(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 43967, 1312]*) (*NotebookOutlinePosition[ 68081, 2185]*) (* CellTagsIndexPosition[ 68037, 2181]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox[ "Numerical Methods: Mathematica 3.0 Notebooks\n(c) John H. Mathews, 1998", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["\n", FontSize->18], StyleBox["To accompany the text:", FontColor->RGBColor[1, 0, 0]], "\n", StyleBox[ "Numerical Methods: for Mathematics, \nScience, and Engineering, 2", FontSize->18, FontColor->RGBColor[0, 0, 1]], StyleBox["nd", FontSize->14, FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" Ed, 1992", FontSize->18, FontColor->RGBColor[0, 0, 1]], StyleBox["\n", FontSize->18], StyleBox[ "Prentice Hall, Inc.\nSimon & Schuster, One Lake Street\nUpper Saddle \ River, New Jersey 07458 U.S.A.", FontSize->18, FontColor->RGBColor[0, 1, 1]], StyleBox["\n", FontSize->18], StyleBox[ "Prentice Hall, Inc.; USA, Canada, Mexico ISBN 0-13-624990-6\nPrentice \ Hall, International Editions: ISBN 0-13-625047-5\nwww.Prenhall.com", FontSize->14, FontColor->RGBColor[0, 1, 0]], StyleBox["\n", FontSize->14], StyleBox[ "This free software is compliments of the author.\nE-mail address: in%\ \"mathews@fullerton.edu\"", FontSize->14, FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[TextData[StyleBox["CONTENTS", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Text", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Chapter 10. Solution of Partial Differential Equations", FontWeight->"Bold"], "\n\t", StyleBox["Algorithm 10.1 ", FontWeight->"Bold"], StyleBox["Finite-Difference Solution for the Wave Equation", FontColor->RGBColor[1, 0, 1]], ".\n\t", StyleBox["Algorithm 10.2 ", FontWeight->"Bold"], StyleBox["Forward-Difference Method for the Heat Equation", FontColor->RGBColor[1, 0, 1]], ".\n\t", StyleBox["Algorithm 10.3 ", FontWeight->"Bold"], StyleBox["Crank-Nicholson Method for the Heat Equation", FontColor->RGBColor[1, 0, 1]], ".\n\t", StyleBox["Algorithm 10.4 ", FontWeight->"Bold"], StyleBox["Dirichlet Method for Laplace's Equation", FontColor->RGBColor[1, 0, 1]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ RowBox[{"\n", RowBox[{ StyleBox[\(Clear["\"]\), "MR"], StyleBox[";", "MR"], "\n", \(Off[General::"\"]\), ";", "\n", \(Clear[a, a0, b, b0, c, c0, CoeffMat, d, d0, Dirichlet, f, g, g1, g2, GridU, i, j, k, m, n, Neumann, q, qList, r, SolveU, t, TriMat, u, w, x]\), ";", "\n", \(On[General::"\"]\), ";"}]}]], "Input", InitializationCell->True], Cell[TextData[{ StyleBox["Algorithm 10.1 ", FontWeight->"Bold"], StyleBox["Finite-Difference Solution for the Wave Equation", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[".", FontWeight->"Bold"], " To approximate the solution for the wave equation:\n", Cell[BoxData[ \(\(u\_tt\) \((x, t)\) = \ c\^2\ \(u\_xx\) \((x, t)\)\)]], " over ", Cell[BoxData[ \(R = {\((x, t)\) : \ 0 \[LessEqual] x \[LessEqual] a, \ \ 0 \[LessEqual] t \[LessEqual] b\ }\)]], ",\nwith ", Cell[BoxData[ \(u \((0, t)\) = 0, \ \ u \((a, t)\) = 0\)]], " for ", Cell[BoxData[ \(0 \[LessEqual] t \[LessEqual] b\)]], " and,\n", Cell[BoxData[ \(u \((x, 0)\) = f \((x)\), \ \(u\_t\) \((x, 0)\) = g \((x)\)\)]], " for ", Cell[BoxData[ \(0 \[LessEqual] x \[LessEqual] a\)]], ".\n", StyleBox["Section 10.1 Hyperbolic Equations Page 507", FontWeight->"Bold"] }], "Text", CellFrame->True, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Step sizes: h is the step in the x-direction, k is the step in the \ y-direction.\nWe will solve the problem on the grid\n", Cell[BoxData[ \(u\[LeftDoubleBracket]i, j\[RightDoubleBracket]\)]], "\twhere\t", Cell[BoxData[ \(i = 1, 2, \[CenterEllipsis], n\)]], " is the extent along the x-axis.\n\t\tand\t", Cell[BoxData[ \(j = 1, 2, \[CenterEllipsis], m\)]], " is the extent along the t-axis.\nThe method proceds by simple iteration.\ \nIt is known that r must satisfy ", Cell[BoxData[ \(r \[LessEqual] 1\)]], " for the method to be stable. " }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Execute this cell to activate", FontWeight->"Bold", FontColor->RGBColor[0, 1, 0]], " ", StyleBox["Finite-Difference Solution for the Wave Equation", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[".\n", FontWeight->"Bold"], StyleBox[ "GridU[n_,m_]\nSolveU[n_,m_]\nn_ is the number of subintervals along the \ x-axis,\nm_ is the number of subintervals along the t-axis.", FontColor->RGBColor[1, 0, 0]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Off[General::"\"]; \)\), \(\(Clear[GridU, n, m, u]; \)\), \(\(GridU[n_, m_] := \n\t Module[{}, \n\t\tClear[u]; \n\t\tu = Table[1, {n}, {m}]; \n\t\t For[i = 1, i \[LessEqual] n, \(i++\), u\[LeftDoubleBracket]i, 1\[RightDoubleBracket] = f[i]; \n\t\t\t u\[LeftDoubleBracket]i, 2\[RightDoubleBracket] = \((1 - r\^2)\)\ f[i] + k\ g[i] + 1\/2\ r\^2\ \((f[i + 1] + f[i - 1])\); ]; \n\t\t For[j = 1, j \[LessEqual] m, \(j++\), \n\t\t\t Module[{}, u\[LeftDoubleBracket]1, j\[RightDoubleBracket] = 0; u\[LeftDoubleBracket]n, j\[RightDoubleBracket] = 0; ]]; \n\t\t Print["\"]; ]; \)\)}], "Input", InitializationCell->True, AspectRatioFixed->False], Cell[BoxData[{ \(\(Clear[SolveU, n, m, u]; \)\), \(SolveU[n_, m_] := \n\t Module[{i, j}, \n\t\t For[j = 3, j \[LessEqual] m, \(j++\), \n\t\t\t Module[{}, \n\t\t\t\t For[i = 2, i \[LessEqual] n - 1, \(i++\), \n\t\t\t\t\t \(u\[LeftDoubleBracket]i, j\[RightDoubleBracket] = \((2 - 2\ r\^2)\)\ u\[LeftDoubleBracket]i, j - 1\[RightDoubleBracket] + r\^2\ \(( u\[LeftDoubleBracket]i + 1, j - 1\[RightDoubleBracket] + u\[LeftDoubleBracket]i - 1, j - 1\[RightDoubleBracket])\) - u\[LeftDoubleBracket]i, j - 2\[RightDoubleBracket]; \)]]]; \n\t\tPrint["\"]; ]; \n \(On[General::"\"]; \)\)}], "Input", InitializationCell->True, AspectRatioFixed->False] }, Open ]] }, Closed]], Cell[TextData[{ "\n", StyleBox["Chapter 10, Example 10.1, Page 504.", FontWeight->"Bold"], " Consider the wave equation where ", Cell[BoxData[ \(c\^2 = \ 1\)]], ". The length of the rod is ", Cell[BoxData[ \(L = 1\)]], ". Assume that the initial position is \n\t", Cell[BoxData[ \(u \((x, 0)\)\ = \ \(f \((x)\)\ = \ sin \((\[Pi]\ x)\) + \ sin \((2 \[Pi]\ x)\)\)\)]], ".\nCompare the solution with the exact solution:\n\t", Cell[BoxData[ \(u \((x, t)\)\ = \ sin \((\[Pi]\ x)\) cos \((\[Pi]\ t)\) + \ sin \((2 \[Pi]\ x)\) cos \((4 \[Pi]\ t)\)\)]], ".\nSolution. We will use ", Cell[BoxData[ \(c = 1, \ h = 0.2, \ k = 0.04\)]], " . This forces ", Cell[BoxData[ RowBox[{"r", " ", "=", " ", RowBox[{\(1\^2\), " ", "=", " ", RowBox[{ FractionBox[ StyleBox["0.04", FontSize->12], StyleBox[\(\((0.2)\)\^2\), FontSize->12]], " ", "=", " ", "1"}]}]}]]], ".\nFirst, enter the boundary conditons:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[c, f, g, h, i, k, m, n, r]; \n f[i_] := N[ Sin[\(\[Pi]\ \((i - 1)\)\)\/\(n - 1\)] + Sin[\(2\ \[Pi]\ \((i - 1)\)\)\/\(n - 1\)]]; \n\(g[i_] := 0.0; \)\)], "Input", AspectRatioFixed->False], Cell["Now set up the table of solutions.", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(\(c = 2.0; \)\), \(\(h = 0.1; \)\), \(\(k = 0.05; \)\), \(\(r = \(c\ k\)\/h; \)\), \(\(n = 11; \)\), \(\(m = 21; \)\), \(GridU[n, m]\)}], "Input", AspectRatioFixed->False], Cell[BoxData[ \(SolveU[n, m]\)], "Input", AspectRatioFixed->False], Cell[BoxData[ \(\(ListPlot3D[u, AxesLabel \[Rule] {"\", "\", "\"}, ViewPoint \[Rule] {4, 2, 3}]; \)\)], "Input", AspectRatioFixed->True], Cell["Compare with the analytic solution.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\(SetOptions[Plot3D, PlotPoints \[Rule] {n, m}]; \)\), \(\(Plot3D[ Sin[\[Pi]\ x]\ Cos[2\ \[Pi]\ t] + Sin[2\ \[Pi]\ x]\ Cos[4\ \[Pi]\ t], { x, 0, 1}, {t, 0, 1}]; \)\)}], "Input", AspectRatioFixed->True], Cell["To see the numerical values enter the command:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(TableForm[Transpose[Chop[N[u, 3]]]]\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Chapter 10, Example 10.2, Page 506.", FontWeight->"Bold"], " Consider the wave equation where ", Cell[BoxData[ \(c\^2 = 1\)]], ". The length of the rod is ", Cell[BoxData[ \(L = 1\)]], ". Assume that the initial position \n", Cell[BoxData[ RowBox[{\(u \((x, 0)\)\), " ", "=", " ", RowBox[{\(f \((x)\)\), " ", "=", RowBox[{ StyleBox["{", FontSize->18], GridBox[{ { RowBox[{ \(x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ for\ \ \ 0\), " ", "\[LessEqual]", " ", "x", " ", "\[LessEqual]", FractionBox[ StyleBox[\(\ 3\), FontSize->10], StyleBox["5", FontSize->10]]}]}, { RowBox[{ RowBox[{ FractionBox[ StyleBox["5", FontSize->12], StyleBox["2", FontSize->12]], "-", RowBox[{ FractionBox[ StyleBox[\(5 x\), FontSize->12], StyleBox["2", FontSize->12]], " ", "for", " ", FractionBox[ RowBox[{" ", StyleBox["3", FontSize->10]}], StyleBox["5", FontSize->10]]}]}], " ", "\[LessEqual]", " ", "x", " ", "\[LessEqual]", " ", "1"}]} }]}]}]}]]], "\nSolution. We will use ", Cell[BoxData[ \(c = 1, \ h = 0.2, \ k = 0.04\)]], " . This forces ", Cell[BoxData[ RowBox[{"r", " ", "=", " ", RowBox[{\(1\^2\), " ", "=", " ", RowBox[{ FractionBox[ StyleBox["0.04", FontSize->12], StyleBox[\(\((0.2)\)\^2\), FontSize->12]], " ", "=", " ", "1"}]}]}]]], ".\nFor the boundary conditons we need 11 function values for ", Cell[BoxData[ \(f \((x)\)\)]], "." }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[ \(Clear[c, f, g, h, i, k, m, n, r]; \nf[1] = 0.0; \ \ f[2] = 0.1; \ \ \ \ f[3] = 0.2; \ f[4] = 0.3; \nf[5] = 0.4; \ \ f[6] = 0.5; \ \ \ \ f[7] = 0.6; \ f[8] = 0.45; \nf[9] = 0.3; \ \ f[10] = 0.15; f[11] = 0.0; \n \(g[i_] := 0.0; \)\)], "Input", AspectRatioFixed->False], Cell["Now set up the table of solutions.", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(\(c = 2.0; \)\), \(\(h = 0.1; \)\), \(\(k = 0.05; \)\), \(\(r = \(c\ k\)\/h; \)\), \(\(n = 11; \)\), \(\(m = 21; \)\), \(GridU[n, m]\)}], "Input", AspectRatioFixed->False], Cell[BoxData[ \(SolveU[n, m]\)], "Input", AspectRatioFixed->False], Cell[BoxData[ \(\(ListPlot3D[u, AxesLabel \[Rule] {"\", "\", "\"}, ViewPoint \[Rule] {4, 2, 3}]; \)\)], "Input", AspectRatioFixed->True], Cell["To see the numerical values enter the command:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(TableForm[Transpose[Chop[N[u, 3]]]]\)], "Input"], Cell[TextData[{ StyleBox["Algorithm 10.2 ", FontWeight->"Bold"], StyleBox["Forward-Difference Method for the Heat Equation", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[".", FontWeight->"Bold"], " \nTo approximate the solution for the heat equation \n ", Cell[BoxData[ \(\(u\_t\) \((x, t)\) = \ c\^2\ \(u\_xx\) \((x, t)\)\)]], " over ", Cell[BoxData[ \(R = {\((x, t)\) : \ 0 \[LessEqual] x \[LessEqual] a, \ \ 0 \[LessEqual] t \[LessEqual] b\ }\)]], " \nwith \n u(x,0) = f(x) for 0 \[LessEqual] x \[LessEqual] a \ and ", Cell[BoxData[ \(0 \[LessEqual] x \[LessEqual] a\)]], " and,\n u(0,t) = ", Cell[BoxData[ \(c\_1\)]], ", u(a,t) = ", Cell[BoxData[ \(c\_2\)]], " for ", Cell[BoxData[ \(0 \[LessEqual] t \[LessEqual] b\)]], ".\n", StyleBox["Section 10.2 Parabolic Equations Page 516", FontWeight->"Bold"] }], "Text", CellFrame->True, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "We will solve the problem on the grid:\n", Cell[BoxData[ \(u\[LeftDoubleBracket]i, j\[RightDoubleBracket]\)]], "\twhere\t", Cell[BoxData[ \(i = 1, 2, \[CenterEllipsis], n\)]], " is the extent along the x-axis.\n\t\tand\t", Cell[BoxData[ \(j = 1, 2, \[CenterEllipsis], m\)]], " is the extent along the t-axis.\nThe function ", Cell[BoxData[ \(g\_1[j]\)]], " furnishes the values ", Cell[BoxData[ \(u\[LeftDoubleBracket]1, j\[RightDoubleBracket]\)]], " and\nthe function ", Cell[BoxData[ \(g\_2[j]\)]], " furnishes the values ", Cell[BoxData[ \(u\[LeftDoubleBracket]n, j\[RightDoubleBracket]\)]], " and\nthe function ", Cell[BoxData[ \(f[i]\)]], " furnishes the initial condition ", Cell[BoxData[ \(u\[LeftDoubleBracket]i, 1\[RightDoubleBracket]\)]], ".\nFor some problems the corners must be modified.\nThe grid ", Cell[BoxData[ \(u\[LeftDoubleBracket]i, j\[RightDoubleBracket]\)]], " is formed with the subroutine ", Cell[BoxData[ \(GridU[n, m]\)]], "." }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Execute this cell to activate", FontWeight->"Bold", FontColor->RGBColor[0, 1, 0]], " ", StyleBox["Forward-Difference Method for the Heat Equation", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[".\n", FontWeight->"Bold"], StyleBox[ "GridU[n_,m_]\nSolveU[n_,m_]\nn_ is the number of subintervals along the \ x-axis,\nm_ is the number of subintervals along the t-axis.", FontColor->RGBColor[1, 0, 0]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Off[General::"\"]; \)\), \(\(GridU[n_, m_] := \n\t Module[{}, \n\t\tClear[u]; \n\t\tu = Table[1, {n}, {m}]; \n\t\t For[i = 1, i \[LessEqual] n, \(i++\), \n\t\t\t u\[LeftDoubleBracket]i, 1\[RightDoubleBracket] = f[i]]; \n\t\t For[j = 1, j \[LessEqual] m, \(j++\), \n\t\t\t Module[{}, \n\t\t\t\t u\[LeftDoubleBracket]1, j\[RightDoubleBracket] = g1[j]; \n\t\t\t\t u\[LeftDoubleBracket]n, j\[RightDoubleBracket] = g2[j]; ]]; \n\t\t Print["\"]; ]; \)\)}], "Input", InitializationCell->True, AspectRatioFixed->False], Cell[BoxData[ \(SolveU[n_, m_] := \n\t Module[{i, j}, \n\t\t For[j = 2, j \[LessEqual] m, \(j++\), \n\t\t\t \(For[i = 2, i \[LessEqual] n - 1, \(i++\), \n\t\t\t\t u\[LeftDoubleBracket]i, j\[RightDoubleBracket] = \((1 - 2\ r)\)\ u\[LeftDoubleBracket]i, j - 1\[RightDoubleBracket] + r\ \((u\[LeftDoubleBracket]i - 1, j - 1\[RightDoubleBracket] + u\[LeftDoubleBracket]i + 1, j - 1\[RightDoubleBracket]) \)]; \)]; \n\t\t Print["\"]; ]; \n \(On[General::"\"]; \)\)], "Input", InitializationCell->True, AspectRatioFixed->False] }, Open ]] }, Closed]], Cell[TextData[{ "\n", StyleBox["Chapter 10, Example 10.3, Page 511. ", FontWeight->"Bold"], "Consider the heat equation where ", Cell[BoxData[ \(c\^2 = 1\)]], ". The length of the rod is ", Cell[BoxData[ \(L = 1\)]], ". Assume that the ends of the rod are held at the temperature ", Cell[BoxData[ \(u = 0\)]], ". Assume that the initial temperature distribution is\n ", Cell[BoxData[ \(u \((x, 0)\)\ = \ \(f \((x)\)\ = \ 4 x\ - \ 4 x\^2\)\)]], ".\nApply the forward difference method with ", Cell[BoxData[ \(r = 1\)]], " and obtain temperature distributions for ", Cell[BoxData[ \(t = 0.04, 0.08, 0.12, \[CenterEllipsis], 0.40\)]], ".\nSolution. We will use ", Cell[BoxData[ \(c = 1, \ h = 0.2, \ k = 0.04\)]], " . This forces ", Cell[BoxData[ RowBox[{"r", " ", "=", " ", RowBox[{\(1\^2\), " ", "=", " ", RowBox[{ FractionBox[ StyleBox["0.04", FontSize->12], StyleBox[\(\((0.2)\)\^2\), FontSize->12]], " ", "=", " ", "1"}]}]}]]], ".\nEnter the boundary conditons:" }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(Clear[c, f, g1, g2, h, i, k, m, n, r, v]; \ng1[j_] := 0.0; \n \(g2[j_] := 0.0; \)\), \(\(f[i_] := N[4\ \((0.2\ \((i - 1)\))\)\ \((1 - 0.2\ \((i - 1)\))\)]; \)\)}], "Input", AspectRatioFixed->False], Cell[BoxData[""], "Input"], Cell["Now set up the table of solutions.", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(\(n = 6; \)\), \(\(m = 11; \)\), \(GridU[n, m]\)}], "Input", AspectRatioFixed->False], Cell["Next, solve it.", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(\(v = u; \)\), \(\(c = 1; \)\), \(\(h = 0.2; \)\), \(\(k = 0.02; \)\), \(\(r = \(c\^2\ k\)\/h\^2; \)\), \(SolveU[n, m]\)}], "Input", AspectRatioFixed->False], Cell[BoxData[ \(\(ListPlot3D[u, AxesLabel \[Rule] {"\", "\", "\"}, ViewPoint \[Rule] {4, 2, 3}]; \)\)], "Input", AspectRatioFixed->True], Cell["To see the numerical values enter the command:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(TableForm[N[Transpose[u], 6]]\)], "Input"], Cell["\<\ Now investigate what happens when the step size is too large.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\(n = 6; \)\), \(\(m = 11; \)\), \(GridU[n, m]\)}], "Input", AspectRatioFixed->False], Cell["Next, solve it.", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(\(v = u; \)\), \(\(c = 1; \)\), \(\(h = 0.2; \)\), \(\(k = 1\/30; \)\), \(\(r = \(c\^2\ k\)\/h\^2; \)\), \(SolveU[n, m]\)}], "Input", AspectRatioFixed->False], Cell[BoxData[ \(\(ListPlot3D[u, AxesLabel \[Rule] {"\", "\", "\"}, ViewPoint \[Rule] {4, 2, 3}]; \)\)], "Input", AspectRatioFixed->True], Cell["To see the numerical values enter the command:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(TableForm[N[Transpose[u], 6]]\)], "Input"], Cell[TextData[{ StyleBox["Algorithm 10.3 ", FontWeight->"Bold"], StyleBox["Crank-Nicholson Method for the Heat Equation", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[".", FontWeight->"Bold"], " \nTo approximate the solution for the heat equation \n", Cell[BoxData[ \(\(u\_t\) \((x, t)\) = \ c\^2\ \(u\_xx\) \((x, t)\)\)]], " over ", Cell[BoxData[ \(R = {\((x, t)\) : \ 0 \[LessEqual] x \[LessEqual] a, \ \ 0 \[LessEqual] t \[LessEqual] b\ }\)]], " \nwith \n", Cell[BoxData[ \(u \((x, 0)\) = \ f \((x)\)\)]], " for 0 \[LessEqual] x \[LessEqual] a and ", Cell[BoxData[ \(0 \[LessEqual] x \[LessEqual] a\)]], " and,\n", Cell[BoxData[ \(u \((0, t)\) = \ c\_1, \ u \((a, t)\) = \ c\_2\)]], " for ", Cell[BoxData[ \(0 \[LessEqual] t \[LessEqual] b\)]], ".\n", StyleBox["Section 10.2 Parabolic Equations Page 517", FontWeight->"Bold"] }], "Text", CellFrame->True, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "We will solve the problem on the grid:\n", Cell[BoxData[ \(u\[LeftDoubleBracket]i, j\[RightDoubleBracket]\)]], "\twhere\t", Cell[BoxData[ \(i = 1, 2, \[CenterEllipsis], n\)]], " is the extent along the x-axis.\n\t\tand\t", Cell[BoxData[ \(j = 1, 2, \[CenterEllipsis], m\)]], " is the extent along the t-axis.\nThe function ", Cell[BoxData[ \(g\_1[j]\)]], " furnishes the values ", Cell[BoxData[ \(u\[LeftDoubleBracket]1, j\[RightDoubleBracket]\)]], " and\nthe function ", Cell[BoxData[ \(g\_2[j]\)]], " furnishes the values ", Cell[BoxData[ \(u\[LeftDoubleBracket]n, j\[RightDoubleBracket]\)]], " and\nthe function ", Cell[BoxData[ \(f[i]\)]], " furnishes the initial condition ", Cell[BoxData[ \(u\[LeftDoubleBracket]i, 1\[RightDoubleBracket]\)]], ".\nFor some problems the corners must be modified.\nThe grid ", Cell[BoxData[ \(u\[LeftDoubleBracket]i, j\[RightDoubleBracket]\)]], " is formed with the subroutine ", Cell[BoxData[ \(GridU[n, m]\)]], "." }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Execute this cell to activate", FontWeight->"Bold", FontColor->RGBColor[0, 1, 0]], " ", StyleBox["Crank-Nicholson Method for the Heat Equation", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[".\n", FontWeight->"Bold"], StyleBox["GridU[n_,m_]", FontColor->RGBColor[1, 0, 0]], StyleBox["\n", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[ "CoeffMat[n_]\nTriMat[a_,d_,c_,b_,n_]\nSolveU[n_,m_]\na_,d_,c_,b_ are \ vectors needed in the solution,", FontColor->RGBColor[1, 0, 0]], "\n", StyleBox[ "n_ is the number of subintervals along the x-axis,\nm_ is the number of \ subintervals along the t-axis.", FontColor->RGBColor[1, 0, 0]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Off[General::"\"]; \)\), \(\(Clear[GridU, n, m, u]; \)\), \(\(GridU[n_, m_] := \n\t Module[{i, j}, \n\t\tClear[u]; \n\t\tu = Table[1, {n}, {m}]; \n\t\t For[i = 1, i \[LessEqual] n, \(i++\), u\[LeftDoubleBracket]i, 1\[RightDoubleBracket] = f[i]]; \n\t\t For[j = 1, j \[LessEqual] m, \(j++\), \n\t\t\t Module[{}, \n\t\t\t\t u\[LeftDoubleBracket]1, j\[RightDoubleBracket] = g1[j]; \n\t\t\t\t u\[LeftDoubleBracket]n, j\[RightDoubleBracket] = g2[j]; ]]; \n\t\t Print["\"]; ]; \)\)}], "Input", InitializationCell->True, AspectRatioFixed->False], Cell[BoxData[{ \(\(Clear[CoeffMat, n]; \)\), \(CoeffMat[n_] := \n\t Module[{}, \n\t\tClear[r]; \n\t\tVa = Table[\(-1\), {n - 1}]; \n\t\t Va\[LeftDoubleBracket]n - 1\[RightDoubleBracket] = 0; \n\t\t Vc = Table[\(-1\), {n - 1}]; \n\t\t Vc\[LeftDoubleBracket]1\[RightDoubleBracket] = 0; \n\t\t Vd = Table[2 + 2\/r, {n}]; \n\t\t Vd\[LeftDoubleBracket]1\[RightDoubleBracket] = 1; \n\t\t Vd\[LeftDoubleBracket]n\[RightDoubleBracket] = 1; \n\t\t Print["\"]; ]\)}], "Input", InitializationCell->True, AspectRatioFixed->False], Cell[BoxData[{ \(\(Clear[TriMat, a, d, c, b, n]; \)\), \(\(TriMat[a_, d_, c_, b_, n_] := \n\t Module[{a0, b0, c0, d0, r, t}, \n\t\ta0 = a; \n\t\tb0 = b; \n\t\t c0 = c; \n\t\td0 = d; \n\t\tClear[x0]; \n\t\tx0 = Table[0, {n}]; \n \t\tFor[r = 2, r \[LessEqual] n, \(r++\), \n\t\t\t \(Module[{}, \n\t\t\t\t t = a0\[LeftDoubleBracket]r - 1\[RightDoubleBracket]\/d0 \[LeftDoubleBracket]r - 1\[RightDoubleBracket]; \n\t\t\t\t d0\[LeftDoubleBracket]r\[RightDoubleBracket] = d0\[LeftDoubleBracket]r\[RightDoubleBracket] - t\ c0\[LeftDoubleBracket]r - 1\[RightDoubleBracket]; \n \t\t\t\tb0\[LeftDoubleBracket]r\[RightDoubleBracket] = b0\[LeftDoubleBracket]r\[RightDoubleBracket] - t\ b0\[LeftDoubleBracket]r - 1\[RightDoubleBracket]]; \)]; \n \t\tx0\[LeftDoubleBracket]n\[RightDoubleBracket] = b0\[LeftDoubleBracket]n\[RightDoubleBracket]\/d0 \[LeftDoubleBracket]n\[RightDoubleBracket]; \n\t\t For[r = n - 1, 1 \[LessEqual] r, \(r--\), \n\t\t\t x0\[LeftDoubleBracket]r\[RightDoubleBracket] = \(b0\[LeftDoubleBracket]r\[RightDoubleBracket] - c0\[LeftDoubleBracket]r\[RightDoubleBracket]\ x0\[LeftDoubleBracket]r + 1\[RightDoubleBracket]\)\/d0 \[LeftDoubleBracket]r\[RightDoubleBracket]]; \n\t\t Return[x0]]; \)\)}], "Input", InitializationCell->True, AspectRatioFixed->False], Cell[BoxData[{ \(\(Clear[SolveU, n, m]; \)\), \(SolveU[n_, m_] := \n\t Module[{i, j}, \n\t\tClear[b, x]; \n\t\tb = Table[0, {n}]; \n\t\t x = Table[0, {n}]; \n\t\t For[j = 2, j \[LessEqual] m, \(j++\), \n\t\t\t \(Module[{}, \n\t\t\t\t b\[LeftDoubleBracket]1\[RightDoubleBracket] = g1[j]; \n\t\t\t\t b\[LeftDoubleBracket]n\[RightDoubleBracket] = g2[j]; \n\t\t\t\t For[i = 2, i \[LessEqual] n - 1, \(i++\), \n\t\t\t\t\t b\[LeftDoubleBracket]i\[RightDoubleBracket] = u\[LeftDoubleBracket]i - 1, j - 1\[RightDoubleBracket] + \((2\/r - 2)\)\ u\[LeftDoubleBracket]i, j - 1\[RightDoubleBracket] + u\[LeftDoubleBracket]i + 1, j - 1\[RightDoubleBracket]]; \n \t\t\t\tx = TriMat[Va, Vd, Vc, b, n]; \n\t\t\t\t For[i = 1, i \[LessEqual] n, \(i++\), u\[LeftDoubleBracket]i, j\[RightDoubleBracket] = x\[LeftDoubleBracket]i\[RightDoubleBracket]]; ]; \)]; \n\t\t Print["\"]]; \n On[General::"\"]; \)}], "Input", InitializationCell->True, AspectRatioFixed->False] }, Open ]] }, Closed]], Cell[TextData[{ "\n", StyleBox["Chapter 10, Example 10.4, Page 515.", FontWeight->"Bold"], " Consider the heat equation where ", Cell[BoxData[ \(c\^2 = 1\)]], ". The length of the rod is ", Cell[BoxData[ \(L = 1\)]], ". Assume that the ends of the rod are held at the temperature ", Cell[BoxData[ \(u = 0\)]], ". Assume that the initial temperature distribution is\n ", Cell[BoxData[ \(u \((x, 0)\)\ = \ \(f \((x)\)\ = \ sin \((\[Pi]\ x)\)\)\)]], ".\nApply the Crank-Nicholson method with ", Cell[BoxData[ \(r = 1\)]], " and obtain temperature distributions for ", Cell[BoxData[ \(t = 0.04, 0.08, 0.12, \[CenterEllipsis], 0.40\)]], ". Compare the solution with the exact solution:\n ", Cell[BoxData[ \(u \((x, t)\)\ = \ sin \((\[Pi]\ x)\) \[ExponentialE]\^\(-t\[Pi]\^2\) + \ \ sin \((3 \[Pi]\ x)\) \[ExponentialE]\^\(\(-9\) t\[Pi]\^2\)\)]], ".\nSolution. We will use ", Cell[BoxData[ \(c = 1, \ h = 0.2, \ k = 0.04\)]], " . This forces This forces ", Cell[BoxData[ RowBox[{"r", " ", "=", " ", RowBox[{\(1\^2\), " ", "=", " ", RowBox[{ FractionBox[ StyleBox["0.04", FontSize->12], StyleBox[\(\((0.2)\)\^2\), FontSize->12]], " ", "=", " ", "1"}]}]}]]], ".\nFirst set up the boundary conditons:" }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(Clear[c, f, g1, g2, h, i, k, m, n, r]; \nClear[g1, g2, f]; \n \(g1[j_] := 0.0; \)\), \(\(g2[j_] := 0.0; \)\), \(\(f[i_] := N[Sin[\[Pi]\ 0.1\ \((i - 1)\)] + Sin[3\ \[Pi]\ 0.1\ \((i - 1)\)]]; \)\)}], "Input", AspectRatioFixed->False], Cell["Now set up the table of solutions.", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(\(n = 11; \)\), \(\(m = 11; \)\), \(GridU[n, m]\)}], "Input", AspectRatioFixed->False], Cell[TextData[{ "Setting up the tri-diagonal matrx with n rows. Indeed, we could get away \ with ", Cell[BoxData[ \(n - 2\)]], " rows, but the implementation is nice this way. The following matrix \ will usually use ", Cell[BoxData[ \(r = 1\)]], "." }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[ \(CoeffMat[n]\)], "Input", AspectRatioFixed->False], Cell["Next, solve it.", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(\(c = 1; \)\), \(\(h = 0.1; \)\), \(\(k = 0.01; \)\), \(\(r = \(c\^2\ k\)\/h\^2; \)\), \(SolveU[n, m]\)}], "Input", AspectRatioFixed->False], Cell[BoxData[ \(\(ListPlot3D[u, AxesLabel \[Rule] {"\", "\", "\"}, ViewPoint \[Rule] {4, 2, 3}]; \)\)], "Input", AspectRatioFixed->True], Cell["Compare with the analytic solution.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\(SetOptions[Plot3D, PlotPoints \[Rule] {n, m}]; \)\), \(\(Plot3D[ Sin[\[Pi]\ x] \[ExponentialE]\^\(\(-0.1\)\ \ t\ \[Pi]\^2\) + \ \ Sin[3 \[Pi]\ x] \[ExponentialE]\^\(\(-0.1\)\ 9 t\ \[Pi]\^2\), {t, 0, 1}, {x, 0, 1}, ViewPoint \[Rule] {4, 2, 3}]; \)\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Algorithm 10.4 ", FontWeight->"Bold"], StyleBox["Dirichlet Method for Laplace's Equation", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[".", FontWeight->"Bold"], " \nTo approximate the solution of \n", Cell[BoxData[ \(\(u\_xx\) \((x, y)\) + \(u\_yy\) \((x, y)\) = 0\)]], " over ", Cell[BoxData[ \(R = {\((x, y)\) : \ 0 \[LessEqual] x \[LessEqual] a, \ \ 0 \[LessEqual] y \[LessEqual] b\ }\)]], " \nwith \n", Cell[BoxData[ \(u \((x, 0)\) = \(f\_1\) \((x)\), \ \ u \((x, b)\) = \(f\_2\) \((x)\)\)]], " for ", Cell[BoxData[ \(0 \[LessEqual] x \[LessEqual] a\)]], " and,\n", Cell[BoxData[ \(u \((0, y)\) = \(f\_3\) \((y)\), \ \ u \((a, y)\) = \(f\_4\) \((y)\)\)]], " for ", Cell[BoxData[ \(0 \[LessEqual] y \[LessEqual] b\)]], ". \nIt is assumed that ", Cell[BoxData[ \(\[CapitalDelta]x = h, \ \[CapitalDelta]y = h\)]], " and that integers n and m exist so that ", Cell[BoxData[ \(a = n\ h\)]], " and ", Cell[BoxData[ \(b = m\ h\)]], ".\n", StyleBox["Section 10.3 Elliptic Equations Page 531", FontWeight->"Bold"] }], "Text", CellFrame->True, Evaluatable->False, AspectRatioFixed->True], Cell["First, execute the following group of cells:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Execute this cell to activate", FontWeight->"Bold", FontColor->RGBColor[0, 1, 0]], " ", StyleBox["Dirichlet Method for Laplace's Equation", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[".", FontWeight->"Bold"], " ", StyleBox["\n", FontWeight->"Bold"], StyleBox[ "Dirichlet[q_List,n_,m_,w_]\nNeumann[q_List,n_,m_,w_]\nq_List are the \ initial starting values,\nw_ is a coefficient to be supplied,", FontColor->RGBColor[1, 0, 0]], "\n", StyleBox[ "n_ is the number of subintervals along the x-axis,\nm_ is the number of \ subintervals along the y-axis.", FontColor->RGBColor[1, 0, 0]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Off[General::"\"]; \)\), \(\(Clear[Dirichlet, q, n, m, w]; \)\), \(\(Dirichlet[q_List, n_, m_, w_] := \n\t Module[{i, j, k}, \n\t\tErr = 1; \n\t\tk = 0; \n\t\t While[Err > 0.001 && k \[LessEqual] 70, \n\t\t\t Module[{}, \n\t\t\t\tErr = 0.0; \n\t\t\t\t For[j = 2, j \[LessEqual] m - 1, \(j++\), \n\t\t\t\t\t Module[{}, \n\t\t\t\t\t\t For[i = 2, i \[LessEqual] n - 1, \(i++\), \n\t\t\t\t\t\t\t Module[{}, \n\t\t\t\t\t\t\t\t Relax = \(1\/4.0\) \((w\ \(( u\[LeftDoubleBracket]i, j + 1\[RightDoubleBracket] + u\[LeftDoubleBracket]i, j - 1\[RightDoubleBracket] + u\[LeftDoubleBracket]i + 1, j \[RightDoubleBracket] + u\[LeftDoubleBracket]i - 1, j \[RightDoubleBracket] - 4.0\ u\[LeftDoubleBracket]i, j \[RightDoubleBracket])\))\); \n \t\t\t\t\t\t\t\t u\[LeftDoubleBracket]i, j\[RightDoubleBracket] = u\[LeftDoubleBracket]i, j\[RightDoubleBracket] + Relax; \n\t\t\t\t\t\t\t\tErr = Max[Err, Abs[Relax]]; ]]]]; \n \t\t\t\tPrint["\", Err]; k = k + 1]]]; \)\)}], "Input", InitializationCell->True, AspectRatioFixed->False], Cell[BoxData[{ \(\(Clear[Neumann, qList, n, m, w]; \)\), \(Neumann[q_List, n_, m_, w_] := \n\t Module[{i, j, k}, \n\t\tErr = 1; \n\t\tk = 0; \n\t\t While[Err > 0.001 && k \[LessEqual] 70, \n\t\t\t Module[{}, \n\t\t\t\tErr = 0.0; \n\t\t\t\t For[j = 2, j \[LessEqual] m - 1, \(j++\), \n\t\t\t\t\t Module[{}, \n\t\t\t\t\t\t For[i = 2, i \[LessEqual] n - 1, \(i++\), \n\t\t\t\t\t\t\t Module[{}, \n\t\t\t\t\t\t\t\t Relax = \(1\/4.0\) \((w\ \(( u\[LeftDoubleBracket]i, j + 1\[RightDoubleBracket] + u\[LeftDoubleBracket]i, j - 1\[RightDoubleBracket] + u\[LeftDoubleBracket]i + 1, j \[RightDoubleBracket] + u\[LeftDoubleBracket]i - 1, j \[RightDoubleBracket] - 4.0\ u\[LeftDoubleBracket]i, j \[RightDoubleBracket])\))\); \n \t\t\t\t\t\t\t\t u\[LeftDoubleBracket]i, j\[RightDoubleBracket] = u\[LeftDoubleBracket]i, j\[RightDoubleBracket] + Relax; \n\t\t\t\t\t\t\t\tErr = Max[Err, Abs[Relax]]; ]]]]; For[j = 2, j \[LessEqual] m - 1, \(j++\), \n\t\t\t\t\t Module[{}, \n\t\t\t\t\t\t Relax = \(1\/4.0\) \((w\ \(( 2\ u\[LeftDoubleBracket]n - 1, j \[RightDoubleBracket] + u\[LeftDoubleBracket]n, j - 1\[RightDoubleBracket] + u\[LeftDoubleBracket]n, j + 1\[RightDoubleBracket] - 4.0\ u\[LeftDoubleBracket]n, j \[RightDoubleBracket])\))\); \n\t\t\t\t\t\t u\[LeftDoubleBracket]n, j\[RightDoubleBracket] = u\[LeftDoubleBracket]n, j\[RightDoubleBracket] + Relax; \n \t\t\t\t\t\tErr = Max[Err, Abs[Relax]]]]; \n\t\t\t\t Print["\", Err]; k = k + 1]]]; \n \(On[General::"\"]; \)\)}], "Input", InitializationCell->True, AspectRatioFixed->False] }, Open ]] }, Closed]], Cell[TextData[{ "\n", StyleBox["Chapter 10, Example 10.7, Page 529.", FontWeight->"Bold"], "\nSolve Laplace's equation over a 9 by 9 grid with boundary conditions\n\ ", StyleBox["Top:", FontWeight->"Bold"], " 180\n", StyleBox["Left:", FontWeight->"Bold"], " 80\n", StyleBox["Bottom:", FontWeight->"Bold"], " 20\n", StyleBox["Right:", FontWeight->"Bold"], " 0" }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell["First set up the boundary values.", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(Clear[i, j, u, w]; \nu = Table[70, {9}, {9}]; \n For[i = 2, i < 9, \(i++\), u\[LeftDoubleBracket]i, 1\[RightDoubleBracket] = 80]\), \(For[i = 2, i < 9, \(i++\), u\[LeftDoubleBracket]i, 9\[RightDoubleBracket] = 0]\), \(For[j = 2, j < 9, \(j++\), u\[LeftDoubleBracket]1, j\[RightDoubleBracket] = 180]\), \(For[j = 2, j < 9, \(j++\), u\[LeftDoubleBracket]9, j\[RightDoubleBracket] = 20]\), \(\(u\[LeftDoubleBracket]1, 1\[RightDoubleBracket] = 1\/2\ \((u\[LeftDoubleBracket]1, 2\[RightDoubleBracket] + u\[LeftDoubleBracket]2, 1\[RightDoubleBracket])\); \)\), \(\(u\[LeftDoubleBracket]1, 9\[RightDoubleBracket] = 1\/2\ \((u\[LeftDoubleBracket]1, 8\[RightDoubleBracket] + u\[LeftDoubleBracket]2, 9\[RightDoubleBracket])\); \)\), \(\(u\[LeftDoubleBracket]9, 1\[RightDoubleBracket] = 1\/2\ \((u\[LeftDoubleBracket]8, 1\[RightDoubleBracket] + u\[LeftDoubleBracket]9, 2\[RightDoubleBracket])\); \)\), \(\(u\[LeftDoubleBracket]9, 9\[RightDoubleBracket] = 1\/2\ \((u\[LeftDoubleBracket]8, 9\[RightDoubleBracket] + u\[LeftDoubleBracket]9, 8\[RightDoubleBracket])\); \)\)}], "Input",\ AspectRatioFixed->False], Cell["Next, solve it.", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(\(w = N[4\/\(2 + \@\(4 - \((Cos[\[Pi]\/8] + Cos[\[Pi]\/8])\)\^2\)\)]; \)\), \(Dirichlet[u, 9, 9, w]\)}], "Input", AspectRatioFixed->False], Cell[BoxData[ \(\(ListPlot3D[Transpose[u], AxesLabel \[Rule] {"\", "\", "\"}, ViewPoint \[Rule] {4, 2, 3}]; \)\)], "Input", AspectRatioFixed->True], Cell["To see the numerical values enter the command:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(TableForm[N[u, 4]]\)], "Input"], Cell[TextData[{ "\n", StyleBox["Chapter 10, Example 10.7, Page 529.", FontWeight->"Bold"], "\nSolve Laplace's equation with the Neuman boundary condition\nover a 9 \ by 9 grid with boundary conditions\n", StyleBox["Top:", FontWeight->"Bold"], " 180\n", StyleBox["Left:", FontWeight->"Bold"], " 80\n", StyleBox["Bottom:", FontWeight->"Bold"], " ", Cell[BoxData[ \(u\_x\)]], " = 0\n", StyleBox["Right:", FontWeight->"Bold"], " 0\nFirst set up the boundary values." }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(Clear[i, j, u, w]; \nu = Table[70, {9}, {9}]; \n For[i = 2, i < 9, \(i++\), u\[LeftDoubleBracket]i, 1\[RightDoubleBracket] = 80]\), \(For[i = 2, i < 9, \(i++\), u\[LeftDoubleBracket]i, 9\[RightDoubleBracket] = 0]\), \(For[j = 2, j < 9, \(j++\), u\[LeftDoubleBracket]1, j\[RightDoubleBracket] = 180]\), \(For[j = 2, j < 9, \(j++\), u\[LeftDoubleBracket]9, j\[RightDoubleBracket] = 20]\), \(\(u\[LeftDoubleBracket]1, 1\[RightDoubleBracket] = 1\/2\ \((u\[LeftDoubleBracket]1, 2\[RightDoubleBracket] + u\[LeftDoubleBracket]2, 1\[RightDoubleBracket])\); \)\), \(\(u\[LeftDoubleBracket]1, 9\[RightDoubleBracket] = 1\/2\ \((u\[LeftDoubleBracket]1, 8\[RightDoubleBracket] + u\[LeftDoubleBracket]2, 9\[RightDoubleBracket])\); \)\), \(\(u\[LeftDoubleBracket]9, 1\[RightDoubleBracket] = 80; \)\), \(\(u\[LeftDoubleBracket]9, 9\[RightDoubleBracket] = 0; \)\), \(\(For[j = 1, j \[LessEqual] 9, \(j++\), \(u\[LeftDoubleBracket]9, j\[RightDoubleBracket] = u\[LeftDoubleBracket]9, 1\[RightDoubleBracket] + 1\/8\ \((j - 1)\)\ \((u\[LeftDoubleBracket]9, 9\[RightDoubleBracket] - u\[LeftDoubleBracket]9, 1\[RightDoubleBracket])\); \)]; \)\)}], "Input", AspectRatioFixed->False], Cell["Next, solve it.", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(\(w = N[4\/\(2 + \@\(4 - \((Cos[\[Pi]\/8] + Cos[\[Pi]\/8])\)\^2\)\)]; \)\), \(Neumann[u, 9, 9, w]\)}], "Input", AspectRatioFixed->False], Cell[BoxData[ \(\(ListPlot3D[Transpose[u], AxesLabel \[Rule] {"\", "\", "\"}, ViewPoint \[Rule] {4, 2, 3}]; \)\)], "Input", AspectRatioFixed->True], Cell["To see the numerical values enter the command:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(TableForm[N[u, 4]]\)], "Input"] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 640}, {0, 452}}, AutoGeneratedPackage->None, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{567, 332}, WindowMargins->{{-1, Automatic}, {Automatic, 5}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of all cells in \ a given style. Make modifications to any definition using commands in the \ Format menu.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], PageWidth->WindowWidth, ScriptMinSize->9], Cell[StyleData[All, "Presentation"], PageWidth->WindowWidth, ScriptMinSize->12, FontSize->16], Cell[StyleData[All, "Condensed"], PageWidth->WindowWidth, CellBracketOptions->{"Margins"->{1, 1}, "Widths"->{0, 5}}, ScriptMinSize->8, FontSize->11], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, ScriptMinSize->5, FontSize->10, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the Notebook level.\ \>", "Text"], Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, CellFrameLabelMargins->6, StyleMenuListing->None] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellMargins->{{12, Inherited}, {20, 40}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subtitle", 0}, {"Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->36, FontWeight->"Bold"], Cell[StyleData["Title", "Presentation"], CellMargins->{{24, 10}, {20, 40}}, LineSpacing->{1, 0}, FontSize->44], Cell[StyleData["Title", "Condensed"], CellMargins->{{8, 10}, {4, 8}}, FontSize->20], Cell[StyleData["Title", "Printout"], CellMargins->{{2, 10}, {15, 30}}, FontSize->24] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{12, Inherited}, {10, 15}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->24], Cell[StyleData["Subtitle", "Presentation"], CellMargins->{{24, 10}, {15, 20}}, LineSpacing->{1, 0}, FontSize->36], Cell[StyleData["Subtitle", "Condensed"], CellMargins->{{8, 10}, {4, 4}}, FontSize->14], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{2, 10}, {10, 15}}, FontSize->18] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], CellMargins->{{12, Inherited}, {10, 20}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, CounterIncrements->"Subsubtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontFamily->"Helvetica", FontSize->14, FontSlant->"Italic"], Cell[StyleData["Subsubtitle", "Presentation"], CellMargins->{{24, 10}, {10, 20}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Subsubtitle", "Condensed"], CellMargins->{{8, 10}, {8, 12}}, FontSize->12], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{2, 10}, {8, 10}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[FilledSquare]", CellMargins->{{25, Inherited}, {8, 24}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->16, FontWeight->"Bold"], Cell[StyleData["Section", "Presentation"], CellMargins->{{40, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Section", "Condensed"], CellMargins->{{18, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Section", "Printout"], CellMargins->{{13, 0}, {7, 22}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 20}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontSize->14, FontWeight->"Bold"], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{36, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->22], Cell[StyleData["Subsection", "Condensed"], CellMargins->{{16, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Subsection", "Printout"], CellMargins->{{9, 0}, {7, 22}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 18}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, CounterIncrements->"Subsubsection", FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{34, 10}, {11, 26}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Subsubsection", "Condensed"], CellMargins->{{17, Inherited}, {6, 12}}, FontSize->10], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{9, 0}, {7, 14}}, FontSize->11] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{12, 10}, {7, 7}}, LineSpacing->{1, 3}, CounterIncrements->"Text"], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}], Cell[StyleData["Text", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}], Cell[StyleData["Text", "Printout"], CellMargins->{{2, 2}, {6, 6}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], CellMargins->{{12, 10}, {6, 6}}, LineSpacing->{1, 3}, CounterIncrements->"SmallText", FontFamily->"Helvetica", FontSize->9], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, 10}, {8, 8}}, LineSpacing->{1, 5}, FontSize->12], Cell[StyleData["SmallText", "Condensed"], CellMargins->{{8, 10}, {5, 5}}, LineSpacing->{1, 2}, FontSize->9], Cell[StyleData["SmallText", "Printout"], CellMargins->{{2, 2}, {5, 5}}, FontSize->7] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output to the \ kernel. Be careful when modifying, renaming, or removing these styles, \ because the front end associates special meanings with these style names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{45, 10}, {5, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Input", "Presentation"], CellMargins->{{72, Inherited}, {8, 10}}, LineSpacing->{1, 0}], Cell[StyleData["Input", "Condensed"], CellMargins->{{40, 10}, {2, 3}}], Cell[StyleData["Input", "Printout"], CellMargins->{{39, 0}, {4, 6}}, FontSize->9] }, Closed]], Cell[StyleData["InputOnly"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", StyleMenuListing->None, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{47, 10}, {7, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Output", FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, Inherited}, {10, 8}}, LineSpacing->{1, 0}], Cell[StyleData["Output", "Condensed"], CellMargins->{{41, Inherited}, {3, 2}}], Cell[StyleData["Output", "Printout"], CellMargins->{{39, 0}, {6, 4}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Message", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Condensed"], ImageSize->{175, 175}, ImageMargins->{{38, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Printout"], ImageSize->{250, 250}, ImageMargins->{{30, Inherited}, {Inherited, 0}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "Condensed"], FontSize->9], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}], Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, ScriptLevel->0, SingleLetterItalics->True, StyleMenuListing->None, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}], Cell[StyleData["DisplayFormula", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard ButtonFunctions, for use \ in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext ButtonBoxes. The \ \"Hyperlink\" style is for links within the same Notebook, or between \ Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"]], Cell[StyleData["Hyperlink", "Condensed"]], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line help \ system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"]], Cell[StyleData["MainBookLink", "Condensed"]], Cell[StyleData["MainBookLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"]], Cell[StyleData["AddOnsLink", "Condensed"]], Cell[StyleData["AddOnLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Condensed"]], Cell[StyleData["RefGuideLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"]], Cell[StyleData["GettingStartedLink", "Condensed"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"]], Cell[StyleData["OtherInformationLink", "Condensed"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder objects in \ palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->Primary, StyleMenuListing->None, DrawHighlighted->True], Cell[StyleData["SelectionPlaceholder", "Presentation"]], Cell[StyleData["SelectionPlaceholder", "Condensed"]], Cell[StyleData["SelectionPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles of most \ cells. 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If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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