(*********************************************************************** 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). 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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[ 300488, 17312]*) (*NotebookOutlinePosition[ 301542, 17348]*) (* CellTagsIndexPosition[ 301498, 17344]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell[TextData["A Note on the Shooting Method"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Allan Hayes"], "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Mathematica in Education\nVol. 3 No.4\nCopyright 1994\n\ TELOS/Springer-Verlag"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "In Volume 2 Number 4, Robert Spall gave an interesting application of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " to fluid dynamics. The mathematical part was concerned with the solution \ of a nonlinear 2-point boundary by the shooting method. The present note \ takes the reduction of programming effort a bit further by using FindRoot \ instead of a While loop. Some printout about progress is lost but the payoff \ is simpler code and better display of the mathematical ideas.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["The system to be solved is ."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "{f[r](f''[r] - 1/r f'[r] + \n\t\t2 V0^2/U Exp[-2/U f[r]] (1.0 - Exp[-2/U \ f[r]])) +\n\t\tV0^2 r^2 (Exp[-4/U f[r] ] - Exp[-2/U f[r]]) == 0,\n\t\tf[b1] \ == 0.5(a1^2 U),\n\t\tf[b2] == d\n\t\t};"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["We fix the parameters other than ", Evaluatable->False, AspectRatioFixed->True], StyleBox["V0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "\nU = 1.0;\na1 = 4.0; b1 = 4.1;\na2 = .1^10; b2 = .1^10;\nd = 0.5(a2^2 \ U);"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["The system has two boundary points ", Evaluatable->False, AspectRatioFixed->True], StyleBox["b1", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["b2 ", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox["with conditions ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f[b1] == 0.5(a1^2 U)", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f[b2] == d", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], StyleBox[" . 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Here is the \ new system with its two parameters displayed and accessible.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "sys[s_,V0_] := \n\t{f[r](f1'[r] - 1/r f1[r] + \n\t\t2 V0^2/U Exp[-2/U f[r]] \ (1.0 - Exp[-2/U f[r]])) +\n\t\tV0^2 r^2 (Exp[-4/U f[r] ] - Exp[-2/U f[r]]) == \ 0,\n\t\tf'[r] == f1[r],\n\t\tf[b1] == 0.5(a1^2 U),\n\t\tf1[b1] == s\n\t};"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["The following function returns the pair ", Evaluatable->False, AspectRatioFixed->True], StyleBox["{f, f1}", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], StyleBox[" for different ", Evaluatable->False, AspectRatioFixed->True], StyleBox["s ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], StyleBox["and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["V0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "g[s_,V0_] := {f,f1} /. 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