) dt \\ +&= \frac{1}{2} \int_0^2 f(t) dt = \frac{1}{2}\left( \int_0^1 2dt + \int_1^2 1dt \right) = \frac{1}{2} (2\times 1 + 1 \times 1) \\ +a_0 &= \frac{3}{2} +\end{align}\]
+

Then, get \(a_n\):

+
+(3.88)\[\begin{align} +a_n &= \frac{2}{T} \int_0^T f(t) \cos (n \omega t) dt \\ +&= \frac{2}{2} \int_0^2 f(t) \cos (n \pi t) dt = \left( \int_0^1 2 \cos (n \pi t) dt + \int_1^2 1 \cos (n \pi t)dt \right) \\ +&= \frac{2}{n \pi} \sin(n \pi t)\Big|_0^1 + \frac{1}{n\pi} \sin(n \pi t)\Big|_1^2 \\ +&= \frac{2}{n \pi}\left(\sin(n\pi) - \sin(0)\right) + \frac{1}{n\pi}\left( sin(2n\pi) - \sin(n\pi)\right) \\ +a_n &= 0 +\end{align}\]
+

Finally, we can calculate \(b_n\):

+
+(3.89)\[\begin{align} +b_n &= \frac{2}{T} \int_0^T f(t) \sin (n \omega t) dt \\ +&= \frac{2}{2} \int_0^2 f(t) \sin (n \pi t) dt = \left( \int_0^1 2 \sin (n \pi t) dt + \int_1^2 1 \sin (n \pi t)dt \right) \\ +&= -\frac{2}{n \pi} \cos(n \pi t)\Big|_0^1 - \frac{1}{n\pi} \cos(n \pi t)\Big|_1^2 \\ +&= -\frac{2}{n \pi}\left(\cos(n\pi) - \cos(0)\right) - \frac{1}{n\pi}\left( cos(2n\pi) - \cos(n\pi)\right) \\ +b_n &= -\frac{2}{n \pi}\left(\cos(n\pi) - 1\right) - \frac{1}{n\pi}\left( 1 - \cos(n\pi)\right) = -\frac{1}{n\pi}\left( \cos(n\pi) - 1\right) +\end{align}\]
+

but recall that \(n = 1, 2, \ldots, \infty\). Then,

+
+(3.90)\[\begin{align} +b_n &= -\frac{1}{n\pi} \times \begin{cases} -2 \text{ if } n \text{ odd} \\0 \text{ if } n \text{ even}\end{cases} \\ +\rightarrow b_n &= \frac{2}{n\pi} \quad n = \text{odd} +\end{align}\]
+

Then, our Fourier series representation for the function shown above is

+
+(3.91)\[\begin{equation} +f(t) = \frac{3}{2} + \sum_{\substack{n = 1 \\n = \text{odd}}}^{\infty} \frac{2}{n\pi} \sin (n \pi t) +\end{equation}\]
+

Now, let’s see how whether this actually works! Let’s start with one term of the infinite sum, then gradually increase.

+
+
+
t = linspace(0, 4);
+
+% maximum number of terms
+n_max = [1, 2, 3, 5, 10, 25, 50, 250, 500];
+
+for i = 1 : length(n_max)
+    N = n_max(i);
+
+    s = 3./2.;
+    for n = 1 : 2 : 2*N
+        s = s + (2. / (n*pi)) .* sin(n * pi * t);
+    end
+    subplot(3, 3, i)
+    plot(t, s); axis([0 4 0 3]); title(sprintf('%d terms', N));
+end
+
+
+
+
+../../_images/fourier-series_8_0.png +
+
+

As we increase the number of terms, adding higher-frequency sine waves, we are better able to match the original rectangular wave. Notice the discrepancies that remain near the sharp corners even after the rest of the series closely resembles the function: these are known as Gibbs phenomena, caused by the Fourier series overshooting or undershooting (or “ringing”) near discontinuities.

+
+
+

3.4.4. Even and Odd Functions

+

We can simplify our work generating a Fourier series if we can identify the given periodic function \(f(t)\) as an even function or an odd function.

+

Even functions are those where \(f(-x) = f(x)\).

+

Odd functions are those where \(f(-x) = -f(x)\).

+
+
+
y = [-1 0 1 0 -1];
+x = [-2 -1 0 1 2];
+subplot(2,1,1); plot(x,y); title('Even function');
+ax = gca; ax.XAxisLocation = 'origin'; ax.YAxisLocation = 'origin';
+
+y = [0 -1 0 1 0];
+subplot(2,1,2); plot(x,y); title('Odd function');
+ax = gca; ax.XAxisLocation = 'origin'; ax.YAxisLocation = 'origin';
+
+
+
+
+../../_images/fourier-series_11_0.png +
+
+

An even function’s Fourier series simplifies to a Fourier cosine series:

+
+(3.92)\[\begin{align} +f(x) &= a_0 + \sum_{n=1}^{\infty} a_n \cos (n \omega x) dx \\ +a_0 &= \frac{2}{T} \int_0^{T/2} f(x) dx \\ +a_n &= \frac{4}{T} \int_0^{T/2} f(x) \cos(n \omega x) dx +\end{align}\]
+

An odd function’s Fourier series simplifies to a Fourier sine series:

+
+(3.93)\[\begin{align} +f(x) &= \sum_{n=1}^{\infty} b_n \sin (n \omega x) dx \\ +b_n &= \frac{4}{T} \int_0^{T/2} f(x) \sin(n \omega x) dx +\end{align}\]
+

Note: not all periodic functions can be considered an even or an odd function.

+
+
+

3.4.5. Application: Inhomogeneous 2nd-order ODE

+

One way we might use a Fourier series is to solve an inhomogeneous 2nd-order ODE, where the forcing term is given by a periodic function not easily expressed using our typical functions.

+
+

3.4.5.1. Undamped mass-spring system

+

For example, let’s consider an undamped mass-spring system, where the forcing is given by a periodic rectangular wave:

+
+(3.94)\[\begin{equation} +y^{\prime\prime} + 4y = f(t) +\end{equation}\]
+

where the forcing function \(f(t)\) is

+
+
+
t = [0 0 1 1 2 2 3 3];
+y = [0 1 1 -1 -1 1 1 0];
+plot(t,y); ylim([-1.5 1.5]);
+title('Periodic rectangular forcing function');
+ax = gca; ax.XAxisLocation = 'origin';
+
+
+
+
+../../_images/fourier-series_14_0.png +
+
+

Using recognizing this as an odd function, we could construct a Fourier sine series to represent the forcing function:

+
+(3.95)\[\begin{equation} +f(t) = \sum_{\substack{n=1\\n=\text{odd}}}^{\infty} \frac{4}{n\pi} \sin(n \pi t) +\end{equation}\]
+

Let’s confirm this works:

+
+
+
t = linspace(0, 3, 500);
+s = 0;
+for n = 1 : 2 : 1000
+    s = s + (4/(n*pi)).*sin(n*pi*t);
+end
+plot(t, s)
+
+
+
+
+../../_images/fourier-series_16_0.png +
+
+

Looks good!

