(************** 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). 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[ 62573, 1755]*) (*NotebookOutlinePosition[ 63234, 1778]*) (* CellTagsIndexPosition[ 63190, 1774]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ A Nonlinear Oscillator, Numerical Solution\ \>", "Title", TextAlignment->Center], Cell[TextData[{ "Problem 3, on P.S. #3, 2002. Numerical check of perturbation \nsolution, \ problem not solved here.\n\nFinds a numerical solution to:\ny'' + y ", Cell[BoxData[ \(TraditionalForm\`\(\(+\[Epsilon]y\^3\)\(=\)\(0\)\(\ \)\)\)]], "\nwith y'(0)=0 and y(0)=1.\nThe period in the numerical solution is then \ compared to the prediction made from the perturbation expansion. The math \ for the\nperturbation expansion is not shown here." }], "Text"], Cell[BoxData[ \(Clear[eps]\)], "Input"], Cell[TextData[{ "The following line was changed to text. You may want to change the \ Format>Style to \"Input\" to execute the statement, and confirm that ", StyleBox["Mathematica", FontSlant->"Italic"], " cannot find an analytical solution:" }], "Text"], Cell["\<\ DSolve[{y''[t]+y[t]+eps*y[t]^3\[Equal]0,y[0]\[Equal]1,y'[0]\[Equal]\ 0},y[t],t]\ \>", "Text", Background->RGBColor[1, 1, 0]], Cell["\<\ So we resort to a numerical solution for a specified value of \ eps:\ \>", "Text"], Cell[BoxData[ \(nonosc[eps_] := NDSolve[{\(y''\)[t] + y[t] + eps*y[t]^3 \[Equal] 0, y[0] \[Equal] 1, \(y'\)[0] \[Equal] 0}, {y}, {t, 0, 10}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(tryit = nonosc[1.5]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"y", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 10.`}}, "<>"]\), False, Editable->False]}], "}"}], "}"}]], "Output"] }, Open ]], Cell["\<\ Plot the analytical solution for eps=0 in black and the numerical solution for eps=0.5 in red:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ 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00<007ooOol0@Woo00=l07ooOol07Goo00=l07ooOol01goo00<007ooOol0;Woo00<007ooOol047oo 00=l07ooOol07Woo00=l07ooOol05Woo000KOol00`00Oomoo`0=Ool00g`0Oomoo`05Ool00`00Oomo o`12Ool00g`0Oomoo`0MOol00g`0Oomoo`07Ool00`00Oomoo`0]Ool00`00Oomoo`0AOol00g`0Oomo o`0MOol00g`0Oomoo`0GOol001]oo`03001oogoo00eoo`03O01oogoo00Aoo`03001oogoo04Aoo`03 O01oogoo01aoo`03O01oogoo00Moo`03001oogoo02eoo`03001oogoo019oo`03O01oogoo01aoo`03 O01oogoo01Moo`006goo0P003Goo00=l07ooOol01Goo00<007ooOol0A7oo00=l07ooOol077oo00=l 07ooOol027oo00<007ooOol0:goo00<007ooOol04goo00=l07ooOol077oo00=l07ooOol05goo000K Ool00`00Oomoo`0"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.07864, -1.13008, \ 0.0394728, 0.0127737}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Here we make a more convenient interface to access the \ interpolating function that contains the numerical solution. Notice the \ replacement rule is inside a nested list, which is accessed by the [[1,1]]:\ \ \>", "Text"], Cell[BoxData[ \(\(yfun[t_] = y[t] /. tryit[\([1, 1]\)];\)\)], "Input"], Cell["Let's give it a try:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(yfun[2]\)], "Input"], Cell[BoxData[ \(\(-0.965200250162025`\)\)], "Output"] }, Open ]], Cell["Let's find t where y(t)=0 in the numerical solution:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindRoot[yfun[t], {t, {0, 1}}]\)], "Input"], Cell[BoxData[ \({t \[Rule] 1.0839591623123357`}\)], "Output"] }, Open ]], Cell["\<\ Now make a module to find the first t where y(t)=0 for a specified value of \[Epsilon].\ \>", "Text"], Cell[BoxData[ \(FirstZero[eps_] := Module[{t, tryit, yfun, rt}, tryit = nonosc[eps]; \[IndentingNewLine]yfun[t_] = y[t] /. tryit[\([1, 1]\)]; rt = FindRoot[yfun[t] \[Equal] 0, {t, 1, 2}]; \[IndentingNewLine]t /. rt[\([1]\)]\[IndentingNewLine]]\)], "Input"], Cell["Try it:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(FirstZero[\(- .1\)]\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[BoxData[ \(1.6334535024077244`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FirstZero[ .1]\)], "Input"], Cell[BoxData[ \(1.515162351023867`\)], "Output"] }, Open ]], Cell["\<\ Make a table of ratio of the period of oscillation to that with \ \[Epsilon]=0.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[FirstZero[x]*2/Pi, {x, \(- .5\), .5, .1}]\)], "Input"], Cell[BoxData[ \({1.274611126722762`, 1.1998531515165958`, 1.1379310613803093`, 1.085354843905023`, 1.0398887968758344`, 0.9999988917456455`, 0.9645823110087438`, 0.9328352499027123`, 0.904144218864812`, 0.8780344030199516`, 0.8541306198179611`}\)], "Output"] }, Open ]], Cell[BoxData[ \(PertResult1[eps_] := 1/\((1 + 3*eps/8)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[PertResult1[x], {x, \(- .5\), .5, .1}]\)], "Input"], Cell[BoxData[ \({1.2307692307692308`, 1.1764705882352942`, 1.1267605633802817`, 1.081081081081081`, 1.0389610389610389`, 1.`, 0.9638554216867469`, 0.9302325581395349`, 0.898876404494382`, 0.8695652173913044`, 0.8421052631578947`}\)], "Output"] }, Open ]], Cell["\<\ Compare the numerical result, which we assume to be exact, (green) \ with the perturbation result (red). \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{FirstZero[x]*2/Pi, PertResult1[x]}, {x, \(- .5\), .5}, PlotStyle \[Rule] {RGBColor[0, 1, 0], RGBColor[1, 0, 0]}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! 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