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x\_j)\)\ \((x\_j - x\_\(1 + j\))\)\)\)\), \(x\_\(\(-1\) + j\) + x\_\(1 + j\)\ \)\/\(\((x\_\(\(-1\) + j\) - x\_j)\)\ \((x\_j - x\_\(1 + j\))\)\), \(-\(1\/\(\ \((x\_\(\(-1\) + j\) - x\_j)\)\ \((x\_j - x\_\(1 + j\))\)\)\)\)}, {\(x\_\(\(-1\) + j\)\ \ x\_j\)\/\(\((x\_\(\(-1\) + j\) - x\_\(1 + j\))\)\ \((x\_j - x\_\(1 + \ j\))\)\), \(-\(\(x\_\(\(-1\) + j\) + x\_j\)\/\(\((x\_\(\(-1\) + j\) - x\_\(1 + j\))\)\ \((x\_j - x\_\(1 + j\))\)\)\)\), 1\/\(\((x\_\(\(-1\) + j\) - x\_\(1 + j\))\)\ \((x\_j - x\_\(1 + \ j\))\)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Out[ 305] . {x\_\(j + 1\) - x\_\(j - 1\), \((x\_\(j + 1\)\^2 - x\_\(j - 1\)\^2)\)\/2, \ \((x\_\(j + 1\)\^3 - x\_\(j - 1\)\^3)\)\/3} // Simplify\)], "Input"], Cell[BoxData[ \({\(-\(\(\((x\_\(\(-1\) + j\) - x\_\(1 + j\))\)\ \((2\ x\_\(\(-1\) + j\) - 3\ x\_j + x\_\(1 + j\))\)\)\/\(6\ \((x\_\(\(-1\) + j\) - x\_j)\)\)\)\), \(-\(\((x\_\(\(-1\) + j\) - x\_\(1 + \ j\))\)\^3\/\(6\ \((x\_\(\(-1\) + j\) - x\_j)\)\ \((x\_j - x\_\(1 + j\))\)\)\)\), \(\((x\_\(\(-1\) + j\) - x\_\(1 + \ j\))\)\ \((x\_\(\(-1\) + j\) - 3\ x\_j + 2\ x\_\(1 + j\))\)\)\/\(6\ \((x\_j - \ x\_\(1 + j\))\)\)}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Page 232.", "Section"], Cell[CellGroupData[{ Cell["Problem 5.4", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{\(2\/\[CapitalDelta]x\^2\), \((2\ \[Chi] - \((x\_\(i - 1\) - \ \[CapitalDelta]x + x\_\(i - 1\) - 2 \[CapitalDelta]x)\))\), \(2\/\[CapitalDelta]x\^2\), \((2 \ \[Chi] - \((x\_\(i - 1\) + x\_\(i - 1\) - \[CapitalDelta]x)\))\), Cell[""]}], ",", \({\[Chi], x\_\(i - 1\) - 2 \[CapitalDelta]x, x\_\(i - 1\)}\)}], "]"}]], "Input"], Cell[BoxData[ \(8\/\(3\ \[CapitalDelta]x\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{\(2\/\[CapitalDelta]x\^2\), \((2\ \[Chi] - \((x\_\(i - 1\) - \ \[CapitalDelta]x + x\_\(i - 1\) - 2 \[CapitalDelta]x)\))\), \(\(-4\)\/\[CapitalDelta]x\^2\), \ \((2 \[Chi] - \((x\_\(i - 1\) + x\_\(i - 1\) - 2 \[CapitalDelta]x)\))\), Cell[""]}], ",", \({\[Chi], x\_\(i - 1\) - 2 \[CapitalDelta]x, x\_\(i - 1\)}\)}], "]"}]], "Input"], Cell[BoxData[ \(\(-\(64\/\(3\ \[CapitalDelta]x\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{\(2\/\[CapitalDelta]x\^2\), \((2\ \[Chi] - \((x\_\(i - 1\) - \ \[CapitalDelta]x + x\_\(i - 1\) - 2 \[CapitalDelta]x)\))\), \(2\/\[CapitalDelta]x\^2\), \((2 \ \[Chi] - \((x\_\(i - 1\) - \[CapitalDelta]x + x\_\(i - 1\) - 2 \[CapitalDelta]x)\))\), Cell[""]}], ",", \({\[Chi], x\_\(i - 1\) - 2 \[CapitalDelta]x, x\_\(i - 1\)}\)}], "]"}]], "Input"], Cell[BoxData[ \(56\/\(3\ \[CapitalDelta]x\)\)], "Output"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Page 234.", "Section"], Cell[CellGroupData[{ Cell["Problem 5.4", "Subsection"], Cell[BoxData[ \(\(tentL[j_] := Which[\[IndentingNewLine]j \[Equal] \(-2\), \ \(2\/\[CapitalDelta]x\^2\) \((\[Chi] - \((x\_i - \[CapitalDelta]x\/2)\))\) \ \((\[Chi] - x\_i)\), \[IndentingNewLine]j \[Equal] \(-1\), \(\(-4\)\/\ \[CapitalDelta]x\^2\) \((\[Chi] - \((x\_i - \[CapitalDelta]x\ )\))\) \ \((\[Chi] - x\_i)\), \[IndentingNewLine]j \[Equal] 0, \(2\/\[CapitalDelta]x\^2\) \((\[Chi] - \((x\_i - \ \[CapitalDelta]x\ )\))\) \((\[Chi] - \((x\_i - \[CapitalDelta]x\/2)\))\)\ \[IndentingNewLine]];\)\)], "Input"], Cell[BoxData[ \(\(tentR[j_] := Which[\[IndentingNewLine]j \[Equal] 2, \(2\/\[CapitalDelta]x\^2\) \((\[Chi] - x\_i)\) \((\[Chi] - \((x\_i + \[CapitalDelta]x\/2)\))\), \ \[IndentingNewLine]j \[Equal] 1, \(\(-4\)\/\[CapitalDelta]x\^2\) \((\[Chi] - x\_i)\) \((\[Chi] - \((x\_i + \[CapitalDelta]x\ )\))\), \ \[IndentingNewLine]j \[Equal] 0, \(2\/\[CapitalDelta]x\^2\) \((\[Chi] - \((x\_i + \ \[CapitalDelta]x\/2)\))\) \((\[Chi] - \((x\_i + \[CapitalDelta]x\ )\))\)\ \[IndentingNewLine]];\)\)], "Input"], Cell["With tent=tentL", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ D[tent[\(-1\)], \[Chi]] D[\ tent[\(-2\)], \[Chi]], {\[Chi], x\_i - \[CapitalDelta]x, x\_i}]\)], "Input"], Cell[BoxData[ \(\(-\(8\/\(3\ \[CapitalDelta]x\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ D[tent[\(-1\)], \[Chi]] D[\ tent[\(-1\)], \[Chi]], {\[Chi], x\_i - \[CapitalDelta]x, x\_i}]\)], "Input"], Cell[BoxData[ \(16\/\(3\ \[CapitalDelta]x\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ D[tent[\(-1\)], \[Chi]] D[\ tent[0], \[Chi]], {\[Chi], x\_i - \[CapitalDelta]x, x\_i}]\)], "Input"], Cell[BoxData[ \(\(-\(8\/\(3\ \[CapitalDelta]x\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ tent[\(-1\)]\ tent[\(-2\)], {\[Chi], x\_i - \[CapitalDelta]x, x\_i}]\)], "Input"], Cell[BoxData[ \(\[CapitalDelta]x\/15\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ tent[\(-1\)] tent[\(-1\)], {\[Chi], x\_i - \[CapitalDelta]x, x\_i}]\)], "Input"], Cell[BoxData[ \(\(8\ \[CapitalDelta]x\)\/15\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ tent[\(-1\)]\ tent[0], {\[Chi], x\_i - \[CapitalDelta]x, x\_i}]\)], "Input"], Cell[BoxData[ \(\[CapitalDelta]x\/15\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(Integrate[ D[tentL[0], \[Chi]] D[tentL[\(-2\)], \[Chi]], {\[Chi], x\_i - \[CapitalDelta]x, x\_i}]\), "\[IndentingNewLine]", \(Integrate[ D[tentL[0], \[Chi]] D[tentL[\(-1\)], \[Chi]], {\[Chi], x\_i - \[CapitalDelta]x, x\_i}]\), "\[IndentingNewLine]", \(Integrate[ D[tentL[0], \[Chi]] D[tentL[0], \[Chi]], {\[Chi], x\_i - \[CapitalDelta]x, x\_i}]\)}], "Input"], Cell[BoxData[ \(1\/\(3\ \[CapitalDelta]x\)\)], "Output"], Cell[BoxData[ \(\(-\(8\/\(3\ \[CapitalDelta]x\)\)\)\)], "Output"], Cell[BoxData[ \(7\/\(3\ \[CapitalDelta]x\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(Integrate[ D[tentR[0], \[Chi]] D[tentR[0], \[Chi]], {\[Chi], x\_i, x\_i + \[CapitalDelta]x}]\), "\[IndentingNewLine]", \(Integrate[ D[tentR[0], \[Chi]] D[tentR[1], \[Chi]], {\[Chi], x\_i, x\_i + \[CapitalDelta]x}]\), "\[IndentingNewLine]", \(Integrate[ D[tentR[0], \[Chi]] D[tentR[2], \[Chi]], {\[Chi], x\_i, x\_i + \[CapitalDelta]x}]\)}], "Input"], Cell[BoxData[ \(7\/\(3\ \[CapitalDelta]x\)\)], "Output"], Cell[BoxData[ \(\(-\(8\/\(3\ \[CapitalDelta]x\)\)\)\)], "Output"], Cell[BoxData[ \(1\/\(3\ \[CapitalDelta]x\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(Integrate[ tentL[0] tentL[\(-2\)], {\[Chi], x\_i - 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gDA, {h, 0, 2}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["u", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}], " ", "h"}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\(1\/4\), " ", RowBox[{"(", RowBox[{ RowBox[{\(-\(2\/3\)\), " ", RowBox[{ SuperscriptBox["u", TagBox[\((0, 3)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}]}], "-", RowBox[{ SuperscriptBox["u", TagBox[\((2, 1)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}]}], ")"}]}], "+", RowBox[{\(1\/4\), " ", RowBox[{"(", RowBox[{ RowBox[{\(2\/3\), " ", RowBox[{ SuperscriptBox["u", TagBox[\((0, 3)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}]}], "+", RowBox[{ SuperscriptBox["u", TagBox[\((2, 1)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}]}], ")"}]}]}], ")"}], " ", \(h\^2\)}], "+", InterpretationBox[\(O[h]\^3\), SeriesData[ h, 0, {}, 1, 3, 1]]}], SeriesData[ h, 0, { Derivative[ 0, 2][ u][ x, y], Plus[ Times[ Rational[ 1, 4], Plus[ Times[ Rational[ -2, 3], Derivative[ 0, 3][ u][ x, y]], Times[ -1, Derivative[ 2, 1][ u][ x, y]]]], Times[ Rational[ 1, 4], Plus[ Times[ Rational[ 2, 3], Derivative[ 0, 3][ u][ x, y]], Derivative[ 2, 1][ u][ x, y]]]]}, 1, 3, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((fAB - fCD)\) h + \((gBC - gDA)\) h // Simplify\)], "Input"], Cell[BoxData[ \(1\/2\ \((\(-4\)\ u[x, y] + u[\(-h\) + x, \(-h\) + y] + u[\(-h\) + x, h + y] + u[h + x, \(-h\) + y] + u[h + x, h + y])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Series[Out[14], {h, 0, 2}] // Simplify\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["u", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}], "+", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}]}], ")"}], " ", \(h\^2\)}], "+", InterpretationBox[\(O[h]\^3\), SeriesData[ h, 0, {}, 2, 3, 1]]}], SeriesData[ h, 0, { Plus[ Derivative[ 0, 2][ u][ x, y], Derivative[ 2, 0][ u][ x, y]]}, 2, 3, 1]]], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Chapter 7", "Subsubtitle"], Cell[BoxData[ \(\(f[x_] := Which[x \[LessEqual] .8, 0, .8 \[LessEqual] x < 1, \((x - .8)\)\/ .2, 1 \[LessEqual] x \[LessEqual] 1.2, \(-\((x - 1.2)\)\)\/ .2, x > 1.2, 0];\)\)], "Input"], Cell[BoxData[ \(\(initialCondition = Plot[f[x], {x, .5, 1.8}, \[IndentingNewLine]PlotStyle \[Rule] {RGBColor[1, 0, 0]}];\)\)], "Input"], Cell[BoxData[{ \(\(n = 12;\)\), "\[IndentingNewLine]", \(\(\[CapitalDelta]x = \((1.7 - 0.5)\)\/n;\)\), "\[IndentingNewLine]", \(\(x = Table[0.5 + \[CapitalDelta]x\ i, {i, 0, n}];\)\)}], "Input"], Cell[BoxData[{ \(\(ic = Map[f, x];\)\), "\[IndentingNewLine]", \(\(ListPlot[ic];\)\)}], "Input"], Cell[CellGroupData[{ Cell["Page 282. Problem 7.1", "Section"], Cell["Numerical scheme to use.", "Text"], Cell[BoxData[ \(\(os[x_, \[Sigma]_] := Identity[ x] - \(\[Sigma]\/2\) \((RotateLeft[x] - RotateRight[x])\);\)\)], "Input"], Cell[BoxData[ \(\(exercise[nts_, \[Sigma]_] := Module[{exact, ts, approx}, \[IndentingNewLine]exact = Plot[f[x - nts\ \[CapitalDelta]x\ \[Sigma]], {x, 0.5, 1.8}, \[IndentingNewLine]PlotStyle \[Rule] RGBColor[0, 1, 0], \[IndentingNewLine]DisplayFunction \[Rule] Identity\[IndentingNewLine]]; \[IndentingNewLine]ts = Nest[os[#, \[Sigma]] &, ic, nts]; \[IndentingNewLine]approx = ListPlot[\[IndentingNewLine]Thread[{x, ts}], \[IndentingNewLine]PlotJoined \[Rule] True, PlotRange \[Rule] All, \[IndentingNewLine]DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[{initialCondition, exact, approx}];\[IndentingNewLine]];\)\)], "Input"], Cell[BoxData[ \(\(\( (*\ \(Debugging\)\(.\)\ *) \)\(\[IndentingNewLine]\)\(exercise[3, 1.0]\)\)\)], "Input"], Cell[BoxData[ \(\(Table[exercise[i, 1.0], {i, 2, 5}];\)\)], "Input"], Cell[BoxData[ \(\(Table[exercise[i, 0.5], {i, 2, 5}];\)\)], "Input"], Cell[BoxData[ \(\(Table[exercise[i, 1.5], {i, 2, 5}];\)\)], "Input"], Cell["\<\ If I make \[Sigma] very small will this delay the istability? No \ still get divergence from the expected wave form.\ \>", "Text"], Cell[BoxData[ \(\(Table[exercise[i, 0.05], {i, 2, 15}];\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Page 282. Problem 7.2", "Section"], Cell["Numerical scheme to use.", "Text"], Cell[BoxData[ \(\(os[x_, \[Sigma]_] := Identity[ x] - \[Sigma] \((Identity[x] - RotateRight[x])\);\)\)], "Input"], Cell[BoxData[ \(\(exercise[nts_, \[Sigma]_] := Module[{exact, ts, approx}, \[IndentingNewLine]exact = Plot[f[x - nts\ \[CapitalDelta]x\ \[Sigma]], {x, 0.5, 1.8}, \[IndentingNewLine]PlotStyle \[Rule] RGBColor[0, 1, 0], \[IndentingNewLine]DisplayFunction \[Rule] Identity\[IndentingNewLine]]; \[IndentingNewLine]ts = Nest[os[#, \[Sigma]] &, ic, nts]; \[IndentingNewLine]approx = ListPlot[\[IndentingNewLine]Thread[{x, ts}], \[IndentingNewLine]PlotJoined \[Rule] True, PlotRange \[Rule] All, \[IndentingNewLine]DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[{initialCondition, exact, approx}];\[IndentingNewLine]];\)\)], "Input"], Cell[BoxData[ \(\(\( (*\ \(Debugging\)\(.