From 9339a004ca90a58f5cde6410d1b50f5da71faa1c Mon Sep 17 00:00:00 2001 From: Kyle Niemeyer Date: Sun, 23 Feb 2020 23:22:17 -0700 Subject: [PATCH] adds two problems for quiz 3 --- _build/bvps/finite-difference.html | 142 ++++++++++++++--------------- _build/images/annular-fin.png | Bin 0 -> 96209 bytes _build/quizzes/quiz2-IVPs.html | 4 +- content/images/annular-fin.png | Bin 0 -> 96209 bytes content/quizzes/quiz3-BVPs.md | 91 ++++++++++++++++++ 5 files changed, 164 insertions(+), 73 deletions(-) create mode 100644 _build/images/annular-fin.png create mode 100644 content/images/annular-fin.png create mode 100644 content/quizzes/quiz3-BVPs.md diff --git a/_build/bvps/finite-difference.html b/_build/bvps/finite-difference.html index 09ccb00..a403213 100644 --- a/_build/bvps/finite-difference.html +++ b/_build/bvps/finite-difference.html @@ -142,10 +142,10 @@

Solving ODEs with finite differenc
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];
+     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];
 
@@ -197,27 +197,27 @@ 

Matlab implementation
-
clear all; clc
+
clear all; clc
 
-dx = 0.5;
+dx = 0.5;
 x = [0 : dx : 2];
 n = length(x);
-A = zeros(n,n); b = zeros(n,1);
+A = zeros(n,n); b = zeros(n,1);
 
-for i = 1 : n
-    if i == 1
+for i = 1 : n
+    if i == 1
         A(1,1) = 1;
-        b(1) = 1;
-    elseif i == n
+        b(1) = 1;
+    elseif i == n
         A(n,n) = 1;
-        b(n) = 8;
-    else
-        A(i, i-1) = 1 - x(i)*dx/2;
+        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
+        b(i) = 2*x(i)*dx^2;
+    end
+end
 y = A \ b;
 plot(x, y, 'o-')
 
@@ -265,27 +265,27 @@

Matlab implementation
-
clear all; clc
+
clear all; clc
 
-dx = 0.001;
+dx = 0.001;
 x = [0 : dx : 2];
 n = length(x);
-A = zeros(n,n); b = zeros(n,1);
+A = zeros(n,n); b = zeros(n,1);
 
-for i = 1 : n
-    if i == 1
+for i = 1 : n
+    if i == 1
         A(1,1) = 1;
-        b(1) = 1;
-    elseif i == n
+        b(1) = 1;
+    elseif i == n
         A(n,n) = 1;
-        b(n) = 8;
-    else
-        A(i, i-1) = 1 - x(i)*dx/2;
+        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
+        b(i) = 2*x(i)*dx^2;
+    end
+end
 y = A \ b;
 plot(x, y)
 
@@ -433,9 +433,9 @@

Example: nonlinear BVP
@@ -534,9 +534,9 @@

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+ 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}}{\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} + + +```python + +```