(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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). 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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[ 71670, 2333]*) (*NotebookOutlinePosition[ 72331, 2356]*) (* CellTagsIndexPosition[ 72287, 2352]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Matched Asymptotic Expansion Compared with the Numerical Solution \ \>", "Title"], Cell[TextData[{ "The following equation is \"too difficult\" for ", StyleBox["Mathematica", FontSlant->"Italic"], " to find an analytical solution:\n\n\[Epsilon] y'' + 2 y' ", Cell[BoxData[ \(TraditionalForm\`\(+\ e\^y\)\)]], "=0\nwith ", Cell[BoxData[ \(TraditionalForm\`y(0) = 0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y(1) = 0\)]], ".\n\nWhen something is too difficult, ", StyleBox["Mathematica", FontSlant->"Italic"], " returns the command that was typed (after saying \"running\" in the top \ bar of the window for about 20 seconds).\n\nSo we solve the equation by a \ discrete numerical method, and by a method of matched asymptotic expansions. \ The ODE is a two-point boundary-value problem, which we solve using a \ shooting scheme." }], "Text"], Cell[BoxData[ \(<< Graphics`Colors`\)], "Input"], Cell[BoxData[ \(Clear[eps, y, x]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[{eps*\(y''\)[x] + 2 \( y'\)[x] + Exp[y[x]] \[Equal] 0, y[0] \[Equal] 0, y[1] \[Equal] 0}, y[x], x]\)], "Input"], Cell[BoxData[ \(Solve::"ifun" \(\(:\)\(\ \)\) "Inverse functions are being used by \!\(Solve\), so some solutions may \ not be found."\)], "Message"], Cell[BoxData[ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{\(\[ExponentialE]\^y[x]\), "+", RowBox[{"2", " ", 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This \ information will be needed when we attempt to find a numerical \ solution.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[ym[t, ep], t] /. t \[Rule] 0 // N\)], "Input"], Cell[BoxData[ \(\(-1.`\) + 1.3862943611198906`\/ep\)], "Output"] }, Open ]], Cell[TextData[{ "To check our solution, we find a numerical solution for a particular value \ of \[Epsilon]. ", StyleBox["Mathematica", FontSlant->"Italic"], " solves ODE's by marching from initial conditions specified at a starting \ point. So we need to iterate to satisfy the condition y[1]=1/2 using what is \ called a \"shooting method\": we use ", Cell[BoxData[ \(TraditionalForm\`\(y'\)[0] = s\/\[Epsilon]\)]], " where we expect s will be a number near 1." }], "Text"], Cell[BoxData[ \(\(eps = .01;\)\)], "Input"], Cell[BoxData[ \(nshoot[eps_, s_] := NDSolve[{eps*\(y''\)[t] + 2*\(y'\)[t] + Exp[y[t]] \[Equal] 0, y[0] \[Equal] 0, \(y'\)[0] \[Equal] s/eps}, {y}, {t, 0, 1}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(tryit = nshoot[eps, 1. ]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"y", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 1.`}}, "<>"]\), False, Editable->False]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "Let's try to use that ", StyleBox["InterpolatingFunction", FontFamily->"Courier"], " thing. The replacement rule for y is in a nested list. 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Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(theroot = FindRoot[yAt1[eps, s] \[Equal] 0, {s, {1. , 2. }}]\)], "Input"], Cell[BoxData[ \({s \[Rule] 1.3773051783223007`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(yAt1[eps, s /. theroot]\)], "Input"], Cell[BoxData[ \(1.898752464971361`*^-11\)], "Output"] }, Open ]], Cell["\<\ So we see that we have a numerical solution that satisfies both \ boundary conditions. Let's plot it out and compare it with our approximate \ solution.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(numsol = nshoot[eps, s /. theroot]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"y", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 1.`}}, "<>"]\), False, Editable->False]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y[ .3] /. numsol[\([1, 1]\)]\)], "Input"], Cell[BoxData[ \(0.43245867493434226`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ym[ .3, eps]\)], "Input"], Cell[BoxData[ \(0.4307829160924542`\)], "Output"] }, Open ]], Cell["\<\ Next we plot both the numerical and the perturbation solution \ together. 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