\)\ *) \)\(\[IndentingNewLine]\)\(exercise[3, 1.0]\)\)\)], "Input"], Cell[BoxData[ \(exercise[3, 0.5]\)], "Input"], Cell[BoxData[ \(exercise[3, 1.5]\)], "Input"], Cell[BoxData[ \(\(Table[exercise[i, 1.0], {i, 2, 5}];\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Page 282. Problem 7.3", "Section"], Cell["Numerical scheme to use.", "Text"], Cell[BoxData[ \(\(os[x_, \[Sigma]_] := Identity[ x] - \[Sigma] \((Identity[x] - RotateRight[x])\);\)\)], "Input"], Cell["Get initial conditions.", "Text"], Cell[BoxData[{ \(\(a = 0.5;\)\), "\[IndentingNewLine]", \(\(b = 1.7;\)\), "\[IndentingNewLine]", \(\(getIC[\[CapitalDelta]xIn_] := Module[{n}, \[IndentingNewLine]\[CapitalDelta]x = \[CapitalDelta]xIn; \ \[IndentingNewLine]n = \((b - a)\)\/\[CapitalDelta]x; \[IndentingNewLine]x = Table[a + \[CapitalDelta]x\ i, {i, 0, n}]; \[IndentingNewLine]ic = Map[f, x]\[IndentingNewLine]];\)\)}], "Input"], Cell[BoxData[ \(getIC[0.05]\)], "Input"], Cell[BoxData[ \(\(\( (*\(Debugging\)\(.\)*) \)\(\[IndentingNewLine]\)\(ListPlot[ ic];\)\)\)], "Input"], Cell[BoxData[ \(\(exercise[nts_, \[Sigma]_] := Module[{exact, ts, approx}, \[IndentingNewLine]exact = Plot[f[x - nts\ \[CapitalDelta]x\ \[Sigma]], {x, a, b}, \[IndentingNewLine]PlotStyle \[Rule] RGBColor[0, 1, 0], \[IndentingNewLine]DisplayFunction \[Rule] Identity\[IndentingNewLine]]; \[IndentingNewLine]ts = Nest[os[#, \[Sigma]] &, ic, nts]; \[IndentingNewLine]approx = ListPlot[\[IndentingNewLine]Thread[{x, ts}], \[IndentingNewLine]PlotJoined \[Rule] True, PlotRange \[Rule] All, \[IndentingNewLine]DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[{initialCondition, exact, approx}];\[IndentingNewLine]];\)\)], "Input"], Cell[BoxData[ \(\(\( (*\ \(Debugging\)\(.\)\ *) \)\(\[IndentingNewLine]\)\(exercise[3, 0.5]\)\)\)], "Input"], Cell[BoxData[{ \(\(\[Sigma]Exercise = 0.5;\)\), "\[IndentingNewLine]", \(exercise[4, \[Sigma]Exercise]\), "\[IndentingNewLine]", \(exercise[8, \[Sigma]Exercise]\), "\[IndentingNewLine]", \(exercise[16, \[Sigma]Exercise]\)}], "Input"], Cell[BoxData[ \(getIC[0.025]\)], "Input"], Cell[BoxData[{ \(\(\[Sigma]Exercise = 0.5;\)\), "\[IndentingNewLine]", \(exercise[4, \[Sigma]Exercise]\), "\[IndentingNewLine]", \(exercise[8, \[Sigma]Exercise]\), "\[IndentingNewLine]", \(exercise[16, \[Sigma]Exercise]\)}], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Chapter 8", "Subsubtitle"], Cell[CellGroupData[{ Cell["General Routines", "Section"], Cell[BoxData[{ \( (*\ Discretize\ the\ interval\ [a, b]\ into\ nCells.\ \ \(Note : \ the\ number\ of\ nodes\ returned\ is\ then\ nCells + 1.\)\ *) \), \(\[IndentingNewLine]\(discretize[a_, b_, nCells_] := Module[{\[CapitalDelta]x}, \[IndentingNewLine]\[CapitalDelta]x = \((b - a)\)\/nCells; \[IndentingNewLine]Return[ Table[N[a + \[CapitalDelta]x\ i], {i, 0, nCells}]]; \[IndentingNewLine]]; \)\)}], "Input"], Cell[BoxData[ \(\(initialCondition[a_, b_, nCells_, iCFunction_] := Module[{xIC, yIC}, \[IndentingNewLine]xIC = discretize[a, b, nCells]; \[IndentingNewLine]yIC = Map[iCFunction, xIC]; \[IndentingNewLine]Return[Thread[{xIC, yIC}]]; \[IndentingNewLine]]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Page 304", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[\(1 - 4 \[Beta]\ Sin[\[Phi]\/2]\^2\)\/E\^\(\(-\[Beta]\)\ \[Phi]\ \^2\), {\[Phi], 0, 7}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "+", \(\((\[Beta]\/12 - \[Beta]\^2\/2)\)\ \[Phi]\^4\), "+", \(\((\(-\(\[Beta]\/360\)\) + \[Beta]\^2\/12 - \[Beta]\^3\/3)\)\ \ \[Phi]\^6\), "+", InterpretationBox[\(O[\[Phi]]\^8\), SeriesData[ \[Phi], 0, {}, 0, 8, 1]]}], SeriesData[ \[Phi], 0, {1, 0, 0, 0, Plus[ Times[ Rational[ 1, 12], \[Beta]], Times[ Rational[ -1, 2], Power[ \[Beta], 2]]], 0, Plus[ Times[ Rational[ -1, 360], \[Beta]], Times[ Rational[ 1, 12], Power[ \[Beta], 2]], Times[ Rational[ -1, 3], Power[ \[Beta], 3]]]}, 0, 8, 1]]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Page 306", "Section"], Cell[BoxData[ \(\(G[\[Sigma]_] := Plot[\@\(Cos[\[Phi]]\^2 + \(\[Sigma]\^2\) Sin[\[Phi]]\^2\), {\[Phi], 0, \[Pi]}, \[IndentingNewLine]DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(Show[{G[0.25], G[0.5], G[0.75]}, DisplayFunction \[Rule] $DisplayFunction];\)\)], "Input"], Cell["These next plots don't look correct???", "Text"], Cell[BoxData[{ \(\(p[\[Sigma]_] := Plot[ArcTan[\[Sigma]\ Tan[\[Phi]]]\/\(\[Sigma]\ \[Phi]\), {\[Phi], 0, \[Pi]}, \[IndentingNewLine]DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[{p[0.25], p[0.5], p[0.75]}, \[IndentingNewLine]DisplayFunction -> \ $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Page 307", "Section"], Cell[BoxData[ \(<< Graphics`\)], "Input"], Cell[BoxData[ \(\(p[\[Sigma]_] := PolarPlot[\@\(1 - 4\ \[Sigma] \((1 - \[Sigma])\)\ \ Sin[\[Phi]\/2]\^2\), {\[Phi], 0, 2\ \[Pi]}, \[IndentingNewLine]DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(Show[{p[0.25], p[0.5], p[0.75]}, \[IndentingNewLine]DisplayFunction -> \ $DisplayFunction];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(temp[x_] := 4 x \((1 - x)\); \ntemp[0.25]\), \(temp[0.75]\)}], "Input"], Cell[BoxData[ \(0.75`\)], "Output"], Cell[BoxData[ \(0.75`\)], "Output"] }, Open ]], Cell["Why is this not correct???", "Text"], Cell[BoxData[ \(p[\[Sigma]_] := PolarPlot[ ArcTan[\((\[Sigma]\ Sin[\[Phi]])\)\/\(1 - \[Sigma] + \[Sigma]\ Cos[\[Phi]]\)]\/\(\[Sigma]\ \[Phi]\), {\[Phi], 0, 2\ \[Pi]}, \[IndentingNewLine]DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[{p[0.25], p[0.5], p[0.75]}, \[IndentingNewLine]DisplayFunction -> $DisplayFunction]; \)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Page 310", "Section"], Cell[BoxData[ \(\(p[\[Sigma]_] := Plot[\@\(1 - 4\ \(\[Sigma]\^2\) \((1 - \[Sigma]\^2)\)\ Sin[\[Phi]\/2]\ \^4\), {\[Phi], 0, \[Pi]}, \[IndentingNewLine]DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(Show[{p[0.25], p[0.5], p[0.75]}, \[IndentingNewLine]DisplayFunction -> \ $DisplayFunction];\)\)], "Input"], Cell["Why is this not correct???", "Text"], Cell[BoxData[ \(p[\[Sigma]_] := Plot[ArcTan[ \((\[Sigma]\ Sin[\[Phi]])\)\/\(1 - 2 \[Sigma]\^2\ Sin[\[Phi]\/2]\^2\)]\/\(\[Sigma]\ \[Phi]\), { \[Phi], 0, \[Pi]}, \[IndentingNewLine]DisplayFunction \[Rule] Identity]; \[IndentingNewLine]\n Show[{p[0.25], p[0.5], p[0.75]}, \[IndentingNewLine]DisplayFunction -> $DisplayFunction]; \)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Page 314", "Section"], Cell[BoxData[ \(pp[\[Beta]_] := Plot[\((2\ \[Beta]\ Cos[\[Phi]] + \@\(1 - 4\ \(\[Beta]\^2\) Sin[\[Phi]]\^2\))\)\/\(1 + 2 \[Beta]\), {\[Phi], 0, \[Pi]}, \[IndentingNewLine]DisplayFunction \[Rule] Identity]; \[IndentingNewLine]\n Show[{pp[0.25], pp[0.5]}, \[IndentingNewLine]DisplayFunction -> $DisplayFunction]; \)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Page 339", "Section"], Cell[CellGroupData[{ Cell["Problem 8.5", "Subsection"], Cell[BoxData[ \(\(uExact[\[Alpha]_, k_, t_] := \[IndentingNewLine]Plot[\(E\^\(\(-\[Alpha]\)\ \(k\^2\) \(\ \[Pi]\^2\) t\)\) Sin[k\ \[Pi]\ x], {x, 0, 1}, \[IndentingNewLine]PlotStyle \[Rule] RGBColor[0, 1, 0], \[IndentingNewLine]DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(temp = Table[uExact[1\/3, 1, \(1\/10\) i], {i, 0, 10}]; \n\n Show[temp, DisplayFunction \[Rule] $DisplayFunction]; \)], "Input"], Cell[BoxData[ \(\(explicitCentral[v_List, \[Beta]_] := Module[{temp}, \[IndentingNewLine]temp = \((\[Beta]\ RotateRight[ v] + \((1 - 2\ \[Beta])\)\ v + \[Beta]\ RotateLeft[ v])\); \[IndentingNewLine]temp[\([1]\)] = 0; \[IndentingNewLine]temp[\([Length[temp]]\)] = 0; \[IndentingNewLine]Return[ temp];\[IndentingNewLine]];\)\)], "Input"], Cell[BoxData[ \(diffusionOneDimOneStep[a_, b_, iCFunction_, nCells_, scheme_, \[Beta]_, nTimeSteps_] := \[IndentingNewLine]\[IndentingNewLine]Module[{ic, icx, icy}, \[IndentingNewLine]ic = initialCondition[a, b, nCells, iCFunction]; \[IndentingNewLine]icx = Map[First, ic]; \[IndentingNewLine]icy = Map[Last, ic]; \[IndentingNewLine]yAtnTimeSteps = FixedPoint[scheme[#, \[Beta]] &, icy, nTimeSteps]; \[IndentingNewLine]Return[ Thread[{icx, yAtnTimeSteps}]];\[IndentingNewLine]]\)], "Input"], Cell[BoxData[ \(\(Exer85[k_, nCells_, \[Beta]_, nTimeSteps_] := Module[{\[Alpha] = 1, \[CapitalDelta]x = \((1 - 