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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[ 1250211, 30303]*) (*NotebookOutlinePosition[ 1251746, 30353]*) (* CellTagsIndexPosition[ 1251608, 30345]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Numerical Solution To The \nSpreaded Given Problems \ For \nLinear Ordinary Differential Equations ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Subtitle", TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell[TextData[StyleBox["Differential Equation Solver ", FontColor->RGBColor[1, 0, 0]]], "Section"], Cell[BoxData[{ RowBox[{\( (*\(:\)\(Name : NumericalMath`NLinearDSolve`\)*) \), " ", "\n", \( (*\(:\)\(Title : Numerical\ solution\ to\ the\ spreaded\ given\ problems\ for\ \ ordinary\ linear\ differential\ equations\)*) \), "\n", \( (*\(\(:\)\(Authors : \ Anry\ Nersessian\ in\ assistance\ with\ Arnak\ Poghosyan\)\), \ Institute\ of\ Mathematics\ \n\t\t\ \ \ of\ National\ Academy\ of\ \ Sciences\ of\ Armenia*) \), "\n", \( (*\(:\)\(Summary : \n\t\tThis\ package\ finds\ the\ solution\ \ to\ the\ spreaded\ given\ \((including\ integral)\) problems\ on\ a\ finite\ interval\ for\ ordinary\ linear\ \ differential\ equations\ with\ given\ precision . \ The\ given\ spreaded\ problem\ modifies\ to\ the\ Volterra\ \ type\ integral\ equation\ which\ solves\ on\ the\ base\ of\ the\ program\ \ NISolve . \ During\ the\ recursion\ systems\ of\ linear\ equations\ solve\ \ for\ determining\ the\ unknown\ parameters . \ The\ solution\ represents\ both\ as\ a\ FunctionInterpolation\ \ \((default)\)\ and\ \ SymbolicSolution\ \(\(objects\)\(.\)\)\)\ \ \[IndentingNewLine]\t*) \), "\n", "\t", "\n", "\t", \( (*\(:\)\(Context : NumericalMath`NLinearDSolve`\)*) \), "\n", "\t", \( (*\(:\)\(Mathematica\ \(Version : \ 4.1\)\)*) \), "\n", "\t", \( (*\(:\)\(Package\ \(Version : \ Alpha\ 1\)\)*) \), "\n", "\t", \( (*\(:\)\(Copyright : \ Copyright\ 2002\)*) \), "\[IndentingNewLine]", "\n", " ", \( (*\(\(:\)\(History : \ \n\t\ Current\ version\ is\ \ developed\ by\ Anry\ \ Nersessian\ in\ assistance\ with\ \ Arnak\ Poghosyan\)\ \), \ Institute\ of\ Mathematics\ of\ National\ Academy\ of\ Sciences\ of\ \ Armenia, \ July\ 2002. \ \t\n\t\ \ *) \), "\[IndentingNewLine]", "\n", " ", \( (*\(\(:\)\(Keywords : \ Differential\ equation\)\), \ Boundary\ Problem, \ Integral\ equation, \ Fast\ Algorithms*) \), "\[IndentingNewLine]", "\n", "\t", RowBox[{"(*", RowBox[{ RowBox[{":", RowBox[{"Source", ":", "\n", StyleBox["\t", FontColor->GrayLevel[0]], RowBox[{ RowBox[{ StyleBox[ RowBox[{ StyleBox["1", FontColor->GrayLevel[0]], "."}]], " ", \(A . 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Absolute Uniform-error (with weight->`4`) equals \ about `3`. For more precision use more MaxGridPoints.\>";\), "\n", \ \(NLinearDSolve::fcrelun = "\`4`) of the solution equals about `2`. Relative Uniform-error \ (with weight->`4`) equals about `3`. For more precision use more \ MaxGridPoints.\>";\), "\n", \(NLinearDSolve::fcabsL2 = "\`4`) of the solution equals about \ `2`. Absolute L2-error (with weight->`4`) equals about `3`. For more \ precision use more MaxGridPoints.\>";\), "\n", \(NLinearDSolve::fcrelL2 = \ "\`4`) of the solution \ equals about `2`. Relative L2-error (with weight->`4`) equals about `3`. For \ more precision use more MaxGridPoints.\>";\), "\n", \ \(NLinearDSolve::unknnopt = "\";\), "\n", \ \(NLinearDSolve::\ errmes = "\Relative.\>";\), "\n", \(NLinearDSolve::symsol = "\False.\>";\), "\n", \(NLinearDSolve::precinf = "\False.\>";\), "\n", \(NLinearDSolve::nrmes = "\L2.\>";\), "\n", \(NLinearDSolve::comp = "\True.\>";\), "\n", \ \(NLinearDSolve::prg = "\5.\>";\), "\n", RowBox[{\(NLinearDSolve::mxgpmes = "\200.\>";\), "\[IndentingNewLine]"}], "\n", RowBox[{\(Options[NLinearDSolve] = {SymbolicSolution \[Rule] False, PrecisionInformation \[Rule] False, PrecisionGoal \[Rule] Automatic, MaxGridPoints \[Rule] Automatic, SolutionNorm \[Rule] L2, Error \[Rule] Relative, SolutionNorm \[Rule] Uniform, Error \[Rule] Absolute, Weight \[Rule] Automatic, InterpolationOrder \[Rule] 6, InterpolationPoints \[Rule] 200, MaxRecursion \[Rule] 6, WorkingPrecision \[Rule] $MachinePrecision, AccuracyGoal \[Rule] $MachinePrecision - 10, InterpolationPrecision -> $MachinePrecision - 10};\), 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If[Length[t3] \[Equal] 0, 0, Apply[Plus, Table[If[Length[t3[\([i]\)]] > 1, Drop[t3[\([i]\)], {Length[t3[\([i]\)]]}], 1], {i, 1, Length[t3]}]]]; \[IndentingNewLine]pp = Append[t2, t4]; \[IndentingNewLine]t5 = Position[eq4, y[x]]; \[IndentingNewLine]t6 = Position[ eq4, \(\(Derivative[i_]\)[y]\)[ x]]; \[IndentingNewLine]rr = \(-Take[ eq4, {1, \((\(If[Length[t5] \[Equal] 0, t6, t5] // First\) // First)\) - 1}]\); \[IndentingNewLine]p = pp/pp[\([1]\)]; \[IndentingNewLine]r = rr/pp[\([1]\)]; \[IndentingNewLine]t7 = Table[ Position[eq4, \(\(Derivative[i]\)[y]\)[x]], {i, 1, ord}]; \[IndentingNewLine]t8 = Position[Reverse[t7], {}]; \[IndentingNewLine]If[ Length[t8] > 0, Table[p = Insert[p, 0, t8[\([i]\)]], {i, 1, Length[t8]}]]; \[IndentingNewLine]{ord, Drop[p, 1], r}\ \[IndentingNewLine]];\), "\n"}], "\[IndentingNewLine]", RowBox[{ StyleBox[\(flst[lf_] := Block[{fy, m}, \[IndentingNewLine]m = Length[lf]; \n\t fy = RotateLeft[lf, Floor[m/2]]; \n RotateRight[Fourier[fy]/\@m, Floor[m/2]]];\), 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Map[dd1[#] &, lk[d]]; \[IndentingNewLine]a1 = Flatten[{\(-1\), lk[\(-0.8\)], lk[\(-0.6\)], lk[\(-0.4\)], lk[\(-0.2\)], lk[0], lk[0.2], lk[0.4], lk[0.6], lk[0.8], 1}]; \[IndentingNewLine]a2 = Flatten[{0, furelist[\(-0.8\)], furelist[\(-0.6\)], furelist[\(-0.4\)], furelist[\(-0.2\)], furelist[0], furelist[0.2], furelist[0.4], furelist[0.6], furelist[0.8], 0}]; \[IndentingNewLine]a3 = Flatten[{dd1[\(-1\)], furelist1[\(-0.8\)], furelist1[\(-0.6\)], furelist1[\(-0.4\)], furelist1[\(-0.2\)], furelist1[0], furelist1[0.2], furelist1[0.4], furelist1[0.6], furelist1[0.8], dd1[1]}]; \[IndentingNewLine]mn = Re[Transpose[{a1, Transpose[{a2, a3}]}]] // N; \[IndentingNewLine]Interpolation[mn, InterpolationOrder \[Rule] 12]\[IndentingNewLine]];\), "\[IndentingNewLine]"}], "\n", RowBox[{\(<< Utilities`FilterOptions`;\), "\[IndentingNewLine]"}], "\n", RowBox[{ RowBox[{\(NLinearDSolve[eqns_, y_, {var_, a_, b_}, opt___]\), ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ "optNLinearDSolve", 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FilterOptions[FunctionInterpolation, opt]\), ";", "\[IndentingNewLine]", \(optNLinearDSolve = \ {SymbolicSolution, \ PrecisionInformation, \ MaxGridPoints, SolutionNorm, Error, InterpolationOrder, InterpolationPrecision, AccuracyGoal, PrecisionGoal, InterpolationPoints, MaxRecursion, WorkingPrecision, Weight}\), ";", "\[IndentingNewLine]", \(optlist = Table[{opt}[\([i]\)] // First, {i, 1, Length[{opt}]}]\), ";", "\[IndentingNewLine]", \(optErrorNLinearDSolve = Complement[optlist, optNLinearDSolve]\), ";", "\[IndentingNewLine]", \(If[Length[optErrorNLinearDSolve] > 0, Table[Message[NLinearDSolve::unknnopt, optErrorNLinearDSolve[\([i]\)]], {i, 1, Length[optErrorNLinearDSolve]}]; Break[]]\), ";", "\[IndentingNewLine]", \(ClearAttributes[NIntegrate, HoldAll]\), ";", "\[IndentingNewLine]", \({irp, pi, pg, mxgp, sn, absrel, wght} = \({SymbolicSolution, PrecisionInformation, PrecisionGoal, MaxGridPoints, SolutionNorm, Error, Weight} /. {opt}\) /. Options[NLinearDSolve]\), ";", "\[IndentingNewLine]", \(If[wght === Automatic, wght = 1]\), ";", "\[IndentingNewLine]", \(If[\((absrel === Relative)\)\ || \ \((absrel === Absolute)\), , Message[\ NLinearDSolve::errmes]; absrel = Relative]\), ";", "\n", " ", \(If[\((irp === True)\) || \((irp === False)\), , Message[NLinearDSolve::symsol]; irp = False]\), ";", "\t", "\[IndentingNewLine]", \(If[\((pi === True)\) || \((pi === False)\), , Message[NLinearDSolve::precinf]; pi = False]\), ";", "\t", "\[IndentingNewLine]", \(If[\((sn === L2)\) || \((sn === Uniform)\), , \ sn = L2; Message[NLinearDSolve::nrmes]]\), ";", "\[IndentingNewLine]", \(If[\ pg\ === \ Automatic, \ pg\ = \ $MachinePrecision - 11, If[Not[IntegerQ[pg]] || Not[Positive[pg]], Message[NLinearDSolve::prg]; pg = 5]]\), ";", "\[IndentingNewLine]", \(If[\ mxgp === Automatic, mxgp = 200, If[Not[IntegerQ[mxgp]] || Not[Positive[mxgp]], \ mxgp\ = \ 200; Message[NLinearDSolve::mxgpmes]]]\), ";", "\[IndentingNewLine]", \(If[absrel === Relative, absrel = 1, absrel = 2]\), ";", "\[IndentingNewLine]", \(If[sn === L2, sn = 2, sn = 0]\), ";", "\[IndentingNewLine]", \(pg = 10\^\(-pg\)\), ";", "\[IndentingNewLine]", "\[IndentingNewLine]", " ", \(eq = eqns // First\), ";", " ", "\[IndentingNewLine]", " ", \(ezr = Drop[eqns, 1] // N\), ";", "\[IndentingNewLine]", \(sss = RecognizeDif[eq, y, var]\), ";", "\[IndentingNewLine]", \(ord = sss[\([1]\)]\), ";", "\[IndentingNewLine]", \(If[Length[ezr] \[Equal] ord, , Message[NLinearDSolve::ndnco, Length[ezr], ord]; Break[]]\), ";", "\[IndentingNewLine]", \(p[x_] = sss[\([2]\)] /. var \[Rule] x\), ";", "\[IndentingNewLine]", \(f[x_] = sss[\([3]\)] /. var \[Rule] x\), ";", "\[IndentingNewLine]", \(k[x_, t_] = \[Sum]\+\(m = 1\)\%ord\(\( p[ x]\)[\([m]\)]\/\(\((m - 1)\)!