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Cell[CellGroupData[{
Cell["How Does Heat Dissipate?", "Title",
  FontWeight->"Bold"],

Cell[TextData[StyleBox[
"Chapter 11, End of Chapter Additional Exercises, Final Part",
  FontFamily->"Arial",
  FontSize->16,
  FontWeight->"Bold"]], "Text"],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell["\<\
OBJECTIVE:  Observe the physical interpretations of the contour and level \
curve plots and see an application of the Fourier Series.\
\>", "Text"],

Cell[TextData[{
  "Do you want to see physical interpretations of the contours and level \
curves you have been drawing? This exploration of the heat equation will \
enable you to do so.  The heat equation is a partial differential equation \
and its solution employs the Fourier series in a meaningful way. \nNote",
  StyleBox[":",
    FontWeight->"Bold"],
  "  Part II requires knowledge of the Fourier series."
}], "Text"],

Cell[CellGroupData[{

Cell["Technology Guidelines", "Subsection",
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Cell[TextData[{
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  StyleBox[
  " before executing a new module.\nTO OPEN CELLS, put your cursor on the \
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  StyleBox["Abort Evaluation.\n",
    FontSlant->"Italic"],
  "ORDER OF EXECUTION\n\tExecute cells in the order given. Do not skip any \
Input cells within a given notebook.\nSAVING NOTEBOOKS\n\tYou can save \
anytime to any directory you choose, and it is wise to save often.\n\t\
However, before you do your final save, delete all your output by selecting \
the \n\t ",
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then shut down ",
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    FontSlant->"Italic"],
  " and start it up again."
}], "Text"]
}, Closed]]
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Cell[CellGroupData[{

Cell["Background", "Section"],

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["L",
    FontSlant->"Italic"],
  " with perfectly insulated sides. The value of the positive constant ",
  Cell[BoxData[
      \(TraditionalForm\`c\^2\)]],
  " is determined by the material from which the rod is made."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Part 1: Fixed End Temperatures", "Section"],

Cell[TextData[{
  "We first examine a very simple solution to the problem where we assume \
that the ends of the rod are immersed in ice and are maintained at a \
temperature of ",
  Cell[BoxData[
      \(TraditionalForm\`0\^0\)]],
  "C. If the rod is of length 4, the 0 degrees at both ends is represented by \
the boundary conditions ",
  Cell[BoxData[
      \(TraditionalForm\`w(0, t)\  = \ \(w(4, t)\  = \ 0\)\)]],
  ". Assume also that the initial temperature distribution within the rod is \
given by ",
  Cell[BoxData[
      \(TraditionalForm\`w(x, 0)\  = \ sin\ 2  \[Pi]x\)]],
  ".\n\nWe first show that ",
  Cell[BoxData[
      FormBox[
        RowBox[{\(w(x, t)\), " ", "=", " ", 
          RowBox[{
            FormBox[\(e\^\(\(-3\) t\)\),
              "TraditionalForm"], \(sin(2  \[Pi]\ x)\)}]}], 
        TraditionalForm]]],
  " is a solution to the differential equation with these boundary and \
initial conditions. The value of the constant",
  StyleBox[" c",
    FontSlant->"Italic"],
  " for this problem is ",
  Cell[BoxData[
      InterpretationBox[\("c = "\[InvisibleSpace]\@3\/\(2\ \[Pi]\)\
\[InvisibleSpace]" or "\[InvisibleSpace]0.27566444771089604`\),
        SequenceForm[ "c = ", 
          Times[ 
            Rational[ 1, 2], 
            Power[ 3, 
              Rational[ 1, 2]], 
            Power[ Pi, -1]], " or ", .27566444771089604],
        Editable->False]]],
  "."
}], "Text"],