0)\)\/nCells, ue, au, lpa}, \[IndentingNewLine]ue = uExact[1, k, \[Beta]\ \ \(\[CapitalDelta]x\^2\/\[Alpha]\) nTimeSteps]; \[IndentingNewLine]ua = \ \[IndentingNewLine]diffusionOneDimOneStep[0, 1, Sin[k\ \[Pi]\ #1] &, nCells, explicitCentral, \[Beta], nTimeSteps]; \[IndentingNewLine]lpa = ListPlot[ua, DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[{ue, lpa}, DisplayFunction \[Rule] $DisplayFunction];\ \[IndentingNewLine]];\)\)], "Input"], Cell[BoxData[{ \(Exer85[1, 30, 1\/3, 5]\), "\[IndentingNewLine]", \(Exer85[1, 30, 1\/3, 10]\)}], "Input"], Cell[BoxData[{ \(Exer85[1, 30, 1\/6, 5]\), "\[IndentingNewLine]", \(Exer85[1, 30, 1\/6, 10]\)}], "Input"], Cell[BoxData[{ \(Exer85[5, 30, 1\/3, 5]\), "\[IndentingNewLine]", \(Exer85[5, 30, 1\/3, 10]\)}], "Input"], Cell[BoxData[{ \(Exer85[5, 30, 1\/6, 5]\), "\[IndentingNewLine]", \(Exer85[5, 30, 1\/6, 10]\)}], "Input"], Cell[BoxData[{ \(Exer85[10, 30, 1\/3, 5]\), "\[IndentingNewLine]", \(Exer85[10, 30, 1\/3, 10]\)}], "Input"], Cell[BoxData[{ \(Exer85[10, 30, 1\/6, 5]\), "\[IndentingNewLine]", \(Exer85[10, 30, 1\/6, 10]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[\((1 - 4\ \[Beta]\ Sin[\[Phi]\/2]\^2)\)\/E\^\(\(-\[Beta]\)\ \ \[Phi]\^2\), {\[Phi], 0, 6}] // Simplify\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "+", \(1\/12\ \((\[Beta] - 6\ \[Beta]\^2)\)\ \[Phi]\^4\), "-", \(1\/360\ \((\[Beta]\ \((1 - 30\ \[Beta] + 120\ \[Beta]\^2)\))\)\ \[Phi]\^6\), "+", InterpretationBox[\(O[\[Phi]]\^7\), SeriesData[ \[Phi], 0, {}, 0, 7, 1]]}], SeriesData[ \[Phi], 0, {1, 0, 0, 0, Times[ Rational[ 1, 12], Plus[ \[Beta], Times[ -6, Power[ \[Beta], 2]]]], 0, Times[ Rational[ -1, 360], \[Beta], Plus[ 1, Times[ -30, \[Beta]], Times[ 120, Power[ \[Beta], 2]]]]}, 0, 7, 1]]], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Page 341", "Section"], Cell["Problem 8.18.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[ ArcSin[\[Sigma]\ Sin[\[Phi]]]\/\(\[Sigma]\ \[Phi]\), {\[Phi], 0, 4}] // Simplify\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "+", \(1\/6\ \((\(-1\) + \[Sigma]\^2)\)\ \[Phi]\^2\), "+", \(1\/120\ \((1 - 10\ \[Sigma]\^2 + 9\ \[Sigma]\^4)\)\ \[Phi]\^4\), "+", InterpretationBox[\(O[\[Phi]]\^5\), SeriesData[ \[Phi], 0, {}, 0, 5, 1]]}], SeriesData[ \[Phi], 0, {1, 0, Times[ Rational[ 1, 6], Plus[ -1, Power[ \[Sigma], 2]]], 0, Times[ Rational[ 1, 120], Plus[ 1, Times[ -10, Power[ \[Sigma], 2]], Times[ 9, Power[ \[Sigma], 4]]]]}, 0, 5, 1]]], "Output"] }, Open ]] }, Closed]] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1024}, {0, 768}}, WindowSize->{787, 623}, WindowMargins->{{99, Automatic}, {9, Automatic}} ] (*********************************************************************** Cached data follows. 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