more eigenvalue content

_build/bvps/finite-difference.html

@@ -11,7 +11,7 @@
 next_page:
   url: /bvps/eigenvalue.html
 suffix: .ipynb
-search: x y delta f t prime equation frac begin end right align boundary left b derivative dx infty m difference xi fin solve equations yi l finite c conditions where solution n heat transfer lets our ac point system condition nonlinear text k p ti differences mathcal o example ode points linear matlab h exact using order second consider gives domain through formula term q dt theta well rightarrow approx into above recursion get bmatrix hand guess temperature d value also accurate five set mathbf implement fixed octave because e control volume method approximations derivatives function approximate forward backward taylor
+search: x y delta f prime equation t frac begin end align right boundary left b derivative dx difference xi fin solve equations yi l infty finite c conditions where solution n heat transfer m lets our ac point system condition text k p differences mathcal o example ode points linear matlab nonlinear h exact using order second consider gives domain through formula term q dt theta well rightarrow approx into above recursion get bmatrix guess temperature d value also accurate five set mathbf implement hand fixed octave because e control volume method approximations derivatives function approximate forward backward taylor series
 
 comment: "***PROGRAMMATICALLY GENERATED, DO NOT EDIT. SEE ORIGINAL FILES IN /content***"
 ---
@@ -699,12 +699,7 @@ <h3 id="Heat-transfer-with-radiation">Heat transfer with radiation<a class="anch
 \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}</p>
-<p>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, moving the nonlinear parts to the right-hand side:
-\begin{align}
-\frac{T_{i-1} - 2T_i + T_{i+1}}{\Delta x^2} - m^2 \left( T_i - T_{\infty} \right) - M^2 \left( T_i^4 - T_{\infty}^4 \right) &amp;= 0 \\
-\frac{T_{i-1} - 2T_i + T_{i+1}}{\Delta x^2} - m^2 T_i &amp;= M^2 \left( T_i^4 - T_{\infty}^4 \right) - m^2 T_{\infty} \\
-T_{i-1} + T_i (-2 - \Delta x^2 m^2
-\end{align}</p>
+<p>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.</p>
 
 </div>
 </div>

_build/bvps/masses.m

@@ -1,4 +1,5 @@
 function dxdt = masses(t, x)
+% this is a function file to calculate the derivatives associated with the system
 
 m1 = 40;
 m2 = 40;

_build/bvps/shooting-method.html

@@ -113,9 +113,9 @@ <h2 id="Example:-linear-ODE">Example: linear ODE<a class="anchor-link" href="#Ex
 
 <div class="inner_cell">
     <div class="input_area">
-<div class=" highlight hl-matlab"><pre><span></span><span class="n">clear</span> <span class="n">all</span><span class="p">;</span> <span class="n">clc</span>
+<div class=" highlight hl-matlab"><pre><span></span><span class="n">clear </span><span class="s">all</span><span class="p">;</span><span class="n"> clc</span>
 
-<span class="c">% target boundary condition</span>
+<span class="s">% target boundary condition</span><span class="p"></span>
 <span class="n">target</span> <span class="p">=</span> <span class="mi">8</span><span class="p">;</span>
 
 <span class="c">% Pick a guess for y&#39;(0) of 1</span>
@@ -234,7 +234,7 @@ <h2 id="Example:-nonlinear-ODE">Example: nonlinear ODE<a class="anchor-link" hre
 <div class="inner_cell">
     <div class="input_area">
 <div class=" highlight hl-matlab"><pre><span></span><span class="c">%%python</span>
-<span class="n">import</span> <span class="n">sympy</span> <span class="n">as</span> <span class="n">sym</span>
+<span class="n">import </span><span class="s">sympy as sym</span><span class="p"></span>
 <span class="n">sym</span><span class="p">.</span><span class="n">init_printing</span><span class="p">()</span>
 <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="p">=</span> <span class="n">sym</span><span class="p">.</span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y u v&#39;</span><span class="p">)</span>
 
@@ -364,9 +364,9 @@ <h2 id="Example:-nonlinear-ODE">Example: nonlinear ODE<a class="anchor-link" hre
 
 <div class="inner_cell">
     <div class="input_area">
-<div class=" highlight hl-matlab"><pre><span></span><span class="n">clear</span> <span class="n">all</span><span class="p">;</span> <span class="n">clc</span>
+<div class=" highlight hl-matlab"><pre><span></span><span class="n">clear </span><span class="s">all</span><span class="p">;</span><span class="n"> clc</span>
 