+

To find the exact solution for our displacement \(y(t)\), we can follow our usual analytical solution approach: find the homogeneous solution \(y_H\), then find the inhomogeneous solution \(y_{IH}\); the overall solution is then \(y(t) = y_H + y_{IH}\). The homogeneous solution is

+
+(3.96)\[\begin{equation} +y_H = c_1 \sin (2t) + c_2 \cos (2t) +\end{equation}\]
+

We then find the inhomogeneous solution using

+
+(3.97)\[\begin{equation} +y^{\prime\prime} + 4y = \frac{4}{n\pi} \sin (n \pi t) \quad n = 1, 3, \ldots, \infty +\end{equation}\]
+

Solving this will give us a specific \(y_{IH, n}\); the complete inhomogeneous solution is then

+
+(3.98)\[\begin{equation} +y_{IH} = \sum_{\substack{n=1\\n=\text{odd}}}^{\infty} y_{IH, n} \;. +\end{equation}\]
+

Recognizing that our forcing function is sinusoidal, we should use the method of undetermined coefficients:

+
+(3.99)\[\begin{equation} +y_{IH, n} = K_1 \sin (n \pi t) + K_2 \cos (n \pi t) +\end{equation}\]
+

Inserting this into the above ODE gives

+
+(3.100)\[\begin{align} +K_1 &= \frac{4}{n \pi (4 - n^2 \pi^2)} \\ +K_2 &= 0 +\end{align}\]
+

Thus, the overall solution is

+
+(3.101)\[\begin{equation} +y(t) = c_1 \sin(2t) + c_2 \cos(2t) + \sum_{\substack{n=1\\n=\text{odd}}}^{\infty} \frac{4}{n \pi (4 - n^2 \pi^2)} \sin (n \pi t) +\end{equation}\]
+
+
+

3.4.5.2. Damped mass-spring system

+

What about a damped mass-spring system? Recall that the homogeneous solution could take one of these three forms:

+
+(3.102)\[\begin{align} +y_H &= c_1 e^{-\lambda_1 t} + c_2 e^{-\lambda_2 t} \\ +y_H &= c_1 e^{-\lambda_1 t} + c_2 t e^{-\lambda_2 t} \text{ or} \\ +y_H &= e^{-\alpha t} (c_1 \sin(\beta t) + c_2 \cos(\beta t)) +\end{align}\]
+

while the inhomogeneous solution, given a Fourier series forcing function, will take the form

+
+(3.103)\[\begin{equation} +y_{IH} = K_1 \sin() + K_2 \cos() +\end{equation}\]
+

The overall solution combines the homogenenous and inhomogeneous solutions. But, the homogeneous solution in this case is transient, because it decays to zero. On the other hand, the inhomogeneous solution remains, and is the steady-state solution.

+
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/second-order/initial-value-problems.html b/docs/content/second-order/initial-value-problems.html new file mode 100644 index 0000000..6bd59a0 --- /dev/null +++ b/docs/content/second-order/initial-value-problems.html @@ -0,0 +1,796 @@ + + + + + + + + 3.2. Initial-Value Problems — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ + +
+
+ +
+ +
+

3.2. Initial-Value Problems

+

This section focuses on analytical solutions for initial-value problems, meaning problems where we know the values of \(y(t)\) and \(\frac{dy}{dt}\) at \(t=0\) (or \(y(x)\) and \(\frac{dy}{dx}\) at \(x=0\)): \(y(0)\) and \(y^{\prime}(0)\).

+
+

3.2.1. Equations with constant coefficents

+

A common category of 2nd-order homogeneous ODEs are equations with constant coefficients, of the form:

+
+(3.24)\[\begin{equation} +y^{\prime\prime} + a y^{\prime} + by = 0 +\end{equation}\]
+

Note that these are unforced, and the right-hand side is zero.

+

Solutions to these equations take the form \(y(x) = e^{\lambda x}\), and inserting this into the ODE gives us the characteristic equation

+
+(3.25)\[\begin{equation} +\lambda^2 + a \lambda + b = 0 +\end{equation}\]
+

which we can solve to find the solution for given coefficients \(a\) and \(b\) and initial conditions. Depending on those coefficients and the solution to the characteristic equation, our solution can fall into one of three cases:

+
    +
  • Real roots: \(\lambda_1\) and \(\lambda_2\). This is an overdamped system and the full solution takes the form

  • +
+
+(3.26)\[\begin{equation} +y(x) = c_1 e^{\lambda_1 x} + c_2 e^{\lambda_2 x} +\end{equation}\]
+
    +
  • Repeated roots: \(\lambda_1 = \lambda_2 = \lambda\). This is a critically damped system and the full solution is

  • +
+
+(3.27)\[\begin{equation} +y(x) = c_1 e^{\lambda x} + c_2 x e^{\lambda x} +\end{equation}\]
+

(Where does that second part come from, you might ask? Well, we know that \(y_1\) is \(e^{\lambda x}\), but the second part cannot also be \(e^{\lambda x}\) because those are linearly dependent. So, we use reduction of order to find \(y_2\), which is \(x e^{\lambda x}\).

+
    +
  • Imaginary roots: \(\lambda = \frac{-a}{2} \pm \beta i\), where \(\beta = \frac{1}{2} \sqrt{4b - a^2}\). This is an underdamped system and the solution takes the form

  • +
+
+(3.28)\[\begin{equation} +y(x) = e^{-ax/2} \left( c_1 \sin \beta x + c_2 \cos \beta x \right) +\end{equation}\]
+

Some examples:

+
    +
  1. \(y^{\prime\prime} + 3 y^{\prime} + 2y = 0\)

  2. +
+
+(3.29)\[\begin{align} +\rightarrow \lambda^2 + 3\lambda + 2 &= 0 \\ +(\lambda + 2)(\lambda + 1) &= 0 \\ +\lambda &= -2, -1 \\ +y(x) &= c_1 e^{-x} + c_2 e^{-2x} +\end{align}\]
+

Then, we would use the initial conditions given for \(y(0)\) and \(y^{\prime}(0)\) to find \(c_1\) and \(c_2\).

+
    +
  1. \(y^{\prime\prime} + 6 y^{\prime} + 9y = 0\)

  2. +
+
+(3.30)\[\begin{align} +\rightarrow \lambda^2 + 6\lambda + 9 &= 0 \\ +(\lambda + 3)(\lambda + 3) &= 0 \\ +\lambda &= -3 \\ +y(x) &= c_1 e^{-3x} + c_2 x e^{-3x} +\end{align}\]
+
    +
  1. \(y^{\prime\prime} + 6 y^{\prime} + 25 y = 0\)

  2. +
+
+(3.31)\[\begin{align} +\rightarrow \lambda^2 + 6\lambda + 25 &= 0 \\ +\lambda &= -3 \pm 4i \\ +y(x) &= e^{-3x} \left( c_1 \sin 4x + c_2 \cos 4x \right) +\end{align}\]
+
+
+

3.2.2. Euler-Cauchy equations

+

Euler-Cauchy equations are of the form

+
+(3.32)\[\begin{equation} +x^2 y^{\prime\prime} + axy^{\prime} + by = 0 +\end{equation}\]
+

Solutions take the form \(y = x^m\), which when plugged into the ODE leads to a different characterisic equation to find \(m\):

+
+(3.33)\[\begin{align} +y &= x^m \\ +y^{\prime} &= m x^{m-1} \\ +y^{\prime\prime} &= m (m-1) x^{m-2} \\ +\rightarrow x^2 m (m-1) x^{m-2} + axmx^{m-1} + bx^m &= 0 \\ +m^2 + (a-1)m + b &= 0 +\end{align}\]
+

This is our new characteristic formula for these problems, and solving for the roots of this equation gives us \(m\) and thus our general solution.

+

Like equations with constant coefficients, we have three solution forms depending on the roots of the characteristic equation:

+
    +
  • Real roots: \(y(x) = c_1 x^{m_1} + c_2 x^{m_2}\)

  • +
  • Repeated roots: \(y(x) = c_1 x^m + c_2 x^m \ln x\)

  • +
  • Imaginary roots: \(m = \alpha \pm \beta i\), and \(y(x) = x^{\alpha} \left[c_1 \cos (\beta \ln x) + c_2 \sin (\beta \ln x)\right]\)

  • +
+
+
+

3.2.3. Inhomogeneous 2nd-order ODEs

+

Inhomogeneous, or forced, 2nd-order ODEs with constant coefficients take the form

+
+(3.34)\[\begin{equation} +y^{\prime\prime} + a y^{\prime} + by = F(t) +\end{equation}\]
+

with initial conditions \(y(0) = y_0\) and \(y^{\prime}(0) = y_0^{\prime}\). Depending on the form of the forcing function \(F(t)\), we can solve with techniques such as

+
    +
  • the method of undetermined coefficients

  • +
  • variation of parameters

  • +
  • LaPlace transforms

  • +
+

The solution in general to inhomogeneous ODEs includes two parts:

+
+(3.35)\[\begin{equation} +y(t) = y_{\text{H}} + y_{\text{IH}} = c_1 y_1 + c_2 y_2 + y_{\text{IH}} \;, +\end{equation}\]
+

where \(y_{\text{H}}\) is the solution from the equivalent homogeneous ODE \(y^{\prime\prime} + a y^{\prime} + b y = 0\).

