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Notebook[{
Cell["\<\
Time-Dependent Solution to the Diffusion Equation in the Plane with \
a Point Source at the Origin.\
\>", "Subtitle"],

Cell[BoxData[
    StyleBox[\(<< Graphics`\),
      FontWeight->"Bold"]], "Input"],

Cell["\<\
These notes are based on notes written by Prof. W.C. Carter for MIT \
subject 3.016.  These are excerpts with some additional comments and \
extensions for MIT subject 3.21. \
\>", "Text",
  CellFrame->True,
  Background->GrayLevel[0.833326]],

Cell["\<\
Note: This notebook will produce some neat animations but they each \
take some time to compute. And, if you save the notebook with the graphics, \
it will require 30MB or so of hard disk space.\
\>", "Text",
  CellFrame->True,
  Background->GrayLevel[0.833326]],

Cell["\<\
You should definitely take the time to learn how to make your own \
animations because the results can be really instructive. You can even save \
animations as Quicktime files and import and play them back in other programs \
such as PowerPoint.\
\>", "Text",
  CellFrame->True,
  Background->GrayLevel[0.833326]],

Cell["Scalar field: concentration", "Subsubtitle"],

Cell["\<\
As an example we will look at the concentration field arising from \
a line source in an isotropic medium (e.g. think of a Cu wire embedded in a \
block of Ni; allow interdiffusion to occur for time t > 0).\
\>", "Text",
  CellFrame->True,
  Background->GrayLevel[0.833326]],

Cell["\<\
Define the concentration c(x,y,t) by the following (we will soon \
learn where this equation comes from):\
\>", "Text",
  CellFrame->True,
  Background->GrayLevel[0.833326]],

Cell[BoxData[
    \(concentration\  = 
      Exp[\(-\((x^2\  + \ y^2)\)\)/t]/\((\(\@\[Pi]\) t)\)\)], "Input"],

Cell["\<\
Illustrate the time dependence of this scalar field with a seqence \
of plots...\
\>", "Text",
  CellFrame->True,
  Background->GrayLevel[0.833326]],

Cell[BoxData[
    StyleBox[\(<< Graphics`Animation`\),
      FontWeight->"Bold"]], "Input"],

Cell["\<\
To see animations,use the menu command to group all the graphics \
cells together, and then animate\
\>", "Text",
  CellFrame->True,
  Background->GrayLevel[0.833326]],

Cell[BoxData[
    \(MoviePlot3D[
      concentration, {x, \(-4\), 4}, {y, \(-4\), 4}, {t, 0.01, 2.51,  .05}, 
      PlotPoints \[Rule] 40, PlotRange \[Rule] {0, 2}, 
      DisplayFunction \[Rule] Identity]\)], "Input"],

Cell["\<\
Gradient of a scalar field and relation to flux in a concentration \
field\
\>", "Subsubtitle"],

Cell[TextData[{
  "Now look at the ",
  StyleBox["gradient",
    FontSlant->"Italic"],
  " of this scalar field.  The gradient at a point is a vector directed \
toward the fastest rate of change (\"steepest ascent\") of the scalar field."
}], "Text",
  CellFrame->True,
  Background->GrayLevel[0.833326]],

Cell["\<\
Flux is a vector that points in the direction of the flow and is a \
measure of how much is flowing per unit time.  This illustration is a simple \
but common situation in which the flux is antiparallel to the concentration \
gradient, so the resulting vector field also illustrates the negative of the \
gradient of the concentration field.\
\>", "Text",
  CellFrame->True,
  Background->GrayLevel[0.833326]],

Cell[BoxData[
    \(<< Graphics`PlotField`\)], "Input"],

Cell[BoxData[
    \(flux\  = \ {\(-D[concentration, x]\), \(-D[concentration, 
            y]\)}\)], "Input"],

Cell[BoxData[
    \(This\ is\ an\ example\ of\ a\ time - 
      dependent\ vector\ field\ \(j\&\[LongRightArrow]\) \((x, y, 
          t)\)\)], "Text"],

Cell[BoxData[
    \({\(2\ \[ExponentialE]\^\(\(\(-x\^2\) - y\^2\)\/t\)\ x\)\/\(\@\[Pi]\ \
t\^2\), \(2\ \[ExponentialE]\^\(\(\(-x\^2\) - y\^2\)\/t\)\ y\)\/\(\@\[Pi]\ \
t\^2\)}\)], "Output"],

Cell[BoxData[
    \(Animate[
      PlotVectorField[flux, {x, \(-2\), 2}, {y, \(-2\), 2}, 
        PlotPoints \[Rule] 20, 
        ScaleFunction \[Rule] \((If[# <  .05, 0, \((100.0  #)\)] &)\), 
        ColorFunction \[Rule] \((Hue[1 - 0.75  #] &)\)], {t, 0.01, 
        4.01,  .05}]\)], "Input"],

Cell["\<\
Divergence and relation to accumulation in a concentration \
field\
\>", "Subsubtitle"],

Cell["\<\
Now look at the accumulation which is the negative of the \
divergence of the flux.\
\>", "Text"],

Cell["\<\
Define a function that takes a two-dimensional vector function of \
(x,y) as an argument and returns its divergence\
\>", "Text"],

Cell[BoxData[
    \(divergence[{xcomp_\ , \ ycomp_}]\  := \ 
      Simplify[D[xcomp, x]\  + \ D[ycomp, y]]\)], "Input"],

Cell[BoxData[
    \(divgradptsource\  = \ divergence[\(-flux\)]\)], "Input"],

Cell[BoxData[
    \(Plotting\ the\ divergence\ of\ the\ gradient\ \((\[Del]\(\(\[CenterDot]\
\)\((\[Del]\ f)\)\)\ is\ the\ ``Laplacian''\ \[Del]\^2\ f, \ 
        sometimes\ indicated\ with\ symbol\ \[CapitalDelta]f)\)\)], "Text",
  FontFamily->"Helvetica"],

Cell[BoxData[
    \(MoviePlot3D[
      divgradptsource, {x, \(-4\), 4}, {y, \(-4\), 4}, {t, 0.01, 2.51,  .05}, 
      PlotPoints \[Rule] 40, PlotRange \[Rule] {\(-2\), 2}, 
      DisplayFunction \[Rule] Identity]\)], "Input"],

Cell["\<\
You should review these animations and ensure that you understand \
their behavior, particularly the time and position dependence of the flux \
vector, and the time and position dependence of the accumulation.\
\>", "Text"]
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WindowTitle->"Lecture 11 MIT 3.016   \[Copyright] W. Craig Carter 2003",
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]

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