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Cell["Math 221 - Numerical PDE's, Lab 2", "Subtitle"],

Cell["\<\
We've seen how finite difference formulas may be derived by various \
means in Lab. 1. Today we'll look at applying these formulas to generate ODE \
solvers and test the solvers.\
\>", "Text"],

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Cell["\<\
1.  Numerically solve an ODE by finite difference approximation of \
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\>", "Section"],

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Cell[CellGroupData[{

Cell["First-order forward difference approximation", "Subsection"],

Cell[CellGroupData[{

Cell["Find the update formula", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[{
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["Define the update rule for the sequence of values Qf2", "Text"],

Cell[CellGroupData[{

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

Cell["Carry out the computation for specific numerical parameters", \
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Cell[BoxData[{
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Cell["Set initial conditions", "Text"],

Cell[BoxData[
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Cell["Carry out computations and compare with exact solution", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
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    \(\(qplot\  = \ 
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}, Closed]],

Cell[CellGroupData[{

Cell["First-order backward difference approximation", "Subsection"],

Cell[CellGroupData[{

Cell["Find the update formula", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Clear[k, a, \[Lambda]]; \ order = 1;\), "\[IndentingNewLine]", 
    \(s1backward[q[t], order]\)}], "Input"],

Cell[BoxData[
    \(\(q[t] - q[\(-k\) + t]\)\/k\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(fmodel[t, q, \[Lambda], a]\)], "Input"],

Cell[BoxData[
    \(\(-\[Lambda]\)\ q[t] + a\ Sin[t]\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(fq\)\(\ \)\(=\)\(\ \)\(First[\ 
        Solve[s1backward[q[t], order]\  \[Equal] \ 
            fmodel[t, q, \[Lambda], a], q[t]]\ ]\)\(\ \)\)\)], "Input"],

Cell[BoxData[
    \({q[t] \[Rule] \(q[\(-k\) + t] + a\ k\ Sin[t]\)\/\(1 + k\ \
\[Lambda]\)}\)], "Output"]
}, Open  ]],

Cell["Define the update rule for the sequence of values Qf2", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Clear[Q, n]\), "\[IndentingNewLine]", 
    \(UpdateRule\  = \ 
      Simplify[\ \(fq\ \  /. \ {t \[Rule] n\ k}\)\  /. \ 
          q[t_] \[Rule] Q[t/k]\ \ ]\)}], "Input"],

Cell[BoxData[
    \({Q[n] \[Rule] \(Q[\(-1\) + n] + a\ k\ Sin[k\ n]\)\/\(1 + k\ \
\[Lambda]\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Clear[Qupdate];\)\), "\[IndentingNewLine]", 
    \(Qupdate[n_]\  = \ Q[n]\  /. \ UpdateRule\)}], "Input"],

Cell[BoxData[
    \(\(Q[\(-1\) + n] + a\ k\ Sin[k\ n]\)\/\(1 + k\ \[Lambda]\)\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Carry out the computation for specific numerical parameters", \
"Subsubsection"],

Cell[BoxData[{
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Cell[CellGroupData[{

Cell["Second-order forward difference approximation", "Subsection"],

Cell[CellGroupData[{

Cell["Find the update formula", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[{
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

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

Cell[BoxData[
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Cell[BoxData[
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Cell["Define the update rule for the sequence of values Qf2", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Clear[Q, n];\)\), "\[IndentingNewLine]", 
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[{
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell["Carry out the computation for specific numerical parameters", \
"Subsubsection"],

Cell[BoxData[{
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Cell["Set initial conditions", "Text"],

Cell[BoxData[
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Cell["Carry out computations and compare with exact solution", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
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    \(\(qplot\  = \ 
        Plot[qmodel[t, \[Lambda], a], {t, 0, T}, 
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    \(\(Show[Qplot, qplot, DisplayFunction \[Rule] $DisplayFunction, 
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}, Closed]],

Cell[CellGroupData[{

Cell["Second-order backward difference approximation", "Subsection"],

Cell[CellGroupData[{

Cell["Find the update formula", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Clear[k, a, \[Lambda]]; \ order = 2;\), "\[IndentingNewLine]", 
    \(s1backward[q[t], order]\)}], "Input"],

Cell[BoxData[
    \(\(3\ q[t] + q[\(-2\)\ k + t] - 4\ q[\(-k\) + t]\)\/\(2\ k\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(fmodel[t, q, \[Lambda], a]\)], "Input"],

Cell[BoxData[
    \(\(-\[Lambda]\)\ q[t] + a\ Sin[t]\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(fq\)\(\ \)\(=\)\(\ \)\(First[\ 
        Solve[s1backward[q[t], order]\  \[Equal] \ 
            fmodel[t, q, \[Lambda], a], q[t]]\ ]\)\(\ \)\)\)], "Input"],

Cell[BoxData[
    \({q[t] \[Rule] \(\(-q[\(-2\)\ k + t]\) + 4\ q[\(-k\) + t] + 2\ a\ k\ \
Sin[t]\)\/\(3 + 2\ k\ \[Lambda]\)}\)], "Output"]
}, Open  ]],

Cell["Define the update rule for the sequence of values Qf2", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Clear[Q, n]\), "\[IndentingNewLine]", 
    \(UpdateRule\  = \ 
      Simplify[\ \(fq\ \  /. \ {t \[Rule] n\ k}\)\  /. \ 
          q[t_] \[Rule] Q[t/k]\ \ ]\)}], "Input"],

Cell[BoxData[
    \({Q[n] \[Rule] \(\(-Q[\(-2\) + n]\) + 4\ Q[\(-1\) + n] + 2\ a\ k\ Sin[k\ \
n]\)\/\(3 + 2\ k\ \[Lambda]\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Clear[Qupdate];\)\), "\[IndentingNewLine]", 
    \(Qupdate[n_]\  = \ Q[n]\  /. \ UpdateRule\)}], "Input"],

Cell[BoxData[
    \(\(\(-Q[\(-2\) + n]\) + 4\ Q[\(-1\) + n] + 2\ a\ k\ Sin[k\ n]\)\/\(3 + 2\
\ k\ \[Lambda]\)\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Carry out the computation for specific numerical parameters", \
"Subsubsection"],

Cell[BoxData[{
    \(\(Clear[Q, k, \[Lambda], a];\)\), "\[IndentingNewLine]", 
    \(k = 1. ; \ \[Lambda] = 0.05; \ a = 1. ;\), "\[IndentingNewLine]", 
    \(Q[n_]\  := \ \(Q[n]\  = \ \ Qupdate[n]\)\)}], "Input"],

Cell["Set initial conditions", "Text"],

Cell[BoxData[
    \(For[n = 0, n < order, \(n++\), 
      Q[n] = qmodel[n\ k, \[Lambda], a]]\)], "Input"],

Cell["Carry out computations and compare with exact solution", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Qvalues = 
        Table[{k\ n, Q[n]}, {n, 0, T/k}];\)\), "\[IndentingNewLine]", 
    \(\(Qplot\  = \ 
        ListPlot[Qvalues, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(qplot\  = \ 
        Plot[qmodel[t, \[Lambda], a], {t, 0, T}, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(Show[Qplot, qplot, DisplayFunction \[Rule] $DisplayFunction, 
        PlotRange \[Rule] {{0, T}, {qmin, qmax}}];\)\)}], "Input"],

Cell[GraphicsData["PostScript", "\<\
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Cell[CellGroupData[{

Cell["2. Stability of the above algorithms", "Section"],

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Cell["First-order forward difference approximation", "Subsection"],

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Cell["Find the update formula", "Subsubsection"],

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Cell[CellGroupData[{

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*)



(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)