-<span class="n">target</span> <span class="p">=</span> <span class="mf">1.0</span><span class="p">;</span>
+<span class="s">target = 1.0</span><span class="p">;</span>
 
 <span class="n">guesses</span> <span class="p">=</span> <span class="nb">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span>
 <span class="n">solutions</span> <span class="p">=</span> <span class="nb">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span>
@@ -443,9 +443,9 @@ <h2 id="Example:-nonlinear-ODE">Example: nonlinear ODE<a class="anchor-link" hre
 
 <div class="inner_cell">
     <div class="input_area">
-<div class=" highlight hl-matlab"><pre><span></span><span class="n">clear</span> <span class="n">all</span><span class="p">;</span> <span class="n">clc</span>
+<div class=" highlight hl-matlab"><pre><span></span><span class="n">clear </span><span class="s">all</span><span class="p">;</span><span class="n"> clc</span>
 
-<span class="n">target</span> <span class="p">=</span> <span class="mf">1.0</span><span class="p">;</span>
+<span class="s">target = 1.0</span><span class="p">;</span>
 
 <span class="c">% get these arrays of stored values started.</span>
 <span class="c">% note: I&#39;m only doing this to make it easier to show a table of values</span>
@@ -474,9 +474,9 @@ <h2 id="Example:-nonlinear-ODE">Example: nonlinear ODE<a class="anchor-link" hre
 
     <span class="c">% we should probably set a maximum number of iterations, just to prevent</span>
     <span class="c">% an infinite while loop in case something goes wrong</span>
-    <span class="k">if</span> <span class="n">num</span> <span class="o">&gt;</span><span class="p">=</span> <span class="mf">1e4</span>
-        <span class="k">break</span>
-    <span class="k">end</span>
+<span class="n">    if </span><span class="s">num &gt;= 1e4</span><span class="p"></span>
+<span class="n">        break</span>
+<span class="n">    </span><span class="s">end</span><span class="p"></span>
 <span class="k">end</span>
 
 <span class="n">table</span><span class="p">(</span><span class="n">tries</span><span class="p">,</span> <span class="n">guesses</span><span class="p">,</span> <span class="n">solutions</span><span class="p">)</span>
@@ -529,7 +529,7 @@ <h2 id="Example:-nonlinear-ODE">Example: nonlinear ODE<a class="anchor-link" hre
     <div class="input_area">
 <div class=" highlight hl-matlab"><pre><span></span><span class="c">%plot -r 200</span>
 <span class="n">plot</span><span class="p">(</span><span class="n">F</span><span class="p">(:,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">eta</span><span class="p">);</span> <span class="n">ylim</span><span class="p">([</span><span class="mi">0</span> <span class="mi">5</span><span class="p">])</span>
-<span class="n">xlabel</span><span class="p">(</span>&quot;<span class="n">f</span>^<span class="p">{</span><span class="o">\</span><span class="n">prime</span><span class="p">}(</span><span class="o">\</span><span class="n">eta</span><span class="p">)</span> <span class="p">=</span> <span class="n">u</span><span class="o">/</span><span class="n">U_</span><span class="p">{</span><span class="o">\</span><span class="n">infty</span><span class="p">}</span>&quot;<span class="p">)</span>
+<span class="n">xlabel</span><span class="p">(</span><span class="s">&quot;f^{\prime}(\eta) = u/U_{\infty}&quot;</span><span class="p">)</span>
 <span class="n">ylabel</span><span class="p">(</span><span class="s">&#39;\eta&#39;</span><span class="p">)</span>
 </pre></div>
 

content/bvps/finite-difference.ipynb

@@ -631,20 +631,8 @@
     "\\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, moving the nonlinear parts to the right-hand side:\n",
-    "\\begin{align}\n",
-    "\\frac{T_{i-1} - 2T_i + T_{i+1}}{\\Delta x^2} - m^2 \\left( T_i - T_{\\infty} \\right) - M^2 \\left( T_i^4 - T_{\\infty}^4 \\right) &= 0 \\\\\n",
-    "\\frac{T_{i-1} - 2T_i + T_{i+1}}{\\Delta x^2} - m^2 T_i &= M^2 \\left( T_i^4 - T_{\\infty}^4 \\right) - m^2 T_{\\infty} \\\\\n",
-    "T_{i-1} + T_i (-2 - \\Delta x^2 m^2\n",
-    "\\end{align}"
+    "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."
    ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
   }
  ],
  "metadata": {

content/bvps/masses.m

@@ -1,4 +1,5 @@
 function dxdt = masses(t, x)
+% this is a function file to calculate the derivatives associated with the system
 
 m1 = 40;
 m2 = 40;