\)\) \((x - t)\)\^\(m - 1\)\), ";", "\[IndentingNewLine]", \(kk[x_, t_] = k[\((b - a)\) x + a, \((b - a)\) t + a]\), ";", "\[IndentingNewLine]", \(kkk = Compile[{x, {t, _Real, 1}}, Evaluate[\((b - a)\) kk[Sin[\(\[Pi]\ x\)\/2]\^2, Sin[\(\[Pi]\ t\)\/2]\^2]]]\), ";", "\[IndentingNewLine]", \(flist[x_] = Flatten[{f[x], Table[\[Sum]\+\(s = m\)\%ord\( p[ x]\)[\([s]\)] \((x - a)\)\^\(s - m\)\/\(\((s - \ m)\)!\), {m, 1, ord}]}]\), ";", "\[IndentingNewLine]", \(fflist[x_] = flist[\((b - a)\) x + a]\), ";", "\[IndentingNewLine]", \(ffflist[x_] = fflist[Sin[\(\[Pi]\ x\)\/2]\^2]\), ";", "\[IndentingNewLine]", \(SetAttributes[ffflist, Listable]\), ";", "\[IndentingNewLine]", \(SetAttributes[fflist, Listable]\), ";", "\[IndentingNewLine]", \(SetAttributes[flist, Listable]\), ";", "\[IndentingNewLine]", "\[IndentingNewLine]", \(sh = 110\), ";", "\[IndentingNewLine]", "\[IndentingNewLine]", \(nnc = Round[\((b - a)\) 31]\), ";", "\[IndentingNewLine]", \(rrnnc = Range[1, nnc]\), ";", "\[IndentingNewLine]", \(tcunf = Table[\ 2. \ s/\((2\ nnc + 1)\), {s, 1, nnc}]\), ";", "\[IndentingNewLine]", \(iop = \ Sin[\[Pi]\ \ tcunf]\ \ Apply[Plus, Map[\(\(1. - \((\(-1\))\)\^#\)\/#\) Sin[\[Pi]\ #\ \ tcunf] &, rrnnc]]\), ";", "\[IndentingNewLine]", \(ll = Table[\((b - a)\) Sin[\[Pi]/2\ \ \(2. \ s\)\/\(2\ nnc + 1\)]\^2 + a, {s, 1, nnc}]\), ";", "\[IndentingNewLine]", \(wlist = Map[\((wght /. var -> #)\) &, ll]\), ";", " ", "\[IndentingNewLine]", \(k2[x_, t_, s_] = \ \(\((b - a)\)\/\(\((s - 1)\)!\)\) \((x - \((b - a)\) Sin[\[Pi]/2\ t]\^2 - \ a)\)\^\(s - 1\)\), ";", "\[IndentingNewLine]", "\[IndentingNewLine]", \(nn = 2\), ";", "\[IndentingNewLine]", \(l = Table[\(2. \ s\)\/\(2\ nn + 1\), {s, 1, nn}]\), ";", "\[IndentingNewLine]", StyleBox[\(rr = Range[1, nn]\), FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[";", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "\[IndentingNewLine]", RowBox[{"aux1", "=", RowBox[{ StyleBox["Map", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["[", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], \(kkk[#, l]\ \ Sin[\[Pi]\ \ l] &, l\), "]"}]}], ";", "\[IndentingNewLine]", StyleBox[\(pop = Compile[{n, {x, _Real, 1}}, Evaluate[\(1. \/n\) \((1 - Cos[\[Pi]\ n\ \ l])\) Sin[\[Pi]\ n\ x]]]\), FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[";", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["\[IndentingNewLine]", FontWeight->"Bold"], StyleBox[\(aux2 = \(2. \/\((2 nn + 1. )\)\) Apply[Plus, Transpose[Map[pop[#, l] &, rr]], {1}]\), FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[";", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "\[IndentingNewLine]", RowBox[{ StyleBox["mat", FontWeight->"Bold"], StyleBox["=", FontWeight->"Bold"], StyleBox[\(IdentityMatrix[Length[l]] + aux1\ \ \ aux2\), FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], StyleBox[";", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "\[IndentingNewLine]", \(ss = Transpose[LinearSolve[mat, ffflist[l]]]\), ";", "\[IndentingNewLine]", \(ezr = ezr /. Integrate \[Rule] NIntegrate\), ";", "\[IndentingNewLine]", \(pos = Position[ezr, NIntegrate]\), ";", "\[IndentingNewLine]", "\[IndentingNewLine]", \(If[ Length[pos] > 0, \[IndentingNewLine]ezrnew = Table[ezr[\([i, 1]\)] - ezr[\([i, 2]\)] + 0. , {i, 1, Length[ezr]}]; \[IndentingNewLine]posit = Table[Position[ezrnew[\([i]\)], y], {i, 1, Length[ezrnew]}]; \[IndentingNewLine]posnew = Table[ Table[{posit[\([j, i, 1]\)]}, {i, 1, Length[posit[\([j]\)]]}], {j, 1, Length[ezrnew]}]; \[IndentingNewLine]ls = \(-Apply[ Plus, Table[ Extract[ezrnew[\([j]\)], posnew[\([j]\)]], {j, 1, Length[ezrnew]}], 1]\); \[IndentingNewLine]rs = Table[Delete[ezrnew[\([j]\)], posnew[\([j]\)]], {j, 1, Length[ezrnew]}]\[IndentingNewLine]]\), ";", "\[IndentingNewLine]", "\[IndentingNewLine]", \(If[ Length[pos] > 0, \[IndentingNewLine]\[IndentingNewLine]clist = Table[c[m], {m, 1, ord}]; \[IndentingNewLine]yu1[x_, s_] = \(2\/\((2\ nn + 1)\)\) ss . \((k2[x, l, s] Sin[\[Pi]\ \ l] Apply[Plus, Map[\(1. \/#\) \((1 - \ \[IndentingNewLine]Cos[\[Pi]\ #\ \ ArcCos[1 - 2\ \(x - a\)\/\(b - a\)]/\[Pi]])\) Sin[\[Pi]\ #\ \ l] &, rr]])\); \[IndentingNewLine]yu2[x_, s_, m_] = Which[m \[Equal] s, 1, m < s, \((x - a)\)\^\(s - m\), m > \ s, 0]\/\(\((s - m)\)!\); \[IndentingNewLine]yu4[x_, s_] = \(yu1[x, ord - s]\)[\([1]\)]; \[IndentingNewLine]\ \[IndentingNewLine]yu3[x_, s_] = Table[yu2[x, ord - s, m], {m, 1, ord}] - Drop[yu1[x, ord - s], 1]; \[IndentingNewLine]l1 = ls /. {\(\(Derivative[k_]\)[y]\)[cc_] \[Rule] yu4[cc, k], y[cc_] \[Rule] yu4[cc, 0]}; \[IndentingNewLine]l2 = ls /. {\(\(Derivative[k_]\)[y]\)[cc_] \[Rule] yu3[cc, k], y[cc_] \[Rule] yu3[cc, 0]}; \[IndentingNewLine]eqss = Table[\((l1 + l2 . clist)\)[\([i]\)] \[Equal] rs[\([i]\)], {i, 1, ord}]; \[IndentingNewLine]res = Solve[eqss, Intersection[ Extract[eqss, Position[eqss, c[i_]]]]]; \[IndentingNewLine]ssnew[ x_, s_] = yu4[x, s] + clist . yu3[x, s], \[IndentingNewLine]clist = Prepend[Table[\(-c[i]\), {i, 1, ord}], 1]; \[IndentingNewLine]\ \ \ solution[x_, s_] := \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\/\((2\ nn + 1)\)\ clist . ss . \((k2[x, l, s] Sin[\[Pi]\ \ l] Apply[Plus, Map[\(1. \/#\) \((1 - \[IndentingNewLine]Cos[\ \[Pi]\ #\ \ ArcCos[1 - 2\ \(x - a\)\/\(b - a\)]/\[Pi]])\) Sin[\[Pi]\ #\ \ l] &, rr]])\) + \[Sum]\+\(m = 1\)\%s\((\(c[ m]\/\(\((s - m)\)!\)\) If[m \[Equal] s, 1, \((x - a)\)\^\(s - m\)])\); \ \[IndentingNewLine]ssnew[x_, s_] = solution[x, ord - s]; \[IndentingNewLine]eqss = ezr /. {\(\(Derivative[k_]\)[y]\)[cc_] \[Rule] ssnew[cc, k], y[cc_] \[Rule] ssnew[cc, 0]}; \[IndentingNewLine]res = Solve[eqss, Intersection[Extract[eqss, Position[eqss, c[i_]]]]]]\), ";", "\[IndentingNewLine]", "\[IndentingNewLine]", \(sol = Compile[{{x, _Real, 1}}, Evaluate[ssnew[x, 0] /. res // First]]\), ";", "\[IndentingNewLine]", \(sollist = sol[ll]\), ";", "\[IndentingNewLine]", \(err = 10\^11\), ";", "\[IndentingNewLine]", \(step = 1\), ";", "\[IndentingNewLine]", RowBox[{"Do", "[", RowBox[{ RowBox[{\(Which[\ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ step == 1, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ nndo = \ Which[\ \ \ 10\^10 <= err/pg, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5 \ + Round[1.5\ \ nn], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10\^\ 8 <= err/pg < 10\^10, \ \ \ \ \ \ \ \ 5 + \ Round[1.4\ \ nn], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10\^6 <= err/pg < 10\^8, \ \ \ \ \ \ \ \ \ \ 5 + \ Round[1.3\ \ nn], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10\^4 <= err/pg < 10\^6, \ \ \ \ \ \ \ \ \ \ 5 + Round[1.2\ nn], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10\^2 <= err/pg < 10\^4, \ \ \ \ \ \ \ \ \ \ 5 + Round[1.1\ \ nn], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ err/pg < 10\^2, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ nn + 6, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ err/pg < 10\ , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ nn + 6\ ], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ step == 2, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ nndo = \ Which[\ 10\^10 <= err/pg, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2 + Round[1.04\ \ nn], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10\^8 <= err/pg < 10\^10, \ \ \ \ \ \ \ \ 2 + Round[1.03\ \ nn], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 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= 1, step = 2]\), ";", "\[IndentingNewLine]", \(ldo = Table[\(2. \ s\)\/\(2\ nndo + 1\), {s, 1, nndo}]\), ";", "\[IndentingNewLine]", StyleBox[\(rrdo = Range[1, nndo]\), FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[";", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "\[IndentingNewLine]", RowBox[{"aux1do", "=", RowBox[{ StyleBox["Map", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["[", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"-> 0}], \(kkk[#, ldo]\ \ Sin[\[Pi]\ \ ldo] &, ldo\), "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"If", "[", RowBox[{\(nndo < sh\), ",", "\[IndentingNewLine]", " ", RowBox[{ StyleBox[\(popdo = Compile[{n, {x, _Real, 1}}, Evaluate[\(1. \/n\) \((1 - Cos[\[Pi]\ n\ \ ldo])\) Sin[\[Pi]\ n\ x]]]\), FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[";", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["\[IndentingNewLine]", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], StyleBox[" ", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(aux2do = \(2. \/\((2 nndo + 1. )\)\) Apply[Plus, Transpose[ Map[popdo[#, ldo] &, rrdo]], {1}]\), FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], "\[IndentingNewLine]", ",", "\[IndentingNewLine]", "\[IndentingNewLine]", " ", RowBox[{ StyleBox[\(bnew[x_, z_] = \(coef[nndo]\)[ Mod[z - x + 1, 2] - 1] + \(coef[nndo]\)[ Mod[z + x + 1, 2] - 1]\), FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[";", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, 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FontWeight->"Bold"], StyleBox["=", FontWeight->"Bold"], StyleBox[\(IdentityMatrix[Length[ldo]] + aux1do\ \ \ aux2do\), FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], StyleBox[";", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "\[IndentingNewLine]", \(ssdo = Transpose[LinearSolve[matdo, ffflist[ldo]]]\), ";", "\[IndentingNewLine]", \(If[ Length[pos] > 0, \[IndentingNewLine]yu1do[x_, s_] = \(2. \/\((2\ nndo + 1)\)\) ssdo . \((k2[x, ldo, s] Sin[\[Pi]\ \ ldo] Apply[Plus, Map[\(1. \/#\) \((1 - \ \[IndentingNewLine]Cos[\[Pi]\ #\ \ ArcCos[1 - 2. \ \(x - a\)\/\(b - a\)]/\[Pi]])\) Sin[\[Pi]\ #\ \ ldo] &, rrdo]])\); \[IndentingNewLine]yu2do[x_, s_, m_] = Which[m \[Equal] s, 1. , m < s, \((x - a)\)\^\(s - \ m\), m > s, 0]\/\(\((s - m)\)!