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Notebook[{
Cell["\<\
Solution of the Heat Equation for transient conduction by LaPlace \
Transform\
\>", "Title"],

Cell[TextData[{
  "This notebook has been written in ",
  StyleBox["Mathematica ",
    FontSlant->"Italic"],
  "by \n\nMark J. McCready\nProfessor and Chair of Chemical Engineering\n\
University of Notre Dame\nNotre Dame IN 46556\nUSA\n\n\
Mark.J.McCready.1@nd.edu\nhttp://www.nd.edu/~mjm/\n\n\nIt is copyrighted to \
the extent allowed by whatever laws pertain to the World Wide Web and the \
Internet.\n\nI would hope that as a professional courtesy, that if you use \
it, that this notice remain visible to other users. \nThere is no charge for \
copying and dissemination   \n\nVersion:  3/17/98\nMore recent versions of \
this notebook should be available at the web site:\n\
http://www.nd.edu/~mjm/heatlaplace.nb"
}], "Text"],

Cell[TextData[{
  StyleBox[
  "This notebook shows how to solve transient heat conduction in a \
semi-infinite slab.  It is intended as a supplement to ",
    FontSize->13,
    FontWeight->"Plain"],
  StyleBox["\n",
    FontSize->13],
  StyleBox[" L. G. Leal  (1992) ", "SmallText",
    FontSize->13],
  StyleBox["Laminar flow and Convective Transport Processes", "SmallText",
    FontSize->13,
    FontWeight->"Plain",
    FontSlant->"Italic"],
  StyleBox[", Butterworth", "SmallText",
    FontSize->13,
    FontWeight->"Plain"],
  StyleBox[" ", "SmallText",
    FontSize->13],
  StyleBox["pp 139-144", "SmallText",
    FontSize->13,
    FontWeight->"Plain"],
  StyleBox[".", "SmallText",
    FontSize->13],
  StyleBox["\n \n", "SmallText",
    FontSize->14],
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    FontSize->8,
    FontWeight->"Plain"],
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    FontSize->13,
    FontWeight->"Plain"],
  StyleBox["linear transform techniques",
    FontSize->13,
    FontWeight->"Plain",
    FontSlant->"Italic"],
  StyleBox[
  " but does not give examples.  Many students are not familiar with these.  \
This notebook is intended to illustrate the use of the LaPlace transform to \
solve a simple PDE, and to show how it is implemented in ",
    FontSize->13,
    FontWeight->"Plain"],
  StyleBox["Mathematica",
    FontSize->13,
    FontWeight->"Plain",
    FontSlant->"Italic"],
  StyleBox[".",
    FontSize->13,
    FontWeight->"Plain"],
  StyleBox["\n \n",
    FontSize->13],
  StyleBox[
  "This problem is the heat transfer analog to the \"Rayleigh\" problem that \
starts on page 91. ",
    FontSize->13,
    FontWeight->"Plain"],
  StyleBox[" ",
    FontSize->14]
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  AspectRatioFixed->True],

Cell["Problem formulation", "Subsection"],

Cell[TextData[{
  "Consider a semi-infinite slab where the distance variable, ",
  StyleBox["y",
    FontSlant->"Italic"],
  ", goes from 0 to \[Infinity].  The temperature is initially uniform within \
the slab and we can consider it to be 0.  At ",
  StyleBox["t",
    FontSlant->"Italic"],
  "=0, the temperature at ",
  StyleBox["y",
    FontSlant->"Italic"],
  "=0 is suddenly increased to 1.  We would like to calculate the temperature \
as a function of time, ",
  StyleBox["t",
    FontSlant->"Italic"],
  ", within the slab.\n\nThis problem is commonly solved by a ",
  StyleBox["similarity variable",
    FontSlant->"Italic"],
  " technique that arises because the absence of a physical length scale.  In \
this notebook we use the ",
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  ", which is an integral transform that effectively converts (linear) PDEs, \
(if we wish to use it on say ",
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  ", which is effectively a ",
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    FontSlant->"Italic"],
  " and the equations need to be solved for all positive ",
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    FontSlant->"Italic"],
  ".                "
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Cell["Equation", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
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  "the boundary conditions are \n\[Theta](",
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  ")=0 ",
  StyleBox["t ",
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  "< 0,\n\[Theta](",
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  ", ",
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  "> 0.  \n"
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Cell[TextData[{
  "initialization of ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", load a package"
}], "Subsubsection"],

Cell[BoxData[
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Cell["Here is the analysis", "Subsubsection"],

Cell[TextData[{
  " If we use a transform technique, we intend to simplify the problem by \
transforming the pde to an ode (or an algebraic equation from an ode).  Once \
the transform is done, we will need to evaluate the integral that arises a \
the boundaries.  So the boundary conditions and the domain of the problem \
must be in a form conducive to this.  The Laplace transform is defined from 0 \
to \[Infinity].  In this problem both of the domains are from 0 to \
\[Infinity], however first try to do the transform in ",
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  ".  In ",
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  ".  "
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  "The first two terms make an ode in the transformed \[Theta]",
  StyleBox["(t,y",
    FontSlant->"Italic"],
  ").  Let's call this ",
  Cell[BoxData[
      \(\(\[Theta]\&^\)\)]],
  ".  The last term, which arose from integration by parts of the \[PartialD]\
\[Theta](",
  StyleBox["t,y",
    FontSlant->"Italic"],
  ")/\[PartialD]t term, must be evaluated from the boundary conditions.  The \
temperature for all ",
  StyleBox["y",
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  " at ",
  StyleBox["t",
    FontSlant->"Italic"],
  "=0 is zero, thus this term is 0.  Note that the ",
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    FontSlant->"Italic"],
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calculation) because it is multiplied by Exp[-",
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  StyleBox["Mathematica",
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  ".  The LaplaceTransform[",
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    FontSlant->"Italic"],
  "] does not affect any ",
  StyleBox["y",
    FontSlant->"Italic"],
  " derivatives.  We can write"
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Cell[TextData["Now get the solution out of the {{ }}'s."], "Text",
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        RowBox[{\(\[ExponentialE]\^\(\(\@s\ y\)\/\@\[Alpha]\)\), " ", 
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  "We see that we cannot stand an exponentially increasing part so that ",
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  "=0.  Now, we need to evaluate ",
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  "(y,s) at ",
  StyleBox["y",
    FontSlant->"Italic"],
  "=0 or at some place that we know it.  Well, we know \[Theta][t,y] @ y=0 \
(=1).  Thus, we can transform this to get a value for \[Theta][y=0,s]."
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Cell[CellGroupData[{

Cell[BoxData[
    \(bc0 = LaplaceTransform[1, t, s]\)], "Input",
  CellLabel->"In[255]:=",
  AspectRatioFixed->True],

Cell[BoxData[
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Cell[TextData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[TextData["Thus our solution in transformed space is"], "Text",
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Cell[CellGroupData[{

Cell[TextData[{
  "Plot solution in frequency space to see what ",
  StyleBox["s",
    FontSlant->"Italic"],
  " does"
}], "Subsubsection"],

Cell[TextData["What does this look like ??"], "Text",
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