Cell[BoxData[{
    \(Off[General::spell]\ \), "\n", 
    \(Off[General::spell1]\ \), "\[IndentingNewLine]", 
    \(\(c = \@3\/\(2  \[Pi]\);\)\), "\n", 
    \(w[x_, t_] := \ \(E\^\(\(-3\) t\)\) Sin[2  \[Pi]\ x]\), "\n", 
    \(Print["\<The temperature at the left end of the rod is \>", 
      w[0, t]]\), "\n", 
    \(Print["\<The temperature at the right end of the rod is \>", 
      w[4, t]]\), "\n", 
    \(Print["\<The initial temperature distribution is \>", 
      w[x, 0]]\), "\n", 
    \(Print["\<The second derivative of the temperature with respect to \
position is \>", wxx\  = D[w[x, t], {x, 2}]]\), "\n", 
    \(Print["\<The first derivative of the temperature with respect to time \
is \>", wt = D[w[x, t], {t, 1}]]\), "\n", 
    \(If[wxx == 1/c\^2\ wt, 
      Print[w[x, t], "\< satisfies the differential equation.\>"], 
      Print["\<The equation is not satisfied.\>"]]\)}], "Input"],

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

Cell[BoxData[
    \(\(xlevelplots = 
        Plot[\ {w[0, t], w[ .8, t], w[1.6, t], w[2.4, t], w[3.2, t], 
            w[4, t]}, {t, 0, 1}, \ 
          PlotStyle -> {RGBColor[1, 0, 0], RGBColor[1, 1, 0], 
              RGBColor[0, 1, 0], RGBColor[0, 1, 1], RGBColor[1, 0, 1], 
              RGBColor[0, 0, 1]}, AxesLabel -> {t, "\<w\>"}, 
          PlotLabel -> "\<X-Level Plots\>"]\ ;\)\)], "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. \
\>", "Text"],

Cell[BoxData[
    \(\(tlevelplots = 
        Plot[{w[x, 0], w[x,  .2], w[x,  .4], w[x,  .6], w[x,  .8], 
            w[x, 1]}, {x, 0, 4}, 
          PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 1], 
              RGBColor[0, 1, 0], RGBColor[0, 1, 1], RGBColor[1, 0, 1], 
              RGBColor[0, 0, 1]}, \ \ PlotRange -> All, 
          AxesLabel -> {x, "\<w\>"}, 
          PlotLabel -> "\<T-Level Plots\>"]\ ;\)\)], "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, 4}, {t, 0, 1}, PlotPoints \[Rule] 40, 
        Axes -> True, AxesLabel -> {x, t}];\)\ \), "\n", 
    \(\(Plot3D[w[x, t], {x, 0, 4}, {t, 0, 1}, PlotPoints \[Rule] 40, 
        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[TextData[{
  "You Try It: Part I",
  Cell[BoxData[
      \(TraditionalForm\`-\)]],
  "Insulated Ends "
}], "Section"],

Cell[TextData[{
  "We will next examine a very simple solution to the problem in which it is \
assumed that the ends of the rod are insulated so that no heat can escape. If \
the rod is of length 4, this is represented by the boundary conditions ",
  Cell[BoxData[
      \(TraditionalForm\`\(w\_x\)(0, t)\  = \ \(\(w\_x\)(4, t)\  = \ 0\)\)]],
  ". \n\nWhy? Can you explain this in your own words?"
}], "Text"],

Cell[TextData[{
  "Assume also that the initial temperature distribution within the rod is \
given by w(x,0) = cos {2\[Pi] x).\nFirst show that ",
  Cell[BoxData[
      FormBox[
        RowBox[{" ", 
          StyleBox[
            RowBox[{\(w \((x, t)\)\), "=", " ", 
              RowBox[{
                FormBox[\(e\^\(\(-3\) t\)\),
                  "TraditionalForm"], "cos", \((2  \[Pi]\ x\)}]}],
            FontSlant->"Italic"], "}"}], TextForm]]],
  " is a solution to the differential equation with these boundary and \
initial conditions. The value of the constant",
  StyleBox[" c",
    FontSlant->"Italic"],
  " for this problem is ",
  Cell[BoxData[
      InterpretationBox[\("c = "\[InvisibleSpace]\@3\/\(2\ \[Pi]\)\
\[InvisibleSpace]" or "\[InvisibleSpace]0.27566444771089604`\),
        SequenceForm[ "c = ", 
          Times[ 
            Rational[ 1, 2], 
            Power[ 3, 
              Rational[ 1, 2]], 
            Power[ Pi, -1]], " or ", .27566444771089604],
        Editable->False]]],
  "\n\nFirst define the function."
}], "Text"],

Cell[BoxData[{
    \(Clear[x, t, w]\), "\n", 
    \(w[x_, t_] := Exp[\(-3\) t] Cos[2\ \[Pi]\ x]\), "\[IndentingNewLine]", 
    \(\(c = \@3\/\(2  \[Pi]\);\)\)}], "Input"],

Cell["\<\
Now find the partial derivatives that appear in the equation.\
\>", "Text"],

Cell[BoxData[{
    \(Print["\<The second derivative of the temperature with respect to \
position is \>", wxx\  = D[w[x, t], {x, 2}]]\), "\n", 
    \(Print["\<
		The first derivative of the temperature with respect to time is \>", 
      wt = D[w[x, t], {t, 1}]]\)}], "Input"],

Cell["\<\
Verify that the function satisfies the partial differential equation and the \
initial and boundary conditions.\
\>", "Text"],

Cell[BoxData[{
    \(Print["\<The change in temperature at the left end of the rod is \>", 
      D[w[x, t], x] /. x -> 0]\), "\n", 
    \(Print["\<The change in temperature at the right end of the rod is \>", 
      D[w[x, t], x] /. x -> 4]\), "\n", 
    \(Print["\<The initial temperature distribution is \>", 
      w[x, 0]]\), "\n", 
    \(If[wxx == 1/c\^2\ wt, 
      Print[w[x, t], "\< satisfies the differential equation.\>"], 
      Print["\<The equation is not satisfied.\>"]]\)}], "Input"],

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

Cell[BoxData[
    \(\(xlevelplots = 
        Plot[\ {w[0, t], w[ .8, t], w[1.6, t], w[2.4, t], w[3.2, t], 
            w[4, t]}, {t, 0, 1}, \ 
          PlotStyle -> {RGBColor[1, 0, 0], RGBColor[1, 1, 0], 
              RGBColor[0, 1, 0], RGBColor[0, 1, 1], RGBColor[1, 0, 1], 
              RGBColor[0, 0, 1]}, AxesLabel -> {t, "\<w\>"}, 
          PlotLabel -> "\<X-Level Plots\>"]\ ;\)\)], "Input"],

Cell[TextData[StyleBox[
"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. \
\>", "Text"],

Cell[BoxData[
    \(\(tlevelplots = 
        Plot[{w[x, 0], w[x,  .2], w[x,  .4], w[x,  .6], w[x,  .8], 
            w[x, 1]}, {x, 0, 4}, 
          PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 1], 
              RGBColor[0, 1, 0], RGBColor[0, 1, 1], RGBColor[1, 0, 1], 
              RGBColor[0, 0, 1]}, \ \ PlotRange -> All, 
          AxesLabel -> {x, "\<w\>"}, 
          PlotLabel -> "\<T-Level Plots\>"]\ ;\)\)], "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, 4}, {t, 0, 1}, PlotPoints \[Rule] 40, 
        Axes -> True, AxesLabel -> {x, t}];\)\ \), "\n", 
    \(\(Plot3D[w[x, t], {x, 0, 4}, {t, 0, 1}, PlotPoints \[Rule] 40, 
        AxesLabel -> {"\<x\>", "\<t\>", "\<w\>"}];\)\)}], "Input"],

Cell[TextData[
"What is happening as t\[Rule]\[Infinity] ? Is this what you would expect? \
Why? \n\nHow does this differ from the temperature function in the rod with \
fixed ends?"], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
Part II: Insulated Ends and Use of the Fourier Series (demonstration only)\
\>", "Section"],

Cell[TextData[StyleBox[
"If you have not learned the Fourier series at the end of Chapter 8 of your \
text, you should not do this part of the module. ",
  FontWeight->"Bold"]], "Text"],