\); \[IndentingNewLine]yu4do[x_, s_] = \(yu1do[x, ord - s]\)[\([1]\)]; \[IndentingNewLine]yu3do[x_, s_] = Table[yu2do[x, ord - s, m], {m, 1, ord}] - Drop[yu1do[x, ord - s], 1]; \[IndentingNewLine]l1do = ls /. {\(\(Derivative[k_]\)[y]\)[cc_] \[Rule] yu4do[cc, k], y[cc_] \[Rule] yu4do[cc, 0]}; \[IndentingNewLine]l2do = ls /. {\(\(Derivative[k_]\)[y]\)[cc_] \[Rule] yu3do[cc, k], y[cc_] \[Rule] yu3do[cc, 0]}; \[IndentingNewLine]eqssdo = Table[\((l1do + l2do . clist)\)[\([i]\)] \[Equal] rs[\([i]\)], {i, 1, ord}]; \[IndentingNewLine]resdo = Solve[eqssdo, Intersection[ Extract[eqssdo, Position[eqssdo, c[i_]]]]]; \[IndentingNewLine]ssnewdo[x_, s_] = yu4do[x, s] + clist . yu3do[x, s], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ solutiondo[x_, s_] = \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(2\/\((2\ nndo + 1)\)\) clist . ssdo . \((k2[x, ldo, s] Sin[\[Pi]\ \ ldo] Apply[Plus, Map[\(1. \/#\) \((1 - \ \[IndentingNewLine]Cos[\[Pi]\ #\ \ ArcCos[1 - 2\ \(x - a\)\/\(b - a\)]/\[Pi]])\) Sin[\[Pi]\ #\ \ ldo] &, rrdo]])\) + \[Sum]\+\(m = 1\)\%s\((\(c[ m]\/\(\((s - m)\)!\)\) If[m \[Equal] s, 1, \((x - a)\)\^\(s - m\)])\); \ \[IndentingNewLine]\[IndentingNewLine]ssnewdo[x_, s_] = solutiondo[x, ord - s]; \[IndentingNewLine]eqssdo = ezr /. {\(\(Derivative[k_]\)[y]\)[cc_] \[Rule] ssnewdo[cc, k], y[cc_] \[Rule] ssnewdo[cc, 0]}; \[IndentingNewLine]\[IndentingNewLine]\ resdo = Solve[eqssdo, Intersection[ Extract[eqssdo, Position[eqssdo, c[i_]]]]]]\), ";", "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", " ", \(soldo = Compile[{{x, _Real, 1}}, Evaluate[\((ssnewdo[x, 0] /. resdo // First)\)]]\), ";", "\[IndentingNewLine]", \(sollistdo = soldo[ll]\), ";", "\[IndentingNewLine]", " ", \(If[ sn \[Equal] 2, \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ errdo = \(\((\(2 \((b - a)\)\)\/\(2\ nnc + 1\))\)\^\(1/ 2\)\) \((Apply[Plus, wlist\ \ Abs[sollistdo - \ sollist]\^2\ iop])\)\^\(1/2\); \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ norm = \(\((\(2 \((b - a)\)\)\/\(2\ nnc + 1\))\)\^\(1/ 2\)\) \((\((Apply[Plus, \ wlist\ \ Abs[sollistdo]\^2\ \ iop])\)\^\(1/2\))\); \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ If[absrel \[Equal] 1, errdo = errdo/norm\ ]\[IndentingNewLine]\ \ , \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ errdo = Max[wlist\ \ Abs[ sollistdo - sollist]]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ norm = Max[wlist\ \ Abs[ sollistdo]]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ If[absrel \[Equal] 1, errdo = errdo/norm\ ]\[IndentingNewLine]\ \ \ \ ]\), ";", "\[IndentingNewLine]", \(If[ errdo \[LessEqual] 3\ pg, Break[]]\), ";", "\[IndentingNewLine]", \(If[ nndo > mxgp, \[IndentingNewLine]If[ absrel \[Equal] 1, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ If[sn \[Equal] 2, Message[NLinearDSolve::fcrelL2, nn, norm, errdo, wght], Message[NLinearDSolve::fcrelun, nn, norm, errdo, wght]], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ If[sn \[Equal] 2, Message[NLinearDSolve::fcabsL2, nn, norm, errdo, wght], Message[NLinearDSolve::fcabsun, nn, norm, errdo, wght]]]; \[IndentingNewLine]Break[]]\), ";", "\[IndentingNewLine]", \(nn = nndo\), ";", "\[IndentingNewLine]", \(sollist = sollistdo\), ";", "\[IndentingNewLine]", \(ssnew[x_, s_] = ssnewdo[x, s]\), ";", "\[IndentingNewLine]", "\[IndentingNewLine]", \(res = resdo\), ";", "\[IndentingNewLine]", "\[IndentingNewLine]", \(ss = ssdo\), ";", "\[IndentingNewLine]", \(l = ldo\), ";", "\[IndentingNewLine]", \(rr = rrdo\), ";", "\[IndentingNewLine]", \(err = errdo\)}], ",", " ", \({50000000}\)}], "]"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", " ", \(func[z_] = ssnew[z, 0] /. res // First\), ";", "\[IndentingNewLine]", " ", \(If[ irp, \[IndentingNewLine]\ \ \(Format[func] := SequenceForm["\", {a, b}, "\<,<>]\>"];\)\ \ \ \[IndentingNewLine], \ \[IndentingNewLine]\(newfunc = FunctionInterpolation[Evaluate[func[x]], {x, a, b}, Evaluate[optfint], InterpolationOrder \[Rule] 6, InterpolationPrecision \[Rule] Min[pg + 1, $MachinePrecision], InterpolationPoints \[Rule] 200, PrecisionGoal \[Rule] Min[pg + 1, $MachinePrecision], AccuracyGoal \[Rule] Min[pg + 1, $MachinePrecision]];\)\ \ \ \ \ \ \ \ \ \ \ \[IndentingNewLine]]\), ";", "\[IndentingNewLine]", " ", \(If[ pi === True, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ If[sn \[Equal] 2, \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Print["\\>", wght, "\<) of solution -> about \>", \ PaddedForm[norm, 3]]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ If[ absrel \[Equal] 1, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Print["\\>", wght, "\<)-> about \>", PaddedForm[\ errdo, 3]]\ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Print["\\>", wght, "\<) -> about \>", \ PaddedForm[\ errdo, 3]]\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]\ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Print["\\>", wght, "\<) of solution -> about \>", \ PaddedForm[norm, 3]]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ If[ absrel \[Equal] 1, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ Print["\ \\>", wght, "\<) -> about \>", PaddedForm[\ errdo, 3]]\ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ Print["\\>", wght, "\<) -> about \>", \ PaddedForm[\ errdo, 3]]\ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]\ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]\ \[IndentingNewLine]\ \ \ ]\), ";", "\[IndentingNewLine]", " ", \(If[ irp, \ \ Return[ func], \ \(Return[newfunc];\)\[IndentingNewLine]\ ]\), ";"}]}], "\[IndentingNewLine]", "]"}]}], ";"}], "\n", \(posit = Position[Options[NLinearDSolve], SolutionNorm \[Rule] Uniform];\), "\n", \(Options[NLinearDSolve] = Drop[Options[NLinearDSolve], posit // First];\), "\n", \(End[];\), "\n", \(EndPackage[];\)}], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Chasing from ", FontColor->GrayLevel[0.333333]], StyleBox["MathSource", FontSlant->"Italic", FontColor->GrayLevel[0.333333]], StyleBox[" (", FontColor->GrayLevel[0.333333]], StyleBox["Authors: Len Pismen, Leonid Braverman, Boris Rubinstein", FontSize->12], StyleBox[" ", FontSize->12, FontColor->GrayLevel[0.333333]], StyleBox[")", FontColor->GrayLevel[0.333333]] }], "Section"], Cell[CellGroupData[{ Cell[BoxData[{ \( (*\ \(:\)\(Name : Chasing`\)*) (*\ \(:\)\(Title : Numerical\ solution\ of\ 1 D\ boundary\ problems\)*) (*\ \(\(:\)\(Authors : Len\ Pismen\)\), Leonid\ Braverman, Boris\ Rubinstein*) (*\ \(:\)\(Summary : This\ package\ solves\ linear\ boundary\ value\ \(\(problems\)\(.\)\ \)\)*) (*\ \(:\)\(Context : Chasing`\)*) (*\ \(:\)\(Package\ \(Version : 3.0\)\)*) (*\ \(:\)\(Copyright : Copyright\ 1996 - 2000\)*) (*\ \(\(:\)\(History\ Version\ 1.0\ August\ 1995\)\), Version\ 2.0\ major\ revision\ of\ the\ method, Boris\ Rubinstein, October\ 1996, Version\ 3.0, some\ refinements, October\ 1998. *) (*\ \(\(:\)\(Keywords : boundary\ value\ problem\)\), eigenvalue, eigenfunction, chasing\ method*) (*\ \(\(:\)\(Source : T . Y . Na\)\), Computational\ Methods\ in\ Engineering\ Boundary\ Value\ Problems, Academic\ Press, NY, 1979. *) (*\ \(:\)\(Mathematica\ \(Version : 2.2\ and\ later\)\)*) BeginPackage["\"]\), "\n", \(\(Off[General::spell];\)\), "\n", \(\(\(Off[General::spell1];\)\(\n\) \)\), "\n", \(\(\(AccuracyGoal::usage = "\";\)\(\n\) \)\), "\n", \(\(\(PrecisionGoal::usage = "\";\)\(\n\) \)\), "\n", \(\(\(WorkingPrecision::usage = "\";\)\(\n\) \)\), "\n", \(\(MaxSteps::usage = "\";\)\n\ \[IndentingNewLine] (*Compiled::usage = "\"; \[IndentingNewLine]SolAccuracyGoal::usage = \ "\"; \ \[IndentingNewLine]SolWorkingPrecision::usage = "\"; \ \[IndentingNewLine]SolCompiled::usage = "\";*) \[IndentingNewLine]\), "\n", \(\(\(Method::usage = Method::usage <> "\< When Method is used as an option for the \ function Chasing it specifys the method of generation of the \ auxillary problem. The default value Method -> Nonlinear \ corresponds to the nonlinear problem. The option value Method -> \ Linear corresponds to the linear problem.\>";\)\(\n\) \)\), "\n", \(\(\(Linear::usage = "\";\)\(\n\) \)\), "\n", \(\(\(Nonlinear::usage = "\";\)\ \(\n\) \)\), "\n", \(\(\(Chasing::usage = "\";\)\(\n\) \)\), "\n", \(\(\(Begin["\<`Private`\>"]\)\(\n\) \)\), "\n", \({f, g}; (*The\ internal\ names\ of\ the\ indexed\ functions\ used\ for\ \ rewriting\ of\ the\ original\ system\ and\ for\ preparation\ of\ the\ \ auxillary\ \(\(system\)\(.\)\)*) Off[General::spell, General::spell1];\), "\n", \(\(\(Chasing::pdif = "\";\)\(\n\) \)\), "\n", \(\(\(Chasing::vform = "\";\)\(\n\ \) \)\), "\n", \(\(\(Chasing::vars = "\";\)\(\n\) \)\), "\n", \(\(\(Chasing::nonlin = "\";\)\(\n\) \)\), "\n", \(\(\(Chasing::ndinf = "\";\)\ \(\n\) \)\), "\n", \(\(\(Chasing::liminf = "\";\)\(\n\) \)\), "\n", \(\(\(Chasing::nend = "\";\)\(\n\) \)\), "\n", \(\(\(Chasing::eqnan = "\";\)\(\n\) \)\), "\n", \(\(\(Chasing::bcnan = "\";\)\(\n\) \)\), "\n", \(\(\(Chasing::nbcs = "\";\)\(\n\) \)\), "\n", \(\(\(Chasing::ndnl = "\";\)\(\n\) \)\), "\n", \(\(\(Chasing::bmet = "\ `1` is not \ valid.