(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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[ 27368, 780]*) (*NotebookOutlinePosition[ 28082, 805]*) (* CellTagsIndexPosition[ 28038, 801]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\( (*\ Preliminary\ \(Setup : \ Define\ Modules\)\ *) \), "\[IndentingNewLine]", RowBox[{ FormBox[ RowBox[{\(CreateRules(EQ_, N_)\), ":=", RowBox[{"Module", "[", RowBox[{\({}\), ",", RowBox[{"Flatten", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{"ToRules", "[", RowBox[{ SuperscriptBox["EQ", TagBox[\((k)\), Derivative], MultilineFunction->None], "(", "0", ")"}], "]"}], ",", \({k, 0, N}\)}], "]"}], "]"}]}], "]"}]}], "TraditionalForm"], "\[IndentingNewLine]", RowBox[{\(CreateRules2[EQ_, N_]\), ":=", RowBox[{"Module", "[", RowBox[{\({}\), ",", RowBox[{"Flatten", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{"ToRules", "[", RowBox[{ SuperscriptBox["EQ", TagBox[\((k)\), Derivative], MultilineFunction->None], "[", "s", "]"}], "]"}], ",", \({k, 0, N - 1}\)}], "]"}], "]"}]}], "]"}]}], "\[IndentingNewLine]", "\[IndentingNewLine]", \( (*\ ROUND\ \(ONE : \ COMPUTE\ THE\ FIRST\ TERM\)\ *) \), "\[IndentingNewLine]", "\[IndentingNewLine]", \( (*\ Set\ i\ to\ be\ the\ top - order\ retained\ in\ expansion\ *) \), "\n", \(i = 2;\), "\[IndentingNewLine]", "\[IndentingNewLine]", \( (*\ Define\ the\ numerical\ method\ here\ *) \), "\[IndentingNewLine]", \(M1[h_]\ := \ \ yb[h]\ \[Equal] \ y0\ + \ h*f[yb[h]];\), "\[IndentingNewLine]", "\[IndentingNewLine]", \( (*\ Compute\ higher\ order\ derivatives\ *) \), "\[IndentingNewLine]", \(R\ = \ CreateRules[M1, i];\), "\[IndentingNewLine]", "\[IndentingNewLine]", \( (*\ Expand\ the\ method\ in\ a\ series\ to\ order\ i\ *) \), "\[IndentingNewLine]", \(y1[h] = \ Simplify[Series[yb[h], {h, 0, i}]\ //. 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