+

The forcing function \(F(t)\) may be

+
    +
  • continuous

  • +
  • periodic

  • +
  • aperiodic/discontinuous

  • +
+
+

3.2.3.1. Continuous \(F(t)\): method of undetermined coefficients

+

For continuous forcing functions, we have two solution methods: the method of undetermined coefficients, and variation of parameters.

+

Generally you’ll want to use the method of undetermined coefficients when possible, which depends on if \(F(t)\) matches one of a set of functions. In that case, the form of the inhomogeneous solution \(y_{\text{IH}}(t)\) follows that of the forcing function \(F(t)\), with one or more unknown constants:

+ + + + + + + + + + + + + + + + + + + + + + + + + + +

\(F(t)\)

\(y_{\text{IH}}(t)\)

constant

\(K\)

\(\cos \omega t\)

\(K_1 \cos \omega t + K_2 \sin \omega t\)

\(\sin \omega t\)

\(K_1 \cos \omega t + K_2 \sin \omega t\)

\(e^{-at}\)

\(K e^{-at}\)

\((A) t\)

\(K_0 + K_1 t\)

\(t^n\)

\(K_0 + K_1 t + K_2 t^2 + \ldots + K_n t^n\)

+

For combinations of these functions, we can combine functions; for example, given

+
+(3.36)\[\begin{align} +F(t) &= e^{-at} \cos \omega t \quad \text{or} e^{-at} \sin \omega t \\ +y_{\text{IH}} &= K_1 e^{-at} \cos \omega t + K_2 e^{-at} \sin \omega t +\end{align}\]
+

(Note how in all the above cases how the inhomogeneous solution follows the functional form of the forcing function; for example, the exponential decay rate \(a\) or the sinusoidal frequency \(\omega\) match.

+

The method of undetermined coefficients works by plugging the candidate inhomogeneous solutionn \(y_{\text{IH}}\) into the full ODE, and solving for the constants (e.g., \(K\))—but not from the initial conditions.

+

For example, let’s solve

+
+(3.37)\[\begin{equation} +y^{\prime\prime} + 2y^{\prime} + y = e^{-x} +\end{equation}\]
+

with initial conditions \(y(0) = y^{\prime}(0) = 0\). First, we should find the solution to the homogeneous equation

+
+(3.38)\[\begin{equation} +y^{\prime\prime} + 2y^{\prime} + y = 0 \;. +\end{equation}\]
+

We can do this by using the associated characteristic formula

+
+(3.39)\[\begin{align} +\lambda^2 + 2 \lambda + 1 &= 0 \\ +(\lambda + 1)(\lambda + 1) &= 0 \\ +\rightarrow y_{\text{H}} &= c_1 e^{-x} + c_2 x e^{-x} +\end{align}\]
+

To find the inhomogeneous solution, we would look at the table above to find what matches the forcing function \(e^{-x}\). Normally, we’d grab \(K e^{-x}\), but that would not be linearly independent from the first part of the homogeneous solution \(y_{\text{H}}\). The same is true for \(K x e^{-x}\), which is linearly dependent with the second part of \(y_{\text{H}}\), but \(K x^2 e^{-x}\) works! Then, we just need to find \(K\) by plugging this into the ODE:

+
+(3.40)\[\begin{align} +y_{\text{IH}} &= K x^2 e^{-x} \\ +y^{\prime} &= K e^{-x} (2x - x^2) \\ +y^{\prime\prime} &= K e^{-x} (x^2 - 4x + 2) \\ +2 K &= 1 \\ +\rightarrow K &= \frac{1}{2} \\ +y_{\text{IH}} &= \frac{1}{2} x^2 e^{-x} +\end{align}\]
+

Thus, the overall general solution is

+
+(3.41)\[\begin{equation} +y(x) = c_1 e^{-x} + c_2 x e^{-x} + \frac{1}{2} x^2 e^{-x} +\end{equation}\]
+

and we would solve for the integration constants \(c_1\) and \(c_2\) using the initial conditions.

+

Important points to remember:

+
    +
  • The constants of the inhomogeneous solution \(y_{\text{IH}}\) come from the ODE, not the initial conditions.

  • +
  • Only solve for the integration constants \(c_1\) and \(c_2\) (part of the homogeneous solution) once you have the full general solution \(y = c_1 y_1 + c_2 y_2 + y_{\text{IH}}\).

  • +
+
+
+

3.2.3.2. Continuous \(F(t)\): variation of parameters

+

We have the variation of parameters approach to solve for inhomogeneous 2nd-order ODEs that are more general:

+
+(3.42)\[\begin{equation} +y^{\prime\prime} + p(x) y^{\prime} + q(x) y = r(x) +\end{equation}\]
+

In this case, we can assume a solution \(y(x) = y_1 u_1 + y_2 u_2\).

+

The solution procedure is:

+
    +
  1. Obtain \(y_1\) and \(y_2\) by solving the homogeneous equation: \(y^{\prime\prime} + p(x) y^{\prime} + q(x) y = 0\)

  2. +
  3. Solve for \(u_1\) and \(u_2\):

  4. +
+
+(3.43)\[\begin{align} +u_1 &= - \int \frac{y_2 r(x)}{W} dx + c_1 \\ +u_2 &= \int \frac{y_1 r(x)}{W} dx + c_2 \\ +W &= \begin{vmatrix} +y_1 & y_2\\ y_1^{\prime} & y_2^{\prime}\\ +\end{vmatrix} = y_1 y_2^{\prime} - y_2 y_1^{\prime} \;, +\end{align}\]
+

where \(W\) is the Wronksian.

+
    +
  1. Then, we have the general solution:

  2. +
+
+(3.44)\[\begin{align} +y &= u_1 y_1 + u_2 y_2 \\ +&= \left( -\int \frac{y_2 r(x)}{W} dx + c_1 \right) y_1 + \left( \int \frac{y_1 r(x)}{W} dx + c_2 \right) y_2 \;, +\end{align}\]
+

where we solve for \(c_1\) and \(c_2\) using the two initial conditions.

+
+

3.2.3.2.1. Example 1: variation of parameters

+

First, let’s try the same example we used for the method of undetermined coefficients above:

+
+(3.45)\[\begin{equation} +y^{\prime\prime} + 2 y^{\prime} + y = e^{-x} +\end{equation}\]
+

We already found the homogeneous solution, so we know that \(y_1 = e^{-x}\) and \(y_2 = x e^{-x}\). +Next, let’s get the Wronksian, and then \(u_1\) and \(u_2\).