\>";\)\(\n\) \)\), "\n", \(\(\(Chasing::endbc = "\";\)\(\n\) \)\), "\n", \(\(\($Partder[ equations_] := \((Message[Chasing::pdif, equations]; {$Failed, $Failed, $Failed})\);\)\(\n\) \)\), "\n", \(\(\($NotValidForm := \((Message[ Chasing::vform]; {$Failed, $Failed, $Failed})\);\)\(\n\) \)\), "\n", \(\(\($NotAllVars[ vars_List] := \((Message[Chasing::vars, vars]; $Failed)\);\)\(\n\) \)\), "\n", \(\(\($InfLimits0 := \((Message[Chasing::ndinf]; $Failed)\);\)\(\n\) \)\), "\n", \(\(\($InfLimits := \((Message[Chasing::liminf]; $Failed)\);\)\(\n\) \)\), "\n", \(\(\($NonLinearProblem := \((Message[ Chasing::nonlin]; $Failed)\);\)\(\n\) \)\), "\n", \(\(\($NoEndIntegration := Message[Chasing::nend];\)\(\n\) \)\), "\n", \(\(\($NonNumericEquation := \((Message[ Chasing::eqnan]; $Failed)\);\)\(\n\) \)\), "\n", \(\(\($NonNumericBC := \((Message[Chasing::bcnan]; $Failed)\);\)\(\n\) \)\), "\n", \(\(\($BadBCNumber := \((Message[Chasing::nbcs]; $Failed)\);\)\(\n\) \)\), "\n", \(\(\($ComplexPts := \((Message[Chasing::ndnl]; $Failed)\);\)\(\n\) \)\), "\n", \(\(\($BadMethod[ meth_] := \((Message[Chasing::bmet, meth]; $Failed)\);\)\(\n\) \)\), "\n", \(\(Clear[MinMaxDer];\)\), "\n", \(\(\(MinMaxDer[vars_List, eqns_List] := Module[{varpos, derpos, varsall, newvars1, newvarsall}, varpos = Map[Union[Cases[eqns, #[_] \[Rule] 0, Infinity]] &, vars]; \[IndentingNewLine]derpos = \((\(Cases[ eqns, \(\(Derivative[n_Integer]\)[#]\)[_] \[Rule] n, Infinity] &\) /@ vars)\); \[IndentingNewLine]varsall = Transpose[{vars, MapThread[ Join[#, #2] &, {varpos, derpos}]}]; \[IndentingNewLine]{newvars1, newvarsall} = Transpose[ DeleteCases[ varsall, {_, {}}]]; \[IndentingNewLine]Transpose[{newvars1, Map[Max, newvarsall]}]]\)\(\n\)\(\[IndentingNewLine]\) \)\), "\n", \(\(Clear[TimeDerivSelect];\)\), "\n", \(\(\(TimeDerivSelect[eq_Equal, timevar_] := Module[{eqq, timeord, timeordder}, eqq = eq /. Equal \[Rule] Subtract; \[IndentingNewLine]timeord = Cases[Expand[eqq] + Null, _. \ \(\(Derivative[n__]\)[_]\)[args__] /; Apply[Plus, Cases[Transpose[{{n}, {args}}], {l_Integer, timevar | timevar[_Integer]} \[Rule] l]] > 0]; \[IndentingNewLine]timeordder = Apply[Plus, timeord]; \[IndentingNewLine]timeordder \[Equal] timeordder - eqq]\)\(\n\) \)\), "\n", \(\(TimeDerivSelect[eqs : {__Equal}, timevar_] := Map[TimeDerivSelect[#, timevar] &, eqs];\)\n\[IndentingNewLine] (*Transformation\ of\ a\ system\ of\ \ ordinary\ differential\ equations\ of\ any\ order\ to\ a\ system\ of\ ODE' s\ of\ a\ first\ order\ by\ introducing\ auxiliary\ dependent\ \ variables . Boundary\ conditions\ are\ also\ \(\(transformed\)\(.\)\)*) \ \[IndentingNewLine]\), "\n", \(\(Off[Solve::svars];\)\), "\n", \(\(Clear[TransformSystem];\)\), "\n", \(\(\(TransformSystem[equations : {__Equal}, vars_List, t_] := Module[{bc, eq, maxeq, newvars, lll, mapder, neweqs, rule0, rules, eqs4, eqs, newbc, eqss, fs, rulefs, eqss1, eqss2}, If[\(! FreeQ[ equations, \(\(Derivative[__]\)[Alternatives @@ vars]\)[ B__ /; \((MemberQ[{B}, t] && Length[{B}] > 1)\)], Infinity]\), Return[$Partder[equations]]]; \[IndentingNewLine]bc = Cases[equations, a_ /; FreeQ[a, t]]; \[IndentingNewLine]eq = Complement[equations, bc]; \[IndentingNewLine]If[\(! FreeQ[ eq, \((\((Alternatives @@ vars)\)[ aa_] | \(\(Derivative[_]\)[Alternatives @@ vars]\)[ aa_])\) /; UnsameQ[aa, t]]\), Return[$NotValidForm]]; \[IndentingNewLine] (*Construction\ of\ \ the\ equations*) maxeq = MinMaxDer[vars, eq]; \[IndentingNewLine]newvars = MapThread[Table[\(#1[i]\)[t], {i, 0, #2 - 1}] &, Transpose[maxeq]]; \[IndentingNewLine]lll = DeleteCases[ Flatten[\(Partition[#, 2, 1] &\) /@ newvars, 1], {}]; \[IndentingNewLine]mapder = \(MapAt[D[#, t] &, #, 1] &\) /@ lll; \[IndentingNewLine]neweqs = \(\((# /. List \[Rule] Equal)\) &\) /@ mapder; \[IndentingNewLine]rule0 = \(\((#[aa_] \[Rule] \(#[0]\)[ aa])\) &\) /@ vars; \[IndentingNewLine]rulemaxder = Map[\((\(\(Derivative[Last[#]]\)[First[#]]\)[t] \[Rule] D[\(\(First[#]\)[Last[#] - 1]\)[t], t])\) &, maxeq]; \[IndentingNewLine]rules = \(\(\(Derivative[ nn_Integer]\)[#]\)[aa_] \[Rule] \(#[nn]\)[aa] &\) /@ vars; \[IndentingNewLine]eqs4 = \(\(eq /. rule0\) /. rulemaxder\) /. rules; \[IndentingNewLine]eqs = TimeDerivSelect[Join[neweqs, eqs4], t]; \[IndentingNewLine]newbc = \(bc /. rule0\) /. rules; \[IndentingNewLine]eqss = Join[eqs, newbc]; \[IndentingNewLine]fs = Array[f, Apply[Plus, Map[Last, maxeq]]]; \[IndentingNewLine]rulefs = Thread[Map[Head, Flatten[newvars]] \[Rule] fs]; \[IndentingNewLine]eqss1 = Map[Expand, eqs /. rulefs, {2}]; \[IndentingNewLine]eqss2 = Sort[eqss1, First[Cases[#, \(\(Derivative[1]\)[_[n_Integer]]\)[_] \[Rule] n, Infinity, 1]] < First[Cases[#2, \(\(Derivative[1]\)[_[ n_Integer]]\)[_] \[Rule] n, Infinity, 1]] &]; \[IndentingNewLine]{Join[eqss2, newbc /. rulefs], fs, rulefs}] /; FreeQ[Names["\"], Apply[Alternatives, Map[ToString, vars]]]\)\(\n\) \)\), "\n", \(\(Clear[SelectBC];\)\), "\n", \(\(SelectBC[bc_List, boundPoints_List] := Map[Cases[bc, a_ /; \(! FreeQ[a, _. \ \(_[_Integer]\)[#]]\)] &, boundPoints];\)\n\[IndentingNewLine]\), "\n", \(Clear[MakeAuxSystem]\), "\n", \(\(\(MakeAuxSystem[actbc_, fs_List, funcs_List, matr_?MatrixQ, var_, m_Integer] := Module[{initc, pos, funcs1, add}, initc = Join[ Map[D[actbc /. \(f[n_]\)[_] \[Rule] f[n], #] &, fs], {actbc /. \(f[n_]\)[_] \[Rule] 0}]; \[IndentingNewLine]pos = Part[Position[Abs[Sign[initc]], 1, {1}, 1], 1, 1]; \[IndentingNewLine]funcs1 = ReplacePart[funcs, m\ Part[initc, pos] + \((1 - m)\)\ Part[funcs, pos], pos]; \[IndentingNewLine]add = m\ Part[matr, pos] . Drop[funcs, \(-1\)]/ Part[initc, pos] /. {Part[funcs, pos] \[Rule] 1}; \[IndentingNewLine]DeleteCases[ Thread[\((D[funcs, var] + matr . Drop[funcs1, \(-1\)] - add\ funcs1)\) \[Equal] 0], True]]\)\(\n\) \)\), "\n", \(Clear[Constructor]\), "\n", \(\(\(Constructor[actbc_, fs_List, gs_List, var_] := Module[{pt}, pt = First[ Cases[{actbc}, \(f[_]\)[a_] \[Rule] a, Infinity, 1]]; \[IndentingNewLine]DeleteCases[ Thread[\((gs /. var \[Rule] pt)\) \[Equal] Join[Map[D[actbc /. \(f[n_]\)[_] \[Rule] f[n], #] &, fs], {actbc /. \(f[n_]\)[_] \[Rule] 0}]], True]]\)\(\n\)\(\[IndentingNewLine]\) \)\), "\n", \(\(Clear[EndPoint];\)\), "\n", \(\(\(EndPoint[syst : {{__Equal} .. }, funcs : {__List}, {var_, pt_, xmax_}, opts___?OptionQ] := MapThread[ EndPoint[#, #2, {var, pt, xmax}, opts] &, {syst, funcs}]\)\(\n\)\(\[IndentingNewLine]\)\( (*This\ old\ version\ is\ \ disabled\ EndPoint[syst : {__Equal}, funcs_?ListQ, {var_, pt_, xmax_}, opts___?OptionQ] := First[NDSolve[syst, funcs, {var, pt, xmax}, opts, WorkingPrecision \[Rule] $MachinePrecision, MaxSteps \[Rule] 2500, AccuracyGoal \[Rule] Automatic, PrecisionGoal \[Rule] Automatic] /. var \[Rule] xmax]*) \)\(\[IndentingNewLine]\)\( (*This\ part\ is\ added\ 12\ \ Oct\ 1998. \ It\ solves\ the\ problem\ of\ divergence\ of\ the\ solution\ \ inside\ the\ interval\ \((pt, xmax)\) . It\ checks\ whether\ the\ integration\ at\ this\ interval\ is\ \ finished\ successfully, and\ if\ not\ it\ switches\ to\ integration\ along\ other\ contour\ \ \((function\ MakeCurve)\) . This\ integration\ consists\ of\ three\ steps\ \(\((functions\ \ Step1, 2, 3)\)\(.\)\)*) \)\(\[IndentingNewLine]\) \)\), "\n", \(\(\(EndPoint[syst : {__Equal}, funcs_?ListQ, {var_, pt_, xmax_}, opts___?OptionQ] := Module[{s0, rs}, s0 = Step0[syst, funcs, {var, pt, xmax}, opts]; \[IndentingNewLine]If[ s0 === $Failed, rs = Abs[xmax - pt]*0.1; \[IndentingNewLine]MakeCurve[syst, funcs, {var, pt, xmax}, rs, opts], s0]]\)\(\n\)\(\[IndentingNewLine]\) \)\), "\n", \(\(\(Clear[Step0, Step1, Step2, Step3, MakeSyst, MakeCurve]\)\(\n\) \)\), "\n", \(\(\(Step0[syst : {__Equal}, funcs_?VectorQ, {var_, xmin_, xmax_}, opts___?OptionQ] := Module[{sol, endpt}, Off[NDSolve::ndsz]; \[IndentingNewLine]sol = NDSolve[syst, funcs, {var, xmin, xmax}, opts, WorkingPrecision \[Rule] $MachinePrecision, MaxSteps \[Rule] 2500, AccuracyGoal \[Rule] Automatic, PrecisionGoal \[Rule] Automatic]; \[IndentingNewLine]endpt = Part[sol, 1, 1, 2, 0, 1, 1, 2]; \[IndentingNewLine]On[ NDSolve::ndsz]; \[IndentingNewLine]If[ Chop[endpt - xmax] \[Equal] 0, First[sol /. var \[Rule] xmax], $Failed]]\)\(\n\)\(\[IndentingNewLine]\)\( (*In\ order\ \ to\ make\ integration\ at\ steps\ 1 - 3\ we\ need\ to\ change\ the\ direction\ of\ integration\ at\ step\ \ 1\ x \[Rule] xmin + I\ x; \((x = 0\ to\ rectside)\)\ at\ step\ 2\ x \[Rule] \(I\ rectside + x\ \((x = xmin\ to\ xmax)\)\ at\ step\ 3\ x \[Rule] xmax + I\ x\ \((x = rectside\ to\ 0)\)\)*) \)\(\[IndentingNewLine]\) \)\), "\n", \(\(\(MakeSyst[syst : {__Equal}, funcs_?VectorQ, {var_, xmin_, xmax_}, rectside_, step_Integer, prevres_List] := Module[{eqs, bc, heads, rvar, lims, derfact, initpt, rin, rout}, eqs = Select[syst, \(! FreeQ[#, var]\) &]; \[IndentingNewLine]bc = Complement[syst, eqs]; \[IndentingNewLine]heads = Map[Head, funcs]; \[IndentingNewLine]{rvar, lims, derfact, initpt} = Switch[step, 1, {var \[Rule] xmin + I\ var, {var, 0, rectside}, \(-I\), 0}, 2, {var \[Rule] I\ rectside + var, {var, xmin, xmax}, 