Cell[TextData[{
  "Suppose that the rod of length 4 has insulated ends, that is, ",
  Cell[BoxData[
      FormBox[
        RowBox[{" ", 
          RowBox[{
            RowBox[{
              FormBox[\(w\_x\),
                "TraditionalForm"], "(", \(0, t\), ")"}], " ", "=", " ", 
            RowBox[{
              RowBox[{
                FormBox[\(w\_x\),
                  "TraditionalForm"], "(", \(4, t\), ")"}], " ", "=", " ", 
              "0"}]}]}], TraditionalForm]]],
  ". Assume a discontinuous initial temperature distribution that is  ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\ \)\(12\)\)\)]],
  " for the left ",
  Cell[BoxData[
      \(TraditionalForm\`1/4\)]],
  " of the rod and ",
  Cell[BoxData[
      \(TraditionalForm\`0\)]],
  " 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. For this part, we will assume that c = 1.\n\
\nFirst, we determine the Fourier Series corresponding to the initial \
temperature state for the rod. "
}], "Text"],

Cell[BoxData[{
    \(a[n_] := \(a[n] = 
        1/2  Integrate[12  Cos[n\ Pi\ x/4], {x, 0, 1}]\)\t\ \), "\n", 
    \(Print["\<The Fourier cosine series out to n = 30 is \>", 
      fourcos[x_, 30] = 
        a[0]/2 + Sum[a[j] Cos[j\ Pi\ x/4], {j, 1, 30}] // N]\), "\n", 
    \(\(Plot[fourcos[x, 30], {x, 0, 4}];\)\ \ \)}], "Input"],

Cell[BoxData[{
    \(TextForm\`If\ you\ were\ in\ a\ course\ that\ studied\ partial\ \
differential\ equations, \ 
    you\ could\ derive\ the\ solution\ to\ the\ heat\ equation\ to\ be\ the\ \
following . \n\t\t\ \(\(\(\ \)\(a \((0)\)\)\)\/2\ \  + \ \ \ \[Sum]\)\+\(\(\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(n = 1\)\)\%\(\(\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\[Infinity]\)\) 
      a \((n)\)\ \(e\^\(\(-\ \((n\[Pi]\/4)\)\^2\) t\)\) 
      Cos \((\(n\[Pi]\/4\) x)\)\), "\n", 
    \(TextForm\`We\ will\ write\ write\ it\ out\ to\ n = 30. \)}], "Text"],

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

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

Cell[BoxData[
    \(\(xlevelplots = 
        Plot[\ {w[0, t], w[ .8, t], w[1.6, t], w[2.4, t], w[3.2, t], 
            w[4, t]}, {t, 0, 5}, \ 
          PlotStyle -> {RGBColor[1, 0, 0], RGBColor[1, 1, 0], 
              RGBColor[0, 1, 0], RGBColor[0, 1, 1], RGBColor[1, 0, 1], 
              RGBColor[0, 0, 1]}, AxesLabel -> {t, "\<w\>"}, 
          PlotLabel -> "\<X-Level Plots\>"]\ ;\)\)], "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[
    \(\(tlevelplots = 
        Plot[{w[x, 0], w[x,  .3], w[x, 6], w[x, 1], w[x, 1.5], w[x, 2]}, {x, 
            0, 4}, PlotStyle -> {RGBColor[1, 0, 0], RGBColor[1, 1, 0], 
              RGBColor[0, 1, 0], RGBColor[0, 1, 1], RGBColor[1, 0, 1], 
              RGBColor[0, 0, 1]}, \ \ PlotRange -> All, 
          AxesLabel -> {x, "\<w\>"}, 
          PlotLabel -> "\<T-Level Plots\>"]\ ;\)\)], "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[{
    \(\(ContourPlot[w[x, t], {x, 0, 4}, {t, 0, 5}, Axes -> True, 
        AxesLabel -> {x, t}];\)\ \), "\n", 
    \(\(Plot3D[w[x, t], {x, 0, 4}, {t, 0, 5}, 
        AxesLabel -> {x, t, "\<w\>"}];\)\)}], "Input"],

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