+
+(3.46)\[\begin{align} +W &= \begin{vmatrix} y_1 & y_2 \\ y_1^{\prime} & y_2^{\prime} \end{vmatrix} = e^{-x} e^{-x}(1-x) - x e^{-x} (-e^{-x}) = e^{-2x} \\ +% +u_1 &= -\int \frac{x e^{-x} e^{-x}}{e^{-2x}} dx + c_1 = -\int x dx + c_1 = -\frac{1}{2} x^2 + c_1 \\ +u_2 &= \int \frac{e^{-x} e^{-x}}{e^{-2x}} dx + c_2 = \int dx + c_2 = x + c_2 \\ +y(x) &= \left(-\frac{1}{2} x^2 + c_1\right) e^{-x} + (x + c_2) x e^{-x} \\ +\end{align}\]
+

After simplifying, we obtain the same solution as via the method of undetermined coefficients (but with a bit more work):

+
+(3.47)\[\begin{equation} +y(x) = x_1 e^{-x} + c_2 x e^{-x} + \frac{1}{2} x^2 e^{-x} +\end{equation}\]
+
+
+

3.2.3.2.2. Example 2: variation of parameters

+

Now let’s try an example that we could not solve using the method of undetermined coefficients, with a forcing term that involves hyperbolic cosine (cosh); recall that \(\cosh(x) = \frac{e^x + e^{-x}}{2}\).

+
+(3.48)\[\begin{equation} +y^{\prime\prime} + 4 y^{\prime} + 4y = \cosh(x) +\end{equation}\]
+

First, we need to find the homogeneous solution:

+
+(3.49)\[\begin{align} +y^{\prime\prime} + 4 y^{\prime} + 4y &= 0 \\ +\lambda^2 + 4 \lambda + 4 &= 0 \\ +\rightarrow \lambda &= -2 +\end{align}\]
+

So our homogeneous solution involves repeated roots:

+
+(3.50)\[\begin{equation} +y_H = c_1 e^{-2x} + c_2 x e^{-2x} +\end{equation}\]
+

where \(y_1 = e^{-2x}\) and \(y_2 = x e^{-2x}\).

+

Then, we need to find \(u_1\) and \(u_2\), so let’s get the Wronksian and then solve

+
+(3.51)\[\begin{align} +W &= \begin{vmatrix} y_1 & y_2 \\ y_1^{\prime} & y_2^{\prime} \end{vmatrix} = e^{-2x} (e^{-2x}) (1 - 2x) - x e^{-2x}(-2 e^{-2x}) = e^{-4x} \\ +% +u_1 &= - \int \frac{x e^{-2x} \cosh x}{e^{-4x}} dx + c_1 = -\int \frac{x \frac{1}{2}(e^x + e^{-x})}{e^{-2x}} dx + c_1 \\ + &= -\frac{1}{2} \int x (e^{3x} + e^x) dx + c_1 = -\frac{1}{2} \left[ \frac{1}{9} e^{3x}(3x-1) + e^x(x-1) \right] + c_1 \\ +u_1 &= -\frac{1}{18} e^{3x}(3x-1) - \frac{1}{2} e^x (x-1) + c_1 \\ +% +u_2 &= \int \frac{e^{-2x} \cosh x}{e^{-4x}} dx + c_2 = \frac{1}{2} \int e^{2x}(e^x + e^{-x}) dx + c_2 = \frac{1}{2} \int (e^{3x} + e^x) dx + c_2 \\ +u_2 &= \frac{1}{6} e^{3x} + \frac{1}{2} e^x + c_2 +\end{align}\]
+

Then, when we put these all together, we get the full (complicated) solution:

+
+(3.52)\[\begin{equation} +y(x) = \left[ -\frac{1}{18} e^{3x} (3x-1) - \frac{1}{2} e^x (x-1) + c_1 \right] e^{-2x} + \left( \frac{1}{6} e^{3x} + \frac{1}{2} e^x + c_2 \right) x e^{-2x} +\end{equation}\]
+
+
+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
+
+ + + + + + \ No newline at end of file diff --git a/docs/content/second-order/numerical-methods.html b/docs/content/second-order/numerical-methods.html new file mode 100644 index 0000000..fe60802 --- /dev/null +++ b/docs/content/second-order/numerical-methods.html @@ -0,0 +1,909 @@ + + + + + + + + 3.3. Numerical methods for 2nd-order ODEs — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+ +
+
+
+ +
+ +
+

3.3. Numerical methods for 2nd-order ODEs

+

We’ve gone over how to solve 1st-order ODEs using numerical methods, but what about 2nd-order or any higher-order ODEs? We can use the same methods we’ve already discussed by transforming our higher-order ODEs into a system of first-order ODEs.

+

Meaning, if we are given a 2nd-order ODE

+
+(3.53)\[\begin{equation} +\frac{d^2 y}{dx^2} = y^{\prime\prime} = f(x, y, y^{\prime}) +\end{equation}\]
+

we can transform this into a system of two 1st-order ODEs that are coupled:

+
+(3.54)\[\begin{align} +\frac{dy}{dx} &= y^{\prime} = u \\ +\frac{du}{dx} &= u^{\prime} = y^{\prime\prime} = f(x, y, u) +\end{align}\]
+

where \(f(x, y, u)\) is the same as that given above for \(\frac{d^2 y}{dx^2}\).

+

Thus, instead of a 2nd-order ODE to solve, we have two 1st-order ODEs:

+
+(3.55)\[\begin{align} +y^{\prime} &= u \\ +u^{\prime} &= f(x, y, u) +\end{align}\]
+

So, we can use all of the methods we have talked about so far to solve 2nd-order ODEs by transforming the one equation into a system of two 1st-order equations.

+
+

3.3.1. Higher-order ODEs

+

This works for higher-order ODEs too! For example, if we have a 3rd-order ODE, we can transform it into a system of three 1st-order ODEs:

+
+(3.56)\[\begin{align} +\frac{d^3 y}{dx^3} &= f(x, y, y^{\prime}, y^{\prime\prime}) \\ +\rightarrow y^{\prime} &= u \\ +u^{\prime} &= y^{\prime\prime} = w \\ +w^{\prime} &= y^{\prime\prime\prime} = f(x, y, u, w) +\end{align}\]
+
+
+

3.3.2. Example: mass-spring problem

+

For example, let’s solve a forced damped mass-spring problem given by a 2nd-order ODE:

+
+(3.57)\[\begin{equation} +y^{\prime\prime} + 5y^{\prime} + 6y = 10 \sin \omega t +\end{equation}\]
+

with the initial conditions \(y(0) = 0\) and \(y^{\prime}(0) = 5\).

+

We start by transforming the equation into two 1st-order ODEs. Let’s use the variables \(z_1 = y\) and \(z_2 = y^{\prime}\):

+
+(3.58)\[\begin{align} +\frac{dz_1}{dt} &= z_1^{\prime} = z_2 \\ +\frac{dz_2}{dt} &= z_2^{\prime} = y^{\prime\prime} = 10 \sin \omega t - 5z_2 - 6z_1 +\end{align}\]
+
+

3.3.2.1. Forward Euler

+

Then, let’s solve numerically using the forward Euler method. Recall that the recursion formula for forward Euler is:

+
+(3.59)\[\begin{equation} +y_{i+1} = y_i + \Delta x f(x_i, y_i) +\end{equation}\]
+

where \(f(x,y) = \frac{dy}{dx}\).

+

Let’s solve using \(\omega = 1\) and with a step size of \(\Delta t = 0.1\), over \(0 \leq t \leq 3\).

+

We can compare this against the exact solution, obtainable using the method of undetermined coefficients:

+
+(3.60)\[\begin{equation} +y(t) = -6 e^{-3t} + 7 e^{-2t} + \sin t - \cos t +\end{equation}\]
+
+
+
% plot exact solution first
+t = linspace(0, 3);
+y_exact = -6*exp(-3*t) + 7*exp(-2*t) + sin(t) - cos(t);
+plot(t, y_exact); hold on
+
+omega = 1;
+
+dt = 0.1;
+t = [0 : dt : 3];
+
+f = @(t,z1,z2) 10*sin(omega*t) - 5*z2 - 6*z1;
+
+z1 = zeros(length(t), 1);
+z2 = zeros(length(t), 1);
+z1(1) = 0;
+z2(1) = 5;
+for i = 1 : length(t)-1
+    z1(i+1) = z1(i) + dt * z2(i);
+    z2(i+1) = z2(i) + dt * f(t(i), z1(i), z2(i));
+end
+
+plot(t, z1, 'o--')
+xlabel('time'); ylabel('displacement')
+legend('Exact', 'Forward Euler', 'Location','southeast')
+
+
+
+
+../../_images/numerical-methods_2_0.png +
+
+
+
+

3.3.2.2. Heun’s Method

+

For schemes that involve more than one stage, like Heun’s method, we’ll need to implement both stages for each 1st-order ODE. For example:

+
+
+
clear
+% plot exact solution first
+t = linspace(0, 3);
+y_exact = -6*exp(-3*t) + 7*exp(-2*t) + sin(t) - cos(t);
+plot(t, y_exact); hold on
+
+omega = 1;
+
+dt = 0.1;
+t = [0 : dt : 3];
+
+f = @(t,z1,z2) 10*sin(omega*t) - 5*z2 - 6*z1;
+
+z1 = zeros(length(t), 1);
+z2 = zeros(length(t), 1);
+z1(1) = 0;
+z2(1) = 5;
+for i = 1 : length(t)-1
+    % predictor
+    z1p = z1(i) + z2(i)*dt;
+    z2p = z2(i) + f(t(i), z1(i), z2(i))*dt;
+
+    % corrector
+    z1(i+1) = z1(i) + 0.5*dt*(z2(i) + z2p);
+    z2(i+1) = z2(i) + 0.5*dt*(f(t(i), z1(i), z2(i)) + f(t(i+1), z1p, z2p));
+end
+plot(t, z1, 'o')
+xlabel('time'); ylabel('displacement')
+legend('Exact', 'Heuns', 'Location','southeast')
+
+
+
+
+../../_images/numerical-methods_4_0.png +
+
+
+
+

3.3.2.3. Runge-Kutta: ode45

+

We can also solve using ode45, by providing a separate function file that defines the system of 1st-order ODEs. In this case, we’ll need to use a single array variable, Z, to store \(z_1\) and \(z_2\). The first column of Z will store \(z_1\) (Z(:,1)) and the second column will store \(z_2\) (Z(:,2)).

+
+
+
%%file mass_spring.m
+function dzdt = mass_spring(t, z)
+    omega = 1;
+    dzdt = zeros(2,1);
+    
+    dzdt(1) = z(2);
+    dzdt(2) = 10*sin(omega*t) - 6*z(1) - 5*z(2);
+end
+
+
+
+
+
Created file '/Users/kyle/projects/ME373/docs/mass_spring.m'.
+
+
+
+
+
+
+
% plot exact solution first
+t = linspace(0, 3);
+y_exact = -6*exp(-3*t) + 7*exp(-2*t) + sin(t) - cos(t);
+plot(t, y_exact); hold on
+
+% solution via ode45:
+[T, Z] = ode45('mass_spring', [0 3], [0 5]);
+
+plot(T, Z(:,1), 'o')
+xlabel('time'); ylabel('displacement')
+legend('Exact', 'ode45', 'Location','southeast')
+
+
+
+
+../../_images/numerical-methods_7_0.png +
+
+
+
+
+

3.3.3. Backward Euler for 2nd-order ODEs

+

We saw how to implement the Backward Euler method for a 1st-order ODE, but what about a 2nd-order ODE? (Or in general a system of 1st-order ODEs?)

+

The recursion formula is the same, except now our dependent variable is an array/vector:

+
+(3.61)\[\begin{equation} +\mathbf{y}_{i+1} = \mathbf{y}_i + \Delta t \, \mathbf{f} \left( t_{i+1} , \mathbf{y}_{i+1} \right) +\end{equation}\]
+

where the bolded \(\mathbf{y}\) and \(\mathbf{f}\) indicate array quantities (in other words, they hold more than one value).

+

In general, we can use Backward Euler to solve 2nd-order ODEs in a similar fashion as our other numerical methods:

+
    +
  1. Convert the 2nd-order ODE into a system of two 1st-order ODEs

  2. +
  3. Insert the ODEs into the Backward Euler recursion formula and solve for \(\mathbf{y}_{i+1}\)

  4. +
+

The main difference is that we will now have a system of two equations and two unknowns: \(y_{1, i+1}\) and \(y_{2, i+1}\).

+

Let’s demonstrate with an example:

+
+(3.62)\[\begin{equation} +y^{\prime\prime} + 6 y^{\prime} + 5y = 10 \quad y(0) = 0 \quad y^{\prime}(0) = 5 +\end{equation}\]
+

where the exact solution is

+
+(3.63)\[\begin{equation} +y(t) = -\frac{3}{4} e^{-5t} - \frac{5}{4} e^{-t} + 2 +\end{equation}\]
+

To solve numerically,

+
    +
  1. Convert the 2nd-order ODE into a system of two 1st-order ODEs:

  2. +
+
+(3.64)\[\begin{gather} +y_1 = y \quad y_1(t=0) = 0 \\ +y_2 = y^{\prime} \quad y_2 (t=0) = 5 +\end{gather}\]
+

Then, for the derivatives of these variables:

+
+(3.65)\[\begin{align} +y_1^{\prime} &= y_2 \\ +y_2^{\prime} &= 10 - 6 y_2 - 5 y_1 +\end{align}\]
+
    +
  1. Then plug these derivatives into the Backward Euler recursion formulas and solve for \(y_{1,i+1}\) and \(y_{2,i+1}\):

  2. +
+
+(3.66)\[\begin{align} +y_{1, i+1} &= y_{1, i} + \Delta t \, y_{2,i+1} \\ +y_{2, i+1} &= y_{2, i} + \Delta t \left( 10 - 6 y_{2, i+1} - 5 y_{1,i+1} \right) \\ +\\ +y_{1, i+1} - \Delta t \, y_{2, i+1} &= y_{1,i} \\ +5 \Delta t \, y_{1, i+1} + (1 + 6 \Delta t) y_{2, i+1} &= y_{2,i} + 10 \Delta t \\ +\text{or} \quad +\begin{bmatrix} 1 & -\Delta t \\ 5 \Delta t & (1+6\Delta t)\end{bmatrix} +\begin{bmatrix} y_{1, i+1} \\ y_{2, i+1} \end{bmatrix} &= +\begin{bmatrix} y_{1,i} \\ y_{2,i} + 10 \Delta t \\ \end{bmatrix} \\ +\mathbf{A} \mathbf{y}_{i+1} &= \mathbf{b} +\end{align}\]
+

To isolate \(\mathbf{y}_{i+1}\) and get a usable recursion formula, we need to solve this system of two equations. We could solve this by hand using the substitution method, or we can use Cramer’s rule:

+
+(3.67)\[\begin{align} +y_{1, i+1} &= \frac{ y_{1,i} (1 + 6 \Delta t) + \Delta t \left( y_{2,i} + 10 \Delta t \right)}{1 + 6 \Delta t + 5 \Delta t^2} \\ +y_{2, i+1} &= \frac{ y_{2,i} + 10 \Delta t - 5 \Delta t y_{1,i}}{1 + 6 \Delta t + 5 \Delta t^2} +\end{align}\]
+

Let’s confirm that this gives us a good, well-behaved numerical solution and compare with the Forward Euler method:

+
+
+
clear
+
+% Exact solution
+t = linspace(0, 5);
+y_exact = @(t) -(3/4)*exp(-5*t) - (5/4)*exp(-t) + 2;
+plot(t, y_exact(t)); hold on
+
+dt = 0.1;
+t = 0 : dt : 5;
+
+% Forward Euler
+f = @(t, y1, y2) 10 - 6*y2 - 5*y1;
+y1 = zeros(length(t), 1); y2 = zeros(length(t), 1);
+y1(1) = 0; y2(1) = 5;
+for i = 1 : length(t) - 1
+    y1(i+1) = y1(i) + dt*y2(i);
+    y2(i+1) = y2(i) + dt*f(t(i), y1(i), y2(i));
+end
+plot(t, y1, '+')
+
+Y = zeros(length(t), 2);
+Y(1,1) = 0;
+Y(1,2) = 5;
+for i = 1 : length(t) - 1
+    D = 1 + 6*dt + 5*dt^2;
+    Y(i+1, 1) = (Y(i,1)*(1 + 6*dt) + dt*(Y(i,2) + 10*dt)) / D;
+    Y(i+1, 2) = (Y(i,2) + 10*dt - Y(i,1)*5*dt) / D;
+end
+plot(t, Y(:,1), 'o')
+legend('Exact', 'Forward Euler', 'Backward Euler', 'location', 'southeast')
+
+fprintf('Maximum error of Forward Euler: %5.3f\n', max(abs(y1(:) - y_exact(t)')));
+fprintf('Maximum error of Backward Euler: %5.3f', max(abs(Y(:,1) - y_exact(t)')));
+
+
+
+
+
Maximum error of Forward Euler: 0.099
+Maximum error of Backward Euler: 0.068
+
+
+../../_images/numerical-methods_9_1.png +
+
+

So, for \(\Delta t = 0.1\), we see that the Forward and Backward Euler methods give an error \(\mathcal{O}(\Delta t)\), as expected since both methods are first-order accurate.

+

Let’s see how they compare for a larger step size:

+
+
+
clear
+
+% Exact solution
+t = linspace(0, 5);
+y_exact = @(t) -(3/4)*exp(-5*t) - (5/4)*exp(-t) + 2;
+plot(t, y_exact(t)); hold on
+
+dt = 0.5;
+t = 0 : dt : 5;
+
+% Forward Euler
+f = @(t, y1, y2) 10 - 6*y2 - 5*y1;
+y1 = zeros(length(t), 1); y2 = zeros(length(t), 1);
+y1(1) = 0; y2(1) = 5;
+for i = 1 : length(t) - 1
+    y1(i+1) = y1(i) + dt*y2(i);
+    y2(i+1) = y2(i) + dt*f(t(i), y1(i), y2(i));
+end
+plot(t, y1, 'o')
+
+% Backward Euler
+
+Y = zeros(length(t), 2);
+Y(1,1) = 0;
+Y(1,2) = 5;
+for i = 1 : length(t) - 1
+    D = 1 + 6*dt + 5*dt^2;
+    Y(i+1, 1) = (Y(i,1)*(1 + 6*dt) + dt*(Y(i,2) + 10*dt)) / D;
+    Y(i+1, 2) = (Y(i,2) + 10*dt - Y(i,1)*5*dt) / D;
+end
+plot(t, Y(:,1), 'o')
+legend('Exact', 'Backward Euler', 'location', 'southeast')
+
+%fprintf('Maximum error of Forward Euler: %5.3f\n', max(abs(y1(:) - y_exact(t)')));
+%fprintf('Maximum error of Backward Euler: %5.3f', max(abs(Y(:,1) - y_exact(t)')));
+
+fprintf('Maximum error of Forward Euler: %5.3f\n', max(abs(y1(:) - y_exact(t)')));
+fprintf('Maximum error of Backward Euler: %5.3f', max(abs(Y(:,1) - y_exact(t)')));
+
+
+
+
+
Maximum error of Forward Euler: 43.242
+Maximum error of Backward Euler: 0.228
+
+
+../../_images/numerical-methods_11_1.png +
+
+

Backward Euler, since it is unconditionally stable, remains well-behaved at this larger step size, while the Forward Euler method blows up.

+

One other thing: instead of using Cramer’s rule to get expressions for \(y_{1,i+1}\) and \(y_{2,i+1}\), we could instead use Matlab to solve the linear system of equations at each time step. To do that, we could replace

+
A = [1  -dt; 5*dt  (1+6*dt)];
+b = [Y(i,1); Y(i,2)+10*dt];
+Y(i+1,:) = (A\b)';
+
+
+

where A\b is equivalent to inv(A)*b, but faster. Let’s confirm that this gives the same answer:

+
+
+
clear
+
+% Exact solution
+t = linspace(0, 5);
+y_exact = @(t) -(3/4)*exp(-5*t) - (5/4)*exp(-t) + 2;
+plot(t, y_exact(t)); hold on
+
+dt = 0.1;
+t = 0 : dt : 5;
+
+Y = zeros(length(t), 2);
+Y(1,1) = 0;
+Y(1,2) = 5;
+for i = 1 : length(t) - 1
+    A = [1  -dt; 5*dt  (1+6*dt)];
+    b = [Y(i,1); Y(i,2)+10*dt];
+    Y(i+1,:) = (A\b)';
+end
+plot(t, Y(:,1), 'o')
+legend('Exact', 'Backward Euler', 'location', 'southeast')
+
+
+
+
+../../_images/numerical-methods_13_0.png +
+
+
+
+

3.3.4. Cramer’s Rule

+

Cramer’s Rule provides a solution method for a system of linear equations, where the number of equations equals the number of unknowns. It works for any number of equations/unknowns, but isn’t really practical for more than two or three. We’ll focus on using it for a system of two equations, with two unknowns \(x_1\) and \(x_2\):

+
+(3.68)\[\begin{gather} +a_{11} + x_1 + a_{12} x_2 = b_1 \\ +a_{21} + x_1 + a_{22} x_2 = b_2 \\ +\text{or } \mathbf{A} \mathbf{x} = \mathbf{b} +\end{gather}\]
+

where

+
+(3.69)\[\begin{gather} +\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \\ +\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \\ +\mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} +\end{gather}\]
+

The solutions for the unknowns are then

+
+(3.70)\[\begin{align} +x_1 &= \frac{ \begin{vmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{vmatrix} }{D} = \frac{b_1 a_{22} - a_{12} b_2}{D} \\ +x_2 &= \frac{ \begin{vmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{vmatrix} }{D} = \frac{a_{11} b_2 - b_1 a_{21}}{D} +\end{align}\]
+

where \(D\) is the determinant of \(\mathbf{A}\):

+
+(3.71)\[\begin{equation} +D = \det(\mathbf{A}) = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11} a_{22} - a_{12} a_{21} +\end{equation}\]
+

This works as long as the determinant does not equal zero.

+
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+ + + + + + \ No newline at end of file diff --git a/docs/content/second-order/power-series.html b/docs/content/second-order/power-series.html new file mode 100644 index 0000000..f267309 --- /dev/null +++ b/docs/content/second-order/power-series.html @@ -0,0 +1,676 @@ + + + + + + + + 3.5. Power Series Solutions — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ +
+

3.5. Power Series Solutions

+

Power series solutions are another technique we can use to solve 2nd-order homogeneous ODEs of the form

+
+(3.104)\[\begin{equation} +y^{\prime\prime} + p(x) y^{\prime} + q(x) y = 0 +\end{equation}\]
+

This is useful for more-general cases where our other techniques fail.

+

For example, how would you find the solution to this ODE?

+
+(3.105)\[\begin{equation} +(1 + x^2) y^{\prime\prime} - 4 x y^{\prime} + 6y = 0 +\end{equation}\]
+

None of the methods we’ve discussed so far would allow us to find an analytical solution to this problem—but we can using a power series solution.

+

Power series solutions will be of the form

+
+(3.106)\[\begin{equation} +y = \sum_{n=0}^{\infty} a_n x^n +\end{equation}\]
+

where the coefficients \(a_n\) are what we need to find.

+
    +
  1. First, for power series to be a valid solution, we need to check whether \(x=0\) is an ordinary point of the ODE: is the ODE continuous and bounded at \(x=0\)?

  2. +
+

Continuous means that there should be no discontinuity at \(x=0\).

+

Bounded means that the solution should be finite at \(x=0\).

+

For example, consider the ODE

+
+(3.107)\[\begin{equation} +y^{\prime\prime} - 4xy^{\prime} + (4x^2 - 2)y = 0 +\end{equation}\]
+

Both \(p(x) = -4x\) and \(q(x) = (4x^2 - 2)\) are continuous and bounded at \(x=0\), so \(x=0\) is an ordinary point.

+

On the other hand, what about

+
+(3.108)\[\begin{equation} +y^{\prime\prime} + x^3 y^{\prime} + \frac{1}{x} y = 0 \text{ ?} +\end{equation}\]
+

In this case, the solution is unbounded at \(x=0\), and so it is not an ordinary point.

+
    +
  1. If \(x=0\) is an ordinary point, then we can find a solution in the form of a power series:

  2. +
+
+(3.109)\[\begin{equation} +y = \sum_{n=0}^{\infty} a_n x^n +\end{equation}\]
+

We then solve for the coefficients \(a_n\) by plugging this in to the ODE. To do that, we’ll need to take advantage of certain properties of power series.

+
+

3.5.1. Properties of power series

+
    +
  • Dummy index rule. We can replace the index variable used in the power series with another index variable arbitrarily:

  • +
+
+(3.110)\[\begin{equation} +\sum_{n=0}^{\infty} a_n x^n = \sum_{m=0}^{\infty} a_m x^m +\end{equation}\]
+

This is because the index variable is just a “dummy” that only has meaning inside the sum.

+
    +
  • Product rule. We can bring variables, including \(x\), multiplying an entire power series into the power series:

  • +
+
+(3.111)\[\begin{equation} +x \sum_{n=0}^{\infty} a_n x^n = \sum_{n=0}^{\infty} a_n x^{n+1} +\end{equation}\]
+
    +
  • Derivatives. We can take derivatives of our power series:

  • +
+
+(3.112)\[\begin{align} +y^{\prime} &= \sum_{n=0}^{\infty} a_n (n) x^{n-1} = \sum_{n=1}^{\infty} a_n (n) x^{n-1} \\ +y^{\prime\prime} &= \sum_{n=0}^{\infty} a_n (n)(n-1) x^{n-2} = \sum_{n=2}^{\infty} a_n (n)(n-1) x^{n-2} +\end{align}\]
+

Notice that we can change where the sums in the power series start, because for \(y^{\prime}\) the term corresponding to \(n=1\) would just be zero, and similar for the first two terms of \(y^{\prime\prime}\).

+
    +
  • Index shift. We can redefine the index used within a sum to shift where it starts. For example, if we let \(m=n-1\), or \(n=m+1\), then:

  • +
+
+(3.113)\[\begin{equation} +\sum_{n=1}^{\infty} a_n (n) x^{n-1} = \sum_{m=0}^{\infty} a_{m+1} (m+1) x^m +\end{equation}\]
+

Or, in other case, if we let \(m=n-2\), or \(n=m+2\), then:

+
+(3.114)\[\begin{equation} +\sum_{n=2}^{\infty} a_n (n)(n-1) x^{n-2} = \sum_{m=0}^{\infty} a_{m+2} (m+2)(m+1) x^m +\end{equation}\]
+

Now, let’s apply these properties to solve ODEs.

+
+
+

3.5.2. Power series example 1

+

Let’s try to apply the power series approach to solve

+
+(3.115)\[\begin{equation} +y^{\prime\prime} + y = 0 \;, +\end{equation}\]
+

where we know the solution will be \(y(x) = c_1 \sin x + c_2 \cos x\).

+
    +
  1. Is \(x=0\) an ordinary point? Yes, the ODE is continuous and bounded at \(x=0\). +So, we can find a solution of the form \(y(x) = \sum_{n=0}^{\infty} a_n x^n\).

  2. +
  3. Now, we solve for the coefficents by plugging the power series into the ODE:

  4. +
+
+(3.116)\[\begin{equation} +\sum_{n=2}^{\infty} a_n (n)(n-1) x^{n-2} + \sum_{n=0}^{\infty} a_n x^n = 0 +\end{equation}\]
+

Let’s use the index shift rule on the first part of that:

+
+(3.117)\[\begin{equation} +\sum_{n=2}^{\infty} a_n (n)(n-1) x^{n-2} \rightarrow \sum_{m=0}^{\infty} a_{m+2} (m+2)(m+1) x^m +\end{equation}\]
+

Then, we can use the dummy index rule to change \(m\) back to \(n\):

+
+(3.118)\[\begin{equation} +\sum_{m=0}^{\infty} a_m (m+2)(m+1) x^m \rightarrow \sum_{n=0}^{\infty} a_n (n+2)(n+1) x^n +\end{equation}\]
+

Now, let’s replace the first term in the ODE with that, merge both terms into a single sum, and simplify:

+
+(3.119)\[\begin{align} +\sum_{n=0}^{\infty} a_n (n+2)(n+1) x^n + \sum_{n=0}^{\infty} a_n x^n &= 0 \\ +\sum_{n=0}^{\infty} x^n \left[ a_{n+2}(n+2)(n+1) + a_n \right] &= 0 +\end{align}\]
+

There are infinite terms in this sum, involving the continuous variable \(x\); the only way that equation can be satisfied is if

+
    +
  • \(x=0\) always, which cannot be true, or

  • +
  • \(a_{n+2}(n+2)(n+1) + a_n = 0\) for all values of \(n\). This is what we can use to find the coefficients of our power series solution.

  • +
+

Use that expression to define a recursive formula for the coefficients:

+
+(3.120)\[\begin{equation} +a_{n+2} = \frac{-a_n}{(n+1)(n+2)} +\end{equation}\]
+

We can see that the even coefficients will be related to each other, and the odd coefficients will be related. Let’s try to identify a pattern with each, starting with the even terms:

+
+(3.121)\[\begin{align} +n=0: \quad a_2 &= \frac{-a_0}{1 \cdot 2} = \frac{-a_0}{2!} \\ +n=2: \quad a_4 &= \frac{-a_2}{3 \cdot 4} = \frac{a_0}{4!} \\ +n=4: \quad a_6 &= \frac{-a_4}{5 \cdot 6} = \frac{-a_0}{6!} +\end{align}\]
+

and the odd terms:

+
+(3.122)\[\begin{align} +n=1: \quad a_3 &= \frac{-a_1}{2 \cdot 3} = \frac{-a_1}{3!} \\ +n=3: \quad a_5 &= \frac{-a_3}{4 \cdot 5} = \frac{a_1}{5!} \\ +n=5: \quad a_7 &= \frac{-a_5}{6 \cdot 7} = \frac{-a_1}{7!} +\end{align}\]
+

Now, let’s put that all together:

+
+(3.123)\[\begin{align} +y(x) &= a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \dots \\ +y &= a_0 \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \dots \right) + a_1 \left( x - \frac{x^3}{3!} + \frac{x^5}{5!} + \dots \right) +\end{align}\]
+

which you might recognize as being the Taylor series expansion of sine and cosine:

+
+(3.124)\[\begin{equation} +y(x) = a_0 \cos x + a_1 \sin x +\end{equation}\]
+

So, our unknown coefficients end up being our integration constants, which we can use our two constraints to find.

+
+
+

3.5.3. Power series example 2

+

Find the solution to the ODE

+
+(3.125)\[\begin{equation} +(1 + x^2) y^{\prime\prime} - 4x y^{\prime} + 6y = 0 +\end{equation}\]
+

First, rearrange into standard form:

+
+(3.126)\[\begin{equation} +y^{\prime\prime} - \frac{4x}{1+x^2} y^{\prime} + \frac{6}{1+x^2} y = 0 +\end{equation}\]
+

Then, check whether \(x=0\) is an ordinary point: yes, it is.

+

Now, let’s insert the power series into the ODE:

+
+(3.127)\[\begin{align} +y^{\prime\prime} + x^2 y^{\prime\prime} - 4 x y^{\prime} + 6 y &= 0 \\ +\sum_{n=2}^{\infty} a_n (n)(n-1)x^{n-2} + x^2 \sum_{n=0}^{\infty} a_n (n)(n-1)x^{n-2} - 4 x \sum_{n=1} a_n (n) x^{n-1} + 6 \sum_{n=0}^{\infty} a_n x^n &= 0 +\end{align}\]
+

First, we’ll use the power rule:

+
+(3.128)\[\begin{equation} +\sum_{n=2}^{\infty} a_n (n)(n-1)x^{n-2} + \sum_{n=2}^{\infty} a_n (n)(n-1)x^{n} - 4 \sum_{n=1} a_n (n) x^{n} + 6 \sum_{n=0}^{\infty} a_n x^n = 0 +\end{equation}\]
+

and then the index shift and dummy index rules on the first term:

+
+(3.129)\[\begin{equation} +\sum_{n=2}^{\infty} a_n (n)(n-1)x^{n-2} \rightarrow \sum_{m=0}^{\infty} a_{m+2} (m+2)(m+1) x^m \rightarrow \sum_{n=0}^{\infty} a_{n+2} (n+2)(n+1) x^n +\end{equation}\]
+

Then, put that back into the full equation and combine the sums:

+
+(3.130)\[\begin{align} +\sum_{n=0}^{\infty} a_{n+2} (n+2)(n+1) x^n + \sum_{n=0}^{\infty} a_n (n)(n-1)x^{n} - 4 \sum_{n=1} a_n (n) x^{n} + 6 \sum_{n=0}^{\infty} a_n x^n &= 0 \\ +\sum_{n=0}^{\infty} x^n \left[ a_{n+2} (n+2)(n+1) + a_n (n)(n-1) - 4a_n (n) + 6a_n \right] &= 0 \\ +a_{n+2} (n+2)(n+1) + a_n (n^2 -5n + 6) &= 0 \\ +a_{n+2} (n+2)(n+1) + a_n (n-3)(n-2) &= 0 \\ +\end{align}\]
+

Thus, our recursion formula for the coefficients \(a_n\) is

+
+(3.131)\[\begin{equation} +a_{n+2} = -a_n \frac{(n-3)(n-2)}{(n+1)(n+2)} +\end{equation}\]
+

Again, we can see that the even terms will be related and the odd terms will be related:

+
+(3.132)\[\begin{align} +n=0: \quad a_2 &= -a_0 \frac{6}{2} = -3 a_0 \\ +n=2: \quad a_4 &= 0 \\ +n=4: \quad a_6 &= -a_4 \frac{2}{30} = 0 \\ +&\ldots +\end{align}\]
+

and the odd terms:

+
+(3.133)\[\begin{align} +n=1: \quad a_3 &= -a_1 \frac{2}{6} = \frac{-a_1}{3} \\ +n=3: \quad a_5 &= 0 \\ +n=5: \quad a_7 &= -a_5 \frac{6}{42} = 0 \\ +&\ldots +\end{align}\]
+

The solution is then

+
+(3.134)\[\begin{align} +y(x) &= a_0 + a_1 x - 3 a_0 x^2 - \frac{a_1}{3} x^3 \\ +y &= a_0 \left(1 - 3x^2 \right) + a_1 \left( x - \frac{x^3}{3} \right) +\end{align}\]
+

where we find \(a_0\) and \(a_1\) using our initial or boundary conditions.

+
+
+ + + + +
+ + + + + + + + +
+
+ +
+ + +
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+ + + + + + \ No newline at end of file diff --git a/docs/content/second-order/second-order.html b/docs/content/second-order/second-order.html new file mode 100644 index 0000000..4fd63e9 --- /dev/null +++ b/docs/content/second-order/second-order.html @@ -0,0 +1,435 @@ + + + + + + + + 3. Second-order Ordinary Differential Equations — Mechanical Engineering Methods notes + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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3. Second-order Ordinary Differential Equations

+

This chapter focuses on analytical and numerical methods for solving 2nd-order ordinary differential equations (ODEs), including initial-value problems (IVPs) and boundary-value problems (BVPs).

+
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Index

+ +
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+ + + + + + \ No newline at end of file diff --git a/docs/index.html b/docs/index.html new file mode 100644 index 0000000..362ff1b --- /dev/null +++ b/docs/index.html @@ -0,0 +1 @@ + diff --git a/docs/objects.inv b/docs/objects.inv new file mode 100644 index 0000000..30617d0 Binary files /dev/null and b/docs/objects.inv differ diff --git a/docs/reports/elliptic.log b/docs/reports/elliptic.log new file mode 100644 index 0000000..605aeee --- /dev/null +++ b/docs/reports/elliptic.log @@ -0,0 +1,44 @@ +Traceback (most recent call last): + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 627, in _async_poll_for_reply + msg = await ensure_async(self.kc.shell_channel.get_msg(timeout=new_timeout)) + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/util.py", line 89, in ensure_async + result = await obj + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/jupyter_client/channels.py", line 230, in get_msg + raise Empty +_queue.Empty + +During handling of the above exception, another exception occurred: + +Traceback (most recent call last): + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution + executenb( + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 1117, in execute + return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/util.py", line 78, in wrapped + return just_run(coro(*args, **kwargs)) + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/util.py", line 57, in just_run + return loop.run_until_complete(coro) + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete + return future.result() + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 558, in async_execute + await self.async_execute_cell( + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 846, in async_execute_cell + exec_reply = await self.task_poll_for_reply + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 651, in _async_poll_for_reply + await self._async_handle_timeout(timeout, cell) + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 701, in _async_handle_timeout + raise CellTimeoutError.error_from_timeout_and_cell( +nbclient.exceptions.CellTimeoutError: A cell timed out while it was being executed, after 30 seconds. +The message was: Cell execution timed out. +Here is a preview of the cell contents: +------------------- +step_sizes = [0.1, 0.05, 0.025, 0.02, 0.0125, 0.01, 0.005]; +n = length(step_sizes); + +nums_jac = zeros(n,1); times_jac = zeros(n,1); num_iter_jac = zeros(n,1); + +for i = 1 : n + [times_jac(i), nums_jac(i), num_iter_jac(i)] = heat_equation_jacobi(step_sizes(i)); +end +------------------- + diff --git a/docs/reports/parabolic.log b/docs/reports/parabolic.log new file mode 100644 index 0000000..79563fe --- /dev/null +++ b/docs/reports/parabolic.log @@ -0,0 +1,39 @@ +Traceback (most recent call last): + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 627, in _async_poll_for_reply + msg = await ensure_async(self.kc.shell_channel.get_msg(timeout=new_timeout)) + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/util.py", line 89, in ensure_async + result = await obj + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/jupyter_client/channels.py", line 230, in get_msg + raise Empty +_queue.Empty + +During handling of the above exception, another exception occurred: + +Traceback (most recent call last): + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution + executenb( + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 1117, in execute + return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/util.py", line 78, in wrapped + return just_run(coro(*args, **kwargs)) + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/util.py", line 57, in just_run + return loop.run_until_complete(coro) + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete + return future.result() + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 558, in async_execute + await self.async_execute_cell( + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 846, in async_execute_cell + exec_reply = await self.task_poll_for_reply + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 651, in _async_poll_for_reply + await self._async_handle_timeout(timeout, cell) + File "/Users/niemeyek/miniconda3/envs/jupyterbook/lib/python3.8/site-packages/nbclient/client.py", line 701, in _async_handle_timeout + raise CellTimeoutError.error_from_timeout_and_cell( +nbclient.exceptions.CellTimeoutError: A cell timed out while it was being executed, after 30 seconds. +The message was: Cell execution timed out. +Here is a preview of the cell contents: +------------------- +['clear all', '', 'dx = 0.1;', 'alpha = 2.3e-1;', ''] +... +[" imwrite(imind,cm,filename,'gif', 'Loopcount',inf, 'DelayTime',1e-3); 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