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Notebook[{

Cell[CellGroupData[{
Cell["Examining Heat Dissipation", "Title",
  FontWeight->"Bold"],

Cell[CellGroupData[{

Cell["The Heat Equation", "Section"],

Cell[CellGroupData[{

Cell["Background", "Subsection"],

Cell[TextData[{
  "The following problems represent solutions to the partial differential \
heat equation:  ",
  Cell[BoxData[
      \(TraditionalForm\`w\_xx\)]],
  "=",
  Cell[BoxData[
      \(TraditionalForm\`1\/c\^2\)]],
  Cell[BoxData[
      \(TraditionalForm\`w\_t\)]],
  ", where ",
  StyleBox["w(x,t)",
    FontSlant->"Italic"],
  " represents the temperature at position ",
  StyleBox["x",
    FontSlant->"Italic"],
  " at time",
  StyleBox[" t",
    FontSlant->"Italic"],
  " in a thin rod of length ",
  StyleBox["1",
    FontSlant->"Italic"],
  " with perfectly insulated sides and ends that are held at constant \
temperatures of -20 degrees and 55 degrees, that is, ",
  Cell[BoxData[
      FormBox[
        RowBox[{" ", 
          RowBox[{
            RowBox[{
              RowBox[{
                FormBox["w",
                  "TraditionalForm"], \((0, t)\)}], " ", "=", " ", \(-20\)}], 
            ",", "  ", 
            RowBox[{
              RowBox[{
                FormBox["w",
                  "TraditionalForm"], "(", \(1, t\), ")"}], " ", "=", 
              "55"}]}]}], TraditionalForm]]],
  ". Assume an initial temperature distribution that is  ",
  Cell[BoxData[
      \(TraditionalForm\`75\)]],
  " throughout the rod. The value of the positive constant ",
  Cell[BoxData[
      \(TraditionalForm\`c\^2\)]],
  " is determined by the material from which the rod is made."
}], "Text"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
The Heat Equation with Fixed Nonhomogeneous Ends and Use of the Fourier Sine \
Series \
\>", "Section"],

Cell[TextData[{
  "First, we determine the Fourier Series corresponding to the initial \
temperature state for the rod minus the linear function introduced because of \
the differing temperatures at each end of the rod:\n",
  StyleBox["75 - (",
    FontSlant->"Italic"],
  Cell[BoxData[
      \(TraditionalForm\`\(\(\(-20\)\  + \ \((55\  + \ 
                  20)\)\ x\)\()\)\)\  = \ 95\  - \ 75  x\)]],
  ". "
}], "Text"],

Cell[BoxData[{
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    \(\(length\  = \ 1;\)\), "\n", 
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        2/length\ \((Integrate[
              Sin[n\ \[Pi]/length\ \ x]\ \ \((95 - 75  x)\), {x, 0, 
                length}])\)\)\t\t\), "\n", 
    \(Print["\<The Fourier sine series out to n = 30 is \>", 
      foursine[x_, 60] = 
        Sum[b[j]\ Sin[j\ \[Pi]\ /length\ x], {j, 1, 60}] // N]\), "\n", 
    \(\(Plot[foursine[x, 60], {x, 0, 1}];\)\ \ \)}], "Input"],

Cell[BoxData[{
    \(TextForm\`The\ solution\ to\ the\ heat\ equation\ can\ now\ be\ written\
\ as\), "\n", 
    \(TextForm\`\t\t\ \(\(\(\ \)\(\[Sum]\)\)\+\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \)\(n = 
            1\)\)\%\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \)\(\[Infinity]\)\)\) a \((n)\)\ \(e\^\(\(-\ \((n\[Pi]\/4)\)\^2\) t\)\) 
      Sin \((\(n\[Pi]\/4\) x)\)\), "\n", 
    \(TextForm\`We\ will\ write\ write\ it\ out\ to\ n = 30. \)}], "Text"],

Cell[BoxData[{
    \(Clear[w, x, t]\ \), "\n", 
    \(w[x_, 
        t_] = \ \(-20\) + 75  x + 
          Sum[b[j] Exp[\(-\((j\ Pi/length)\)^2\)\ *\ t] 
              Sin[j\ Pi\ x/length], {j, 1, 60}]\  // N\)}], "Input"],

Cell["\<\
Plot the temperature function at various positions along the rod as a \
function of time.  \
\>", "Text"],

Cell[BoxData[
    \(\(xlevelplots = 
        Plot[\ {w[0, t], w[ .2, t], w[ .4, t], w[ .6, t], w[ .8, t], 
            w[1, t]}, {t, 0,  .1}, \ 
          PlotStyle -> {RGBColor[1, 0, 0], RGBColor[1, 1, 0], 
              RGBColor[0, 1, 0], RGBColor[0, 1, 1], RGBColor[1, 0, 1], 
              RGBColor[0, 0, 1]}, AxesLabel -> {t, "\<w\>"}, 
          PlotLabel -> "\<X-Level Plots\>"]\ ;\)\)], "Input"],

Cell["\<\
Interpret these plots. What is happening to the temperature along different \
parts of the rod?\
\>", "Text"],

Cell["\<\
Plot the temperature function at various times as a function of the position \
on the rod.  Interpret the results.\
\>", "Text"],

Cell[BoxData[
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        Plot[{w[x, 0], w[x,  .01], w[x,  .04], w[x,  .07], w[x,  .1], 
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          PlotStyle -> {RGBColor[1, 0, 0], RGBColor[1, 1, 0], 
              RGBColor[0, 1, 0], RGBColor[0, 1, 1], RGBColor[1, 0, 1], 
              RGBColor[0, 0, 1]}, \ \ PlotRange -> All, 
          AxesLabel -> {x, "\<w\>"}, 
          PlotLabel -> "\<T-Level Plots\>"]\ ;\)\)], "Input"],

Cell["\<\
Interpret the results. What is happening to the temperature function as time \
goes on?\
\>", "Text"],

Cell["\<\
Plot the contours for the temperature function and the temperature function \
as a surface.\
\>", "Text"],

Cell[BoxData[{
    \(\(ContourPlot[w[x, t], {x, 0, 1}, {t, 0,  .15}, Axes -> True, 
        AxesLabel -> {x, t}];\)\ \), "\n", 
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        AxesLabel -> {x, t, "\<w\>"}];\)\)}], "Input"],

Cell[TextData[
"What is happening as t\[Rule]\[Infinity] ? Is this what you would expect? \
Why?"], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Rod with Insulated Ends and Use of the Fourier Series", "Section"],

Cell[TextData[{
  "Now suppose that the rod of length 4 has insulated ends, that is, ",
  Cell[BoxData[
      \(TraditionalForm\`w\_x\)]],
  "(0,t) = ",
  Cell[BoxData[
      \(TraditionalForm\`w\_x\)]],
  "(4,t) = 0. Assume a discontinuous initial temperature distribution - 12 \
for the left 1/4 of the rod and 0 elsewhere. This discontinuous initial \
temperature function necessitates the use of the Fourier series in order to \
get a continuous approximation to this discontinuous initial state. The \
following is a sketch of how it might appear.\nFirst determine the Fourier \
Series corresponding to the initial temperature state for the rod. "
}], "Text"],

Cell[BoxData[{
    \(\na[n_] := \(a[n] = 
        1/2  Integrate[12  Cos[n\ Pi\ x/4], {x, 0, 1}]\)\t\ \), "\n", 
    \(fourcos[x_, n_] := 
      a[0]/2 + Sum[a[j] Cos[j\ Pi\ x/4], {j, 1, n}]\), "\n", 
    \(\(Plot[fourcos[x, 50], {x, 0, 4}];\)\ \ \)}], "Input"],

Cell["\<\
Now write the solution to the heat equation allowing a variable number of \
terms of the Fourier series to be used. The symbol \"n\" represents the \
number of terms in the Fourier series to be used. More terms are needed when \
t is closer to 0 in order to get a good approximation.\
\>", "Text"],

Cell[BoxData[{
    \(Clear[w, xlevelplots, tlevelplots]\ \), "\n", 
    \(w[x_, t_, 
        n_] := \ \(\(a[0]/2\)\(+\)\(Sum[
          a[j] Exp[\(-\((j\ Pi/4)\)^2\)\ *\ t] Cos[j\ Pi\ x/4], {j, 1, 
            n}]\)\(\ \)\)\)}], "Input"],

Cell["\<\
Plot the temperature function at various positions along the rod as a \
function of time.  Interpret the results.\
\>", "Text"],

Cell[BoxData[
    \(\(xlevelplots = 
        Plot[\ {w[0, t, 10], w[ .8, t, 10], w[1.6, t, 10], w[2.4, t, 10], 
            w[3.2, t, 10], w[4, t, 10]}, {t, 0, 5}, \ 
          PlotStyle -> {RGBColor[1, 0, 0], RGBColor[1, 1, 0], 
              RGBColor[0, 1, 0], RGBColor[0, 1, 1], RGBColor[1, 0, 1], 
              RGBColor[0, 0, 1]}]\ ;\)\)], "Input"],

Cell["\<\
Plot the temperature function at various times as a function of the position \
on the rod.  Interpret the results.\
\>", "Text"],

Cell[BoxData[
    \(\(tlevelplots = 
        Plot[{w[x, 0, 70], w[x,  .5, 20], w[x, 1, 10], w[x, 1.5, 10], 
            w[x, 2, 10], w[x, 2.5, 10]}, {x, 0, 4}, 
          PlotStyle -> {RGBColor[1, 0, 0], RGBColor[1, 1, 0], 
              RGBColor[0, 1, 0], RGBColor[0, 1, 1], RGBColor[1, 0, 1], 
              RGBColor[0, 0, 1]}, \ \ PlotRange -> All]\ ;\)\)], "Input"],

Cell["\<\
Plot the contours for the temperature function and the temperature function \
as a surface.\
\>", "Text"],

Cell[BoxData[{
    \(\(ContourPlot[w[x, t, 30], {x, 0, 4}, {t, 0, 5}];\)\ \), "\n", 
    \(\(Plot3D[w[x, t, 50], {x, 0, 4}, {t, 0, 5}];\)\)}], "Input"],

Cell["\<\
What is happening as t\[Rule]\[Infinity] ? Is this what you would expect? \
Why?\
\>", "Text"]
}, Closed]]
}, Open  ]]
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