1, xmin}, 3, {var \[Rule] xmax + I\ var, {var, rectside, 0}, \(-I\), rectside}]; \[IndentingNewLine]rin = Map[{#[var] \[Rule] #, \(#'\)[var] \[Rule] derfact*#'} &, heads] // Flatten; \[IndentingNewLine]rout = Map[{# \[Rule] #[var], #' \[Rule] \(#'\)[var]} &, heads] // Flatten; \[IndentingNewLine]eqs = \(\(eqs /. rin\) /. rvar\) /. rout; \[IndentingNewLine]bc = Map[\(Head[First[#]]\)[initpt] \[Equal] Last[#] &, prevres]; \[IndentingNewLine]Join[eqs, bc]]\)\(\n\) \)\), "\n", \(\(\(Step1[syst : {__Equal}, funcs_?VectorQ, {var_, xmin_, xmax_}, rectside_, opts___?OptionQ] := Module[{syst1, sol, endpt}, Off[NDSolve::ndsz]; \[IndentingNewLine]syst1 = MakeSyst[syst, funcs, {var, xmin, xmax}, rectside, 1, Select[syst, FreeQ[#, var] &]]; \[IndentingNewLine]sol = NDSolve[syst1, funcs, {var, 0, rectside}, opts, WorkingPrecision \[Rule] $MachinePrecision, MaxSteps \[Rule] 2500, AccuracyGoal \[Rule] Automatic, PrecisionGoal \[Rule] Automatic]; \[IndentingNewLine]endpt = Part[sol, 1, 1, 2, 0, 1, 1, 2]; \[IndentingNewLine]On[ NDSolve::ndsz]; \[IndentingNewLine]If[ Chop[endpt - rectside] \[Equal] 0, First[sol /. var \[Rule] rectside] // Chop, $Failed]]\)\(\n\) \)\), "\n", \(\(\(Step2[syst : {__Equal}, funcs_?VectorQ, {var_, xmin_, xmax_}, rectside_, outstep1_, opts___?OptionQ] := Module[{syst2, sol, endpt}, If[outstep1 === $Failed, Return[$Failed]]; \[IndentingNewLine]Off[ NDSolve::ndsz]; \[IndentingNewLine]syst2 = MakeSyst[syst, funcs, {var, xmin, xmax}, rectside, 2, outstep1]; \[IndentingNewLine]sol = NDSolve[syst2, funcs, {var, xmin, xmax}, opts, WorkingPrecision \[Rule] $MachinePrecision, MaxSteps \[Rule] 2500, AccuracyGoal \[Rule] Automatic, PrecisionGoal \[Rule] Automatic]; \[IndentingNewLine]endpt = Part[sol, 1, 1, 2, 0, 1, 1, 2]; \[IndentingNewLine]On[ NDSolve::ndsz]; \[IndentingNewLine]If[ Chop[endpt - xmax] \[Equal] 0, First[sol /. var \[Rule] xmax] // Chop, $Failed]]\)\(\n\) \)\), "\n", \(\(\(Step3[syst : {__Equal}, funcs_?VectorQ, {var_, xmin_, xmax_}, rectside_, outstep2_, opts___?OptionQ] := Module[{syst3, sol, endpt}, If[outstep2 === $Failed, Return[$Failed]]; \[IndentingNewLine]Off[ NDSolve::ndsz]; \[IndentingNewLine]syst3 = MakeSyst[syst, funcs, {var, xmin, xmax}, rectside, 3, outstep2]; \[IndentingNewLine]sol = NDSolve[syst3, funcs, {var, rectside, 0}, opts, WorkingPrecision \[Rule] $MachinePrecision, MaxSteps \[Rule] 2500, AccuracyGoal \[Rule] Automatic, PrecisionGoal \[Rule] Automatic]; \[IndentingNewLine]endpt = Part[sol, 1, 1, 2, 0, 1, 1, 1]; \[IndentingNewLine]On[ NDSolve::ndsz]; \[IndentingNewLine]If[Chop[endpt] \[Equal] 0, Thread[\((Map[First, First[sol]] /. var \[Rule] xmax)\) \[Rule] \((Map[Last, First[sol]] /. var \[Rule] 0)\)] // Chop, $Failed]]\)\(\n\) \)\), "\n", \(\(\(MakeCurve[syst : {__Equal}, funcs_?VectorQ, {var_, xmin_, xmax_}, rectside_, opts___?OptionQ] := Module[{outstep1, outstep2}, outstep1 = Step1[syst, funcs, {var, xmin, xmax}, rectside, opts]; \[IndentingNewLine]outstep2 = Step2[syst, funcs, {var, xmin, xmax}, rectside, outstep1, opts]; \[IndentingNewLine]Step3[syst, funcs, {var, xmin, xmax}, rectside, outstep2, opts]]\)\(\n\) \)\), "\n", \(\(\(MakeMainSystem[groupauxsyst_List, groupgs_List, boundpts_List, gs_List, lsortbounds_List, var_, xmax_, newfuncs_List, {wp_, maxstep_, accgoal_, precgoal_, acc_}] := Module[{finalgsrules, gmatr, mainmatr, mainfreeterm}, finalgsrules = Flatten[MapThread[ EndPoint[#, #2, {var, #3, xmax}, WorkingPrecision \[Rule] wp, MaxSteps \[Rule] maxstep, AccuracyGoal \[Rule] accgoal, PrecisionGoal \[Rule] precgoal] &, {groupauxsyst, groupgs, boundpts}]]; \[IndentingNewLine]gmatr = \(gs /. var \[Rule] xmax\) /. finalgsrules; \[IndentingNewLine]mainmatr = Join[Map[Drop[#, \(-1\)] &, gmatr], Outer[D, \(lsortbounds /. Equal \[Rule] Subtract\) /. \(f[n_]\)[_] \[Rule] f[n], newfuncs]]; \[IndentingNewLine]mainfreeterm = Join[Map[Last, gmatr], \(lsortbounds /. Equal \[Rule] Subtract\) /. \(f[n_]\)[_] \[Rule] 0]; \[IndentingNewLine]Chop[{mainmatr, mainfreeterm}, acc]]\)\(\n\) \)\), "\n", \(Clear[Resolvent]\), "\n", \(\(\(Resolvent[matr_?MatrixQ] := Module[{es, ces, eval, efun, cefun}, es = Chop[Eigensystem[matr]]; \[IndentingNewLine]ces = Chop[Eigensystem[Transpose[matr]]]; \[IndentingNewLine]{eval, efun} = Simplify[ Transpose[ DeleteCases[ Transpose[es], {0, _List}]]]; \[IndentingNewLine]cefun = Simplify[ Last[Transpose[ DeleteCases[ Transpose[ces], {0, _List}]]]]; \[IndentingNewLine]Apply[ Plus, MapThread[\(Outer[Times, #, #2]/# . #2\)/#3 &, {efun, cefun, eval}]]]\)\(\n\)\(\[IndentingNewLine]\) \)\), "\n", \(Clear[conjugate]\), "\n", \(\(conjugate[expr_] := expr /. Complex[a_, b_] \[Rule] Complex[a, \(-b\)];\)\n\[IndentingNewLine]\), "\n", \(\(Clear[Chasing];\)\), "\n", \(\(\(Options[Chasing] = {Tolerance \[Rule] 10, WorkingPrecision \[Rule] $MachinePrecision, MaxSteps \[Rule] 1000, AccuracyGoal \[Rule] Automatic, PrecisionGoal \[Rule] Automatic, Method \[Rule] Nonlinear};\)\(\n\) \)\), "\n", \(\(\(Chasing[eqns_List, func : \((_Symbol | _Symbol[var_])\), {var_, xmin_?NumericQ, xmax_?NumericQ}, opts___?OptionQ] := Chasing[eqns, {func}, {var, xmin, xmax}, {0}, opts]\)\(\n\) \)\), "\n", \(\(\(Chasing[eqns_List, funcs_List, {var_, xmin_?NumericQ, xmax_?NumericQ}, opts___?OptionQ] := Chasing[eqns, funcs, {var, xmin, xmax}, {0}, opts]\)\(\n\) \)\), "\n", \(\(\(Chasing[eqns_List, func : \((_Symbol | _Symbol[var_])\), {var_, xmin_?NumericQ, xmax_?NumericQ}, lamset : {_Symbol, _?NumericQ, _?NumericQ}, opts___?OptionQ] := Chasing[eqns, {func}, {var, xmin, xmax}, {lamset}, opts]\)\(\n\)\(\[IndentingNewLine]\) \)\), "\n", \(\(\(Chasing[eqns_List, funcs_List, {var_, xmin_?NumericQ, xmax_?NumericQ}, lamset : {_Symbol, _?NumericQ, _?NumericQ}, opts___?OptionQ] := Chasing[eqns, funcs, {var, xmin, xmax}, {lamset}, opts]\)\(\n\)\(\[IndentingNewLine]\) \)\), "\n", \(\(\(Chasing[eqns_List, funcs_List, {var_, xmin_?NumericQ, xmax_?NumericQ}, lamset : \(({{_Symbol, _?NumericQ, _?NumericQ} .. } | {0})\), opts___?OptionQ] := Module[{func1, eqns1, newfuncs, rulefs, needrulefs, dim, eqs, bounds, boundpts, sortbounds, eqs0, testnum, eqnumtest, lamtestrule, bcnumtest, boundlens, leadcoefs, jacob, jacobbc, tol, method, meth, wp, maxstep, accgoal, precgoal, eigenflag, lamrule, lamset1, lam, acc, freeterm, fullmatr, gs1, gs, auxsyst, groupauxsyst, groupgs, finalgsrules, gmatr, mainmatr, mainfreeterm, endsol, newfs, initcond, fsf, fsb, ic, fdrop, finalsol}, func1 = Map[If[FreeQ[#, var], #, Head[#]] &, funcs]; \[IndentingNewLine]{eqns1, newfuncs, rulefs} = TransformSystem[eqns, func1, var]; \[IndentingNewLine]If[ eqns1 === $Failed, Return[$Failed]]; \[IndentingNewLine]needrulefs = Map[Reverse, Cases[rulefs, Literal[_[0] \[Rule] _]]]; \[IndentingNewLine]dim = Length[newfuncs] + 1; \[IndentingNewLine]eqs0 = Cases[eqns1, a_ /; \(! FreeQ[a, var]\)]; \[IndentingNewLine]bounds = Complement[eqns1, eqs0]; \[IndentingNewLine]leadcoefs = Map[First, eqs0] /. \(f[_Integer]'\)[var] \[Rule] 1; \[IndentingNewLine]eqs = Thread[Map[First, eqs0]/leadcoefs \[Equal] Map[Last, eqs0]/leadcoefs]; \[IndentingNewLine]If[ Abs[xmin] === Infinity || Abs[xmax] === Infinity, Return[$InfLimits0]]; \[IndentingNewLine] (*We\ need\ check\ for\ \ numerical\ value\ of\ the\ equations\ coefficients*) testnum = xmin + 0.1*\((xmax - xmin)\)*I; \[IndentingNewLine]lamtestrule = If[lamset =!= {0}, Map[ Part[#, 1] \[Rule] \((Part[#, 3] - Part[#, 2])\)*0.1*I + Part[#, 2] &, lamset], {}]; \[IndentingNewLine]eqnumtest = N[\(\(Map[Last, eqs] /. \(f[_Integer]\)[var] \[Rule] 1\) /. var \[Rule] testnum\) /. lamtestrule]; \[IndentingNewLine]If[ Not[Apply[And, Map[NumberQ, eqnumtest]]], Return[$NonNumericEquation]]; \[IndentingNewLine]boundpts = Union[Cases[bounds, \(f[_Integer]\)[a_] \[Rule] a, Infinity]]; \[IndentingNewLine]If[ Not[Apply[And, Map[NumberQ, N[boundpts]]]], Return[$NonNumericBC]]; \[IndentingNewLine]If[\(! FreeQ[ N[Join[boundpts, {xmin, xmax}]], _Complex]\), Return[$ComplexPts]]; \[IndentingNewLine]xmaxbc = Max[boundpts]; \[IndentingNewLine]If[Abs[xmaxbc] === Infinity, Return[$InfLimits]]; \[IndentingNewLine]sortbounds = SelectBC[bounds, boundpts]; \[IndentingNewLine]bcnumtest = N[\(\(Flatten[sortbounds] /. Equal \[Rule] Subtract\) /. \(f[_Integer]\)[_] \[Rule] testnum\) /. lamtestrule]; \[IndentingNewLine]If[ Not[Apply[And, Map[NumberQ, bcnumtest]]], Return[$NonNumericBC]]; \[IndentingNewLine]boundlens = Map[Length, sortbounds]; \[IndentingNewLine]If[ Apply[Plus, boundlens] \[NotEqual] Length[eqs], Return[$BadBCNumber]]; \[IndentingNewLine]jacob = Outer[D, Map[Last, eqs] /. \(f[n_Integer]\)[_] \[Rule] f[n], newfuncs] // Transpose; \[IndentingNewLine]jacobbc = Outer[D, \(sortbounds /. Equal \[Rule] Subtract\) /. \(f[n_Integer]\)[_] \[Rule] f[n], newfuncs]; \[IndentingNewLine]If[\(! FreeQ[jacob, Alternatives @@ newfuncs]\) || \(! FreeQ[jacobbc, Alternatives @@ newfuncs]\), Return[$NonLinearProblem]]; \[IndentingNewLine]method = \(\(Method /. \ {opts}\) /. Options[Chasing]\) /. {Linear \[Rule] 0, Nonlinear \[Rule] 1}; \[IndentingNewLine]If[ method =!= 0 && method =!= 1, Return[$BadMethod[ method]]]; \[IndentingNewLine]tol = \(Tolerance /. {opts}\) /. Options[Chasing]; \[IndentingNewLine]acc = 10^\((\(-tol\))\); \[IndentingNewLine]{wp, maxstep, accgoal, precgoal} = \({WorkingPrecision, MaxSteps, AccuracyGoal, PrecisionGoal} /. {opts}\) /. Options[Chasing]; \[IndentingNewLine]freeterm = Map[Last, eqs] /. \(f[n_Integer]\)[_] \[Rule] 0; \[IndentingNewLine]eigenflag = If[\((MatchQ[freeterm, {\((0)\) .. }])\) && lamset =!= {0}, Length[lamset], 0]; \[IndentingNewLine]fullmatr = Join[jacob, {freeterm}]; \[IndentingNewLine]gs1 = Array[\(g[#, #2]\)[var] &, {dim - Last[boundlens] - 1, dim}]; \[IndentingNewLine]gs = If[MatchQ[freeterm, {\((0)\) .. }] && MatchQ[Map[Last, bounds], {\((0)\) .. }], Map[ReplacePart[#, 0, \(-1\)] &, gs1], gs1]; \[IndentingNewLine]auxsyst = MapThread[ Join[MakeAuxSystem[#, newfuncs, #2, fullmatr, var, method], Constructor[#, newfuncs, #2, var]] &, {Flatten[ Drop[sortbounds, \(-1\)]] /. Equal \[Rule] Subtract, gs}] // Chop; \[IndentingNewLine]groupauxsyst = First[Fold[{Append[First[#], Take[Last[#], #2]], Drop[Last[#], #2]} &, {{}, auxsyst}, Drop[boundlens, \(-1\)]]]; \[IndentingNewLine]groupgs = First[Fold[{Append[First[#], Take[Last[#], #2]], Drop[Last[#], #2]} &, {{}, Map[DeleteCases[#, 0] &, gs]}, Drop[boundlens, \(-1\)]]]; \[IndentingNewLine]lamrule = If[eigenflag \[Equal] 0, {}, ttts = Map[ToExpression[ToString[ttt] <> ToString[#]] &, Range[eigenflag]]; \[IndentingNewLine]lamset1 = MapThread[ ReplacePart[#, #2, 1] &, {lamset, ttts}]; \[IndentingNewLine]lams = Map[First, lamset]; \[IndentingNewLine]Clear[ foo]; \[IndentingNewLine]foo[gh_] := Module[{det}, det = Det[ First[\((Apply[ Function, {mmm[groupauxsyst, groupgs, Drop[boundpts, \(-1\)], gs, Last[sortbounds], var, xmaxbc, newfuncs, {wp, maxstep, accgoal, precgoal, acc}]} /. Thread[ lams \[Rule] Array[Slot, eigenflag]]] /. mmm \[Rule] MakeMainSystem)\) @@ gh]]; \[IndentingNewLine]If[eigenflag \[Equal] 1, Re[det], {Re[det], Im[det]}]]; \[IndentingNewLine]\(FindRoot[foo[#], ##2, AccuracyGoal \[Rule] 15] &\) @@ Prepend[lamset1, ttts]] /. Thread[ttts \[Rule] lams]; \[IndentingNewLine]{mainmatr, mainfreeterm} = MakeMainSystem[groupauxsyst /. lamrule, groupgs, Drop[boundpts, \(-1\)], gs, Last[sortbounds] /. lamrule, var, xmaxbc, newfuncs, {wp, maxstep, accgoal, precgoal, acc}]; \[IndentingNewLine]endsol = Which[MatchQ[mainfreeterm, {\((0)\) .. }] && Chop[Det[mainmatr], acc] \[Equal] 0, Message[Chasing::endbc]; \[IndentingNewLine]Chop[ First[NullSpace[mainmatr, ZeroTest \[Rule] \((Chop[#, acc] \[Equal] 0 &)\)]], acc], MatchQ[mainfreeterm, {\((0)\) .. }] && Chop[Det[mainmatr], acc] \[NotEqual] 0, Message[Chasing::endbc]; \[IndentingNewLine]mainfreeterm, (*Not[ MatchQ[mainfreeterm, {\((0)\) .. }]] && Chop[Det[mainmatr], acc] \[Equal] 0, Message[Chasing::endbc]; \[IndentingNewLine]\(-Resolvent[ mainmatr] . mainfreeterm\), *) True, Off[LinearSolve::nosol]; \[IndentingNewLine]\(-LinearSolve[ mainmatr, mainfreeterm]\)]; \[IndentingNewLine]If[ OffQ[LinearSolve::nosol], On[LinearSolve::nosol]]; \[IndentingNewLine]If[ Head[endsol] =!= List, Return[{}]]; \[IndentingNewLine]newfs = Through[newfuncs[var]]; \[IndentingNewLine]initcond = Thread[\((newfs /. var \[Rule] xmaxbc)\) \[Equal] endsol]; \[IndentingNewLine] (*The\ backward\ integration\ \ process\ depends\ on\ the\ relation\ between\ xmax\ and\ xmaxbc . There\ are\ three\ cases . Case\ 1. \ \((xmax \[Equal] xmaxbc)\) \[Equal] > backward\ integration\ is\ made\ from\ xmaxbc\ to\ xmin . Case\ 2. \ \((xmax > xmaxbc)\) \[Equal] > Two\ function\ are\ produced - one\ is\ by\ backward\ integration\ from\ xmaxbc\ to\ xmin, and\ another\ by\ forward\ integration\ from\ xmaxbc\ to\ xmax . Then\ these\ functions\ are\ merged . Case\ 3. \ \((xmax < xmaxbc)\) \[Equal] > There\ are\ two\ backward\ integration\ \(steps : first\ is\ from\ xmaxbc\ to\ xmax\); the\ results\ are\ used\ as\ initial\ conditions\ for\ the\ second\ \ step - from\ xmax\ to\ \(\(xmin\)\(.\)\)*) finalsol = Which[Chop[xmax - xmaxbc] \[Equal] 0, (*Case\ 1*) First[ NDSolve[Join[eqs /. lamrule, initcond], newfs, {var, xmax, xmin}, WorkingPrecision \[Rule] wp, MaxSteps \[Rule] maxstep, AccuracyGoal \[Rule] accgoal, PrecisionGoal \[Rule] precgoal]], xmax > xmaxbc, (*Case\ 2*) fsb = First[NDSolve[Join[eqs /. lamrule, initcond], newfs, {var, xmaxbc, xmin}, WorkingPrecision \[Rule] wp, MaxSteps \[Rule] maxstep, AccuracyGoal \[Rule] accgoal, PrecisionGoal \[Rule] precgoal]]; \[IndentingNewLine]fsf = First[NDSolve[Join[eqs /. lamrule, initcond], newfs, {var, xmaxbc, xmax}, WorkingPrecision \[Rule] wp, MaxSteps \[Rule] maxstep, AccuracyGoal \[Rule] accgoal, PrecisionGoal \[Rule] precgoal]]; \[IndentingNewLine]MapThread[ MIF[#1, #2] &, {fsb, fsf}], xmax < xmaxbc, (*Case\ 3*) fsdrop = First[NDSolve[Join[eqs /. lamrule, initcond], newfs, {var, xmaxbc, xmax}, WorkingPrecision \[Rule] wp, MaxSteps \[Rule] maxstep, AccuracyGoal \[Rule] accgoal, PrecisionGoal \[Rule] precgoal]]; \[IndentingNewLine]If[ Chop[Part[fsdrop, 1, 2, 0, 1, 1, 1] - xmax] \[NotEqual] 0, (*Produce\ a\ warning\ message*) $NoEndIntegration; Return[$Failed]]; \[IndentingNewLine]ic = \(fsdrop /. var \[Rule] xmax\) /. Rule \[Rule] Equal; \[IndentingNewLine]First[ NDSolve[Join[eqs /. lamrule, ic], newfs, {var, xmax, xmin}, WorkingPrecision \[Rule] wp, MaxSteps \[Rule] maxstep, AccuracyGoal \[Rule] accgoal, PrecisionGoal \[Rule] precgoal]]]; (*hcihw*) (*NOTE - Backward\ integration\ may\ also\ fail\ due\ to\ sensitivity\ of\ \ the\ problem\ to\ the\ initial\ conditions, divergency\ of\ the\ integration\ process\ itself, and\ other\ reasons . 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Some comparisons with NDSolve and Chasing ", "Subsection"], Cell[CellGroupData[{ Cell["1.1 ", "Subsubsection"], Cell[TextData[{ "Not all linear equations with pointwise conditions can be solved by the \ ", StyleBox["NDSolve", FontWeight->"Bold"], " or ", StyleBox["Chasing", FontWeight->"Bold"], " .\n For example evaluatiion of following problems generates a lot of \ messages and no result. 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Examples with pointwise given conditions that are unsolvable by NDSolve \ and Chasing as some conditions are containing more than one point\ \>", "Subsection"], Cell[CellGroupData[{ Cell[" 2.1 ", "Subsubsection"], Cell["\<\ Consider the following problem in the interval (2,3)\ \>", "Text"], Cell[BoxData[ \(\(x\^2\) y'' \((x)\) + \(y \((x)\)\)\/\(ln\ x\) = x\ \(e\^x\) \((2 + x\ ln\ x)\), \[IndentingNewLine]y \((2)\) - 3 y' \((2.3)\) + 1.5 y'' \((3)\) = 1, \[IndentingNewLine]y \((2.1)\) + y' \((3)\) = \(-1\)\)], "Input"], Cell[" Here is the solution", "Text"], Cell[BoxData[ \(\(sol[x_] = \(E\^x\) Log[x] - 19.864405477858284`\ Log[x] - 9.778619278759349`\ Log[ x]\ \(\[Integral]\_2\%x\( 1\/Log[t]\^2\) \[DifferentialD]t\);\)\)], "Input"], Cell[BoxData[ RowBox[{ StyleBox["NLinearDSolve", FontWeight->"Bold"], " ", "solves", " ", "this", " ", "problem", " ", "rather", " ", "fast", " ", "and", " ", "with", " ", "high", " ", "precision"}]], "Input", FontFamily->"Times New Roman", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[CellGroupData[{ Cell[BoxData[ \(w = NLinearDSolve[{x\^2\ \ \(y''\)[x] + y[x]\/Log[x] \[Equal] \ x\ \ \(E\^x\) \((2 + x\ \ Log[x])\), y[2] - 3 \( y'\)[2.3] + 1.5 y[3] \[Equal] \ 1, \[IndentingNewLine]y[2.1] + \(y'\)[3] \[Equal] \(-1\)}, y, {x, 2, 3}, \[IndentingNewLine]PrecisionInformation \[Rule] True, \[IndentingNewLine]PrecisionGoal \[Rule] 14, \[IndentingNewLine]SolutionNorm \[Rule] L2, \[IndentingNewLine]SymbolicSolution \[Rule] True\[IndentingNewLine]] // Timing\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"L2 norm (with weight->\"\>", "\[InvisibleSpace]", "1", "\[InvisibleSpace]", "\<\") of solution -> about \"\>", "\[InvisibleSpace]", TagBox[ InterpretationBox["\<\" 13.1\"\>", 13.148521916510306, AutoDelete->True], (PaddedForm[ #, 3]&)]}], SequenceForm[ "L2 norm (with weight->", 1, ") of solution -> about ", PaddedForm[ 13.148521916510306, 3]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Relative L2 error (with weight->\"\>", "\[InvisibleSpace]", "1", "\[InvisibleSpace]", "\<\")-> about \"\>", "\[InvisibleSpace]", TagBox[ InterpretationBox[\(" 1.97"\[Times]10\^"-15"\), .19687067078301452*^-14, AutoDelete->True], (PaddedForm[ #, 3]&)]}], SequenceForm[ "Relative L2 error (with weight->", 1, ")-> about ", PaddedForm[ .19687067078301452*^-14, 3]], Editable->False]], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{\(0.561000000000007`\ Second\), ",", InterpretationBox[\("SymbolicSolution["\[InvisibleSpace]{2, 3}\[InvisibleSpace]",<>]"\), SequenceForm[ "SymbolicSolution[", {2, 3}, ",<>]"], Editable->False]}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ " Here are the actual ", Cell[BoxData[ \(L\_2\)]], "error and ", Cell[BoxData[ \(L\_2\)]], "norm of the solution which coincides with the information message" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Sqrt[ NIntegrate[Evaluate[Abs[\(w[\([2]\)]\)[x] - sol[x]]\^2], {x, 2, 3}]/ NIntegrate[Abs[sol[x]]\^2, {x, 2, 3}]]\)], "Input"], Cell[BoxData[ \(NIntegrate::"slwcon" \(\(:\)\(\ \)\) "Numerical integration converging too slowly; suspect one of the \ following: singularity, value of the integration being 0, oscillatory \ integrand, or insufficient WorkingPrecision. If your integrand is oscillatory \ try using the option Method->Oscillatory in NIntegrate."\)], "Message"], Cell[BoxData[ \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \ recursive bisections in \!\(x\) near \!\(x\) = \!\(2.52734375`\)."\)], \ "Message"], Cell[BoxData[ \(1.9621676569328115`*^-15\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Sqrt[NIntegrate[Abs[sol[x]]\^2, {x, 2, 3}]]\)], "Input"], Cell[BoxData[ \(13.148521916510298`\)], "Output"] }, Open ]], Cell[" Here are the discrepancies of the conditions", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({y[2] - 3 \( y'\)[2.3] + 1.5 y[3] - 1, y[2.1] + Limit[\(y'\)[x], x \[Rule] 3] + 1} /. y \[Rule] w[\([2]\)]\)], "Input"], Cell[BoxData[ \({\(-4.742872761198669`*^-12\), \(-7.993605777301127`*^-13\)}\)], \ "Output"] }, Open ]], Cell[BoxData[ RowBox[{ "Evaluating", " ", "this", " ", "simple", " ", "examples", " ", "by", " ", StyleBox["NDSolve", FontWeight->"Bold"], " ", "or", " ", StyleBox["Chasing", FontWeight->"Bold"], " ", "generates", " ", "no", " ", \(\(result\)\(.\)\)}]], "Input", FontFamily->"Times New Roman", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[CellGroupData[{ Cell[BoxData[ \(w = NDSolve[{x\^2\ \ \(y''\)[x] + y[x]\/Log[x] \[Equal] \ x\ \ \(E\^x\) \((2 + x\ \ Log[x])\), y[2] - 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The actual precision \ is much higher than it is required by the default settings.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Abs[ NIntegrate[x\ E\^\(5 I\ x\), {x, 0, 2}] - NIntegrate[\(w[\([2]\)]\)[x], {x, 0, 2}]]\)], "Input"], Cell[BoxData[ \(1.4603792998549202`*^-12\)], "Output"] }, Open ]], Cell[" ", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[" 3.2", "Subsubsection"], Cell["\<\ Consider the following problem on the interval (1,2) with mixtured \ conditions \ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{\(x\^\(\(3\)\(\ \)\)\), RowBox[{ SuperscriptBox["y", TagBox[\((4)\), Derivative], MultilineFunction->None], "[", "x", "]"}]}], "+", RowBox[{"2", " ", \(x\^2\), " ", RowBox[{ SuperscriptBox["y", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "x", "]"}]}], "-", " ", RowBox[{"x", " ", RowBox[{ SuperscriptBox["y", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", \(\(y'\)[x]\), "-", \(\(x\^3\) y[x]\)}], "\[Equal]", "0"}], ",", \(\[Integral]\_1\%3 y[x] Sin[x] \[DifferentialD]x + y[1] == 9.42433203172718`\[InvisibleSpace] + 4.65938872738068`\ \[ImaginaryI]\), ",", \(\[Integral]\_1\%2\( y''\)[x] x\ \[DifferentialD]x + 2 \(\[Integral]\_1\%3\( y'\)[x] x\^2\ \[DifferentialD]x\) == 77.16466705374886`\[InvisibleSpace] + 41.744923777903146`\ \[ImaginaryI]\), ",", \(\(y'\)[2] - 2 \(\[Integral]\_1\%2\( y''\)[x] Log[x + 2] \[DifferentialD]x\) + \[Integral]\_2\%3 y[ x] \[DifferentialD]x == 2.962288718805786`\[InvisibleSpace] + 2.377497782948788`\ \[ImaginaryI]\), ",", \(\(y'''\)[1] == 1.7320907974056439`\[InvisibleSpace] + 0.42941143422544675`\ \[ImaginaryI]\)}]], "Input"], Cell[BoxData[ \(Here\ is\ the\ exact\ solution\)], "Input", FontFamily->"Times New Roman", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(\(sol[x_] = BesselJ[0, x] - BesselY[0, x] + 2\ BesselJ[0, I\ x] + BesselY[0, I\ x];\)\)], "Input"], Cell[TextData[{ "For solution of the problem we use both ", StyleBox["Integrate", FontWeight->"Bold"], " and ", StyleBox["NIntegrate", FontWeight->"Bold"], " functions. In the last one you can use the options of the NIntegrate \ function to calculate integrals more precise" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"w", "=", RowBox[{ RowBox[{"NLinearDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{\(x\^\(\(3\)\(\ \)\)\), RowBox[{ SuperscriptBox["y", TagBox[\((4)\), Derivative], MultilineFunction->None], "[", "x", "]"}]}], "+", RowBox[{"2", " ", \(x\^2\), " ", RowBox[{ SuperscriptBox["y", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "x", "]"}]}], "-", " ", RowBox[{"x", " ", RowBox[{ SuperscriptBox["y", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", \(\(y'\)[x]\), "-", \(\(x\^3\) y[x]\)}], "\[Equal]", "0"}], ",", \(\[Integral]\_1\%3 y[x] Sin[x] \[DifferentialD]x + y[1] == 9.42433203172718`\[InvisibleSpace] + 4.65938872738068`\ \[ImaginaryI]\), ",", \(NIntegrate[\(y''\)[x] x, {x, 1, 2}, Method \[Rule] GaussKronrod, GaussPoints \[Rule] 11, PrecisionGoal \[Rule] 7] + 2 \(\[Integral]\_1\%3\( y'\)[x] x\^2\ \[DifferentialD]x\) == 77.16466705374886`\[InvisibleSpace] + 41.744923777903146`\ \[ImaginaryI]\), ",", \(\(y'\)[2] - 2 NIntegrate[\(y''\)[x] Log[x + 2], {x, 1, 2}, Method \[Rule] GaussKronrod, GaussPoints \[Rule] 11, PrecisionGoal \[Rule] 7] + \[Integral]\_2\%3 y[ x] \[DifferentialD]x == 2.962288718805786`\[InvisibleSpace] + 2.377497782948788`\ \[ImaginaryI]\), ",", \(\(y'''\)[1] == 1.7320907974056439`\[InvisibleSpace] + 0.42941143422544675`\ \[ImaginaryI]\)}], "}"}], ",", "y", ",", \({x, 1, 3}\), ",", "\[IndentingNewLine]", \(PrecisionInformation \[Rule] True\), ",", "\[IndentingNewLine]", \(Weight \[Rule] \(\((x - 2.5)\)\^2\) \((3 - x)\)\^\(1/2\)\), ",", "\[IndentingNewLine]", \(SymbolicSolution \[Rule] True\), ",", "\[IndentingNewLine]", \(Error \[Rule] Absolute\), ",", "\[IndentingNewLine]", \(SolutionNorm \[Rule] Uniform\), ",", "\[IndentingNewLine]", \(PrecisionGoal \[Rule] 8\)}], "]"}], "//", "Timing"}]}]], "Input", CellTags->"S1.6.4"], Cell[BoxData[ \(NIntegrate::"ploss" \(\(:\)\(\ \)\) "Numerical integration stopping due to loss of precision. Achieved \ neither the requested PrecisionGoal nor AccuracyGoal; suspect one of the \ following: highly oscillatory integrand or the true value of the integral is \ 0. If your integrand is oscillatory try using the option Method->Oscillatory \ in NIntegrate."\)], "Message"], Cell[BoxData[ \(NIntegrate::"ploss" \(\(:\)\(\ \)\) "Numerical integration stopping due to loss of precision. Achieved \ neither the requested PrecisionGoal nor AccuracyGoal; suspect one of the \ following: highly oscillatory integrand or the true value of the integral is \ 0. If your integrand is oscillatory try using the option Method->Oscillatory \ in NIntegrate."\)], "Message"], Cell[BoxData[ \(NIntegrate::"ploss" \(\(:\)\(\ \)\) "Numerical integration stopping due to loss of precision. Achieved \ neither the requested PrecisionGoal nor AccuracyGoal; suspect one of the \ following: highly oscillatory integrand or the true value of the integral is \ 0. If your integrand is oscillatory try using the option Method->Oscillatory \ in NIntegrate."\)], "Message"], Cell[BoxData[ \(General::"stop" \(\(:\)\(\ \)\) "Further output of \!\(NIntegrate :: \"ploss\"\) will be suppressed \ during this calculation."\)], "Message"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Uniform norm (with weight->\"\>", "\[InvisibleSpace]", \(\@\(3 - x\)\ \((\(-2.5`\) + x)\)\^2\), "\[InvisibleSpace]", "\<\") of solution -> about \"\>", "\[InvisibleSpace]", TagBox[ InterpretationBox["\<\" 10.2\"\>", 10.170233189479426, AutoDelete->True], (PaddedForm[ #, 3]&)]}], SequenceForm[ "Uniform norm (with weight->", Times[ Power[ Plus[ 3, Times[ -1, x]], Rational[ 1, 2]], Power[ Plus[ -2.5, x], 2]], ") of solution -> about ", PaddedForm[ 10.170233189479426, 3]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Absolute Uniform error (with weight->\"\>", "\[InvisibleSpace]", \(\@\(3 - x\)\ \((\(-2.5`\) + x)\)\^2\), "\[InvisibleSpace]", "\<\") -> 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This is helpful while calculating the derivatives of \ the solution \ \>", "Text"], Cell[BoxData[ \(\(g[x_] = Chop[\(w[\([2]\)]\)[x] // TrigExpand, 10\^\(-12\)] // Simplify;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(g[x]\)], "Input"], Cell[BoxData[ \(\((\(\(3.005144024962477`\)\(\[InvisibleSpace]\)\) + 0.99999989462202`\ \[ImaginaryI])\) - \((\(\(0.05314018723984648`\)\ \(\[InvisibleSpace]\)\) - 1.0319747391629441`*^-6\ \[ImaginaryI])\)\ x - \ \((\(\(0.8495167725072921`\)\(\[InvisibleSpace]\)\) - 0.2499952644132646`\ \[ImaginaryI])\)\ x\^2 + \ \((\(\(3.124243584955702`\)\(\[InvisibleSpace]\)\) + 0.000013540504167797174`\ \[ImaginaryI])\)\ x\^3 - \ \((\(\(8.968973182412059`\)\(\[InvisibleSpace]\)\) - 0.015597941052879576`\ \[ImaginaryI])\)\ x\^4 + \ \((\(\(20.728243910389175`\)\(\[InvisibleSpace]\)\) + 0.00004017719299136738`\ \[ImaginaryI])\)\ x\^5 - \ \((\(\(35.75929756225628`\)\(\[InvisibleSpace]\)\) - 0.000388031870299539`\ \[ImaginaryI])\)\ x\^6 + \ \((\(\(47.068899126935584`\)\(\[InvisibleSpace]\)\) + 0.00004157740791913686`\ \[ImaginaryI])\)\ x\^7 - \ \((\(\(48.181877073554716`\)\(\[InvisibleSpace]\)\) + 0.000023355929973086063`\ \[ImaginaryI])\)\ x\^8 + \ \((\(\(38.888603301190564`\)\(\[InvisibleSpace]\)\) + 0.00001768615941511424`\ \[ImaginaryI])\)\ x\^9 - \ \((\(\(24.955137887595498`\)\(\[InvisibleSpace]\)\) + 8.376442012097621`*^-6\ \[ImaginaryI])\)\ x\^10 + \ \((\(\(12.774622893411966`\)\(\[InvisibleSpace]\)\) + 3.282477901892807`*^-6\ \[ImaginaryI])\)\ x\^11 - \ \((\(\(5.208956485221075`\)\(\[InvisibleSpace]\)\) + 1.0347082424295017`*^-6\ \[ImaginaryI])\)\ x\^12 + \ \((\(\(1.6806819021959567`\)\(\[InvisibleSpace]\)\) + 2.6263698135247834`*^-7\ \[ImaginaryI])\)\ x\^13 - \ \((\(\(0.42358185434743084`\)\(\[InvisibleSpace]\)\) + 5.2829484304206336`*^-8\ \[ImaginaryI])\)\ x\^14 + \ \((\(\(0.08160717510196983`\)\(\[InvisibleSpace]\)\) + 8.232480604428164`*^-9\ \[ImaginaryI])\)\ x\^15 - \ \((\(\(0.01160384900113586`\)\(\[InvisibleSpace]\)\) + 9.576357205482968`*^-10\ \[ImaginaryI])\)\ x\^16 + \ \((\(\(0.0011473391081682704`\)\(\[InvisibleSpace]\)\) + 7.818741416732017`*^-11\ \[ImaginaryI])\)\ x\^17 - \ \((\(\(0.00007042922925230639`\)\(\[InvisibleSpace]\)\) + 3.990335731111905`*^-12\ \[ImaginaryI])\)\ x\^18 + 2.020896342970046`*^-6\ x\^19\)], "Output"] }, Open ]], Cell["Here are the discrepancies of the conditions", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({NIntegrate[\(w[\([2]\)]\)[x] Sin[x], {x, 1, 3}] + \(w[\([2]\)]\)[1] - 9.42433203172718` - \ 4.65938872738068`\ \[ImaginaryI], NIntegrate[\(w[\([2]\)]''\)[x] x\ , {x, 1, 2}, Method \[Rule] GaussKronrod, GaussPoints \[Rule] 11, PrecisionGoal \[Rule] 7] + 2 NIntegrate[\(w[\([2]\)]'\)[x] x\^2, {x, 1, 3}] - 77.16466705374886`\[InvisibleSpace] - 41.744923777903146`\ \[ImaginaryI], \(g'\)[2] - 2 NIntegrate[\(g''\)[x] Log[x + 2], {x, 1, 2}, Method \[Rule] GaussKronrod, GaussPoints \[Rule] 11, PrecisionGoal \[Rule] 7] + NIntegrate[\(w[\([2]\)]\)[x], {x, 2, 3}] - 2.962288718805786` - \ 2.377497782948788`\ \[ImaginaryI], \(g'''\)[1] - 1.7320907974056439`\[InvisibleSpace] - 0.42941143422544675`\ \[ImaginaryI]}\)], "Input"], Cell[BoxData[ \({\(\(0.`\)\(\[InvisibleSpace]\)\) + 0.`\ \[ImaginaryI], \(-7.915744504316535`*^-9\) - 2.1316282072803006`*^-14\ \[ImaginaryI], 2.4174329649184756`*^-8 + 8.180487838238548`*^-7\ \[ImaginaryI], \ \(\(0.00005135071794870605`\)\(\[InvisibleSpace]\)\) - 4.341916270966806`*^-10\ \[ImaginaryI]}\)], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["4. Singular Coefficients and general type conditions", "Subsection"], Cell[CellGroupData[{ Cell["4.1. Bessel's equation ", "Subsubsection"], Cell["\<\ Consider the following problem with singular coefficients for well known \ Bessel equation on the interval (0,1)\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{\(z\^2\), " ", RowBox[{ SuperscriptBox["y", "\[Prime]\[Prime]", MultilineFunction->None], "[", "z", "]"}]}], "+", RowBox[{"z", " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "z", "]"}]}], "+", \(\((z\^2 - l\^2)\)\ y[z]\)}], "=", "0"}], ",", "\[IndentingNewLine]", " ", \(y \((0)\) = 0\), ",", " ", "\[IndentingNewLine]", \(y \((1)\) = 0.31166404072415316\)}]], "Input"], Cell["with exact solution y(x)=BesselJ[1.3,z].", "Text"], Cell[TextData[{ "Here is the solution of the problem in the weighted space ", Cell[BoxData[ \(L\_2\)]], "(", Cell[BoxData[ \(\@x\)]], ",0,1)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"w", "=", RowBox[{ RowBox[{"NLinearDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{\(z\^2\), " 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The \ result may be inaccurate at a low working precision."\)], "Message"], Cell[BoxData[ \(NDSolve::"eqnsb" \(\(:\)\(\ \)\) "Equation not solvable by the chasing method."\)], "Message"], Cell[BoxData[ RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{\(\((\(-1.6900000000000002`\) + z\^2)\)\ y[z]\), "+", RowBox[{"z", " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "z", "]"}]}], "+", RowBox[{\(z\^2\), " ", RowBox[{ SuperscriptBox["y", "\[Prime]\[Prime]", MultilineFunction->None], "[", "z", "]"}]}]}], "==", "0"}], ",", \(y[0] == 0\), ",", \(y[1] == 0.31166404072415316`\)}], "}"}], ",", "y", ",", \({z, 0, 1}\)}], "]"}]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Chasing", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{\(z\^2\), " ", RowBox[{ SuperscriptBox["y", "\[Prime]\[Prime]", MultilineFunction->None], "[", "z", "]"}]}], "+", RowBox[{"z", " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "z", "]"}]}], "+", \(\((z\^2 - 1.3\^2)\)\ y[z]\)}], "\[Equal]", "0"}], ",", \(y[0] \[Equal] 0\), ",", " ", \(y[1] \[Equal] 0.31166404072415316\)}], "}"}], ",", "y", ",", \({z, 0, 1}\)}], "]"}]], "Input"], Cell[BoxData[ \(Power::"infy" \(\(:\)\(\ \)\) "Infinite expression \!\(1\/0.`\) encountered."\)], "Message"], Cell[BoxData[ \(Power::"infy" \(\(:\)\(\ \)\) "Infinite expression \!\(1\/0.`\^2\) encountered."\)], "Message"], Cell[BoxData[ \(Power::"infy" \(\(:\)\(\ \)\) "Infinite expression \!\(1\/0.`\^2\) encountered."\)], "Message"], Cell[BoxData[ \(General::"stop" \(\(:\)\(\ \)\) "Further output of \!\(Power :: \"infy\"\) will be suppressed during \ this calculation."\)], "Message"], Cell[BoxData[ \(NDSolve::"ndnum" \(\(:\)\(\ \)\) "Encountered non-numerical value for a derivative at \!\(z\) == \ \!\(1.436167957735481`*^-271\)."\)], "Message"], Cell[BoxData[ \(NDSolve::"ndnum" \(\(:\)\(\ \)\) "Encountered non-numerical value for a derivative at \!\(z\) == \ \!\(1.436249191482395`*^-271\)."\)], "Message"], Cell[BoxData[ \(ReplaceAll::"reps" \(\(:\)\(\ \)\) "\!\({\({$Failed}\)}\) is neither a list of replacement rules nor a \ valid dispatch table, and so cannot be used for replacing."\)], "Message"], Cell[BoxData[ \(ReplaceAll::"reps" \(\(:\)\(\ \)\) "\!\({$Failed}\) is neither a list of replacement rules nor a valid \ dispatch table, and so cannot be used for replacing."\)], "Message"], Cell[BoxData[ \(Join::"heads" \(\(:\)\(\ \)\) "Heads \!\(ReplaceAll\) and \!\(List\) at positions \!\(1\) and \!\(2\) \ are expected to be the same."\)], "Message"], Cell[BoxData[ \(LinearSolve::"lsv" \(\(:\)\(\ \)\) "Second argument \!\(Join[\(\(\(\(\(\({\(\(\(\(Chasing`Private`g[\(\(1, \ 1\)\)]\)\)[1]\)\), \(\(\(\(Chasing`Private`g[\(\(1, 2\)\)]\)\)[1]\)\), \(\(\(\ \(Chasing`Private`g[\(\(1, 3\)\)]\)\)[1]\)\)}\)\) /. \[InvisibleSpace] \ $Failed\)\), \(\({\(-\(\(\[LeftSkeleton] 20 \ \[RightSkeleton]\)\)\)}\)\)\)\)]\) is neither a non-empty vector nor a list \ of vectors of equal length."\)], "Message"], Cell[BoxData[ \({}\)], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["4.2. Equation for JacobiP[n,a,b,x] polynomials", "Subsubsection"], Cell["Consider the following problem on the interval (-1,1)", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{\((1 - x\^2)\), RowBox[{ SuperscriptBox["w", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", RowBox[{\((b - a - \((a + b + 2)\) x)\), RowBox[{ SuperscriptBox["w", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", \(n \((n + a + b + 1)\) w[x]\)}], "=", "0"}], ",", \(\(-1\) \[LessEqual] x \[LessEqual] 1\)}], "\n", \(w[0] - \(w'\)[0] + 2 \(\[Integral]\_\(\(-1\)/2\)\%\(1/2\)w[x] Sin[x] \[DifferentialD]x\) \[Equal] \ 0.19785845544020186`, \[IndentingNewLine]w[\(-1\)] \[Equal] \ \(-0.4656172500000001`\), \[IndentingNewLine]n = 5, \[IndentingNewLine]a = .2, b = \(- .3\)\)}], "Input"], Cell[TextData[{ "with exact solution ", StyleBox["w(x)=JacobiP[n,a,b,x].", FontSlant->"Italic"] }], "Text"], Cell["\<\ Here is the solution of the problem with the well known classic weight which \ is singular in the end x=-1 and wanishing in the end x=1\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{\(n = 5;\), "\n", \(a = .2;\), "\n", \(b = \(- .3\);\), "\n", RowBox[{"jac", "=", RowBox[{ RowBox[{"NLinearDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{\((1 - x\^2)\), RowBox[{ SuperscriptBox["w", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", RowBox[{\((b - a - \((a + b + 2)\) x)\), RowBox[{ SuperscriptBox["w", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", "\[IndentingNewLine]", \(n \((n + a + b + 1)\) w[x]\)}], "\[Equal]", " ", "0"}], ",", " ", \(w[\(-1\)] \[Equal] \(-0.4656172500000001`\)\), ",", "\[IndentingNewLine]", \(w[0] - \(w'\)[0] + 2 NIntegrate[Evaluate[w[x] Sin[x]], {x, 0, 1/2}, Method \[Rule] GaussKronrod, GaussPoints \[Rule] 11] \[Equal] \ \(-1.5239163778012081`\)\)}], "}"}], ",", "w", ",", \({x, \(-1\), 1}\), ",", "\[IndentingNewLine]", \(SymbolicSolution \[Rule] True\), ",", "\[IndentingNewLine]", \(Weight -> \(\((1 - x)\)\^a\) \((1 + x)\)\^\(\(b\)\(\ \)\)\)}], "]"}], "//", "Timing"}]}]}], "Input"], Cell[BoxData[ \(NIntegrate::"ploss" \(\(:\)\(\ \)\) "Numerical integration stopping due to loss of precision. Achieved \ neither the requested PrecisionGoal nor AccuracyGoal; suspect one of the \ following: highly oscillatory integrand or the true value of the integral is \ 0. If your integrand is oscillatory try using the option Method->Oscillatory \ in NIntegrate."\)], "Message"], Cell[BoxData[ \(NIntegrate::"ploss" \(\(:\)\(\ \)\) "Numerical integration stopping due to loss of precision. Achieved \ neither the requested PrecisionGoal nor AccuracyGoal; suspect one of the \ following: highly oscillatory integrand or the true value of the integral is \ 0. If your integrand is oscillatory try using the option Method->Oscillatory \ in NIntegrate."\)], "Message"], Cell[BoxData[ \(NIntegrate::"ploss" \(\(:\)\(\ \)\) "Numerical integration stopping due to loss of precision. Achieved \ neither the requested PrecisionGoal nor AccuracyGoal; suspect one of the \ following: highly oscillatory integrand or the true value of the integral is \ 0. If your integrand is oscillatory try using the option Method->Oscillatory \ in NIntegrate."\)], "Message"], Cell[BoxData[ \(General::"stop" \(\(:\)\(\ \)\) "Further output of \!\(NIntegrate :: \"ploss\"\) will be suppressed \ during this calculation."\)], "Message"], Cell[BoxData[ RowBox[{"{", RowBox[{\(0.9510000000000218`\ Second\), ",", InterpretationBox[\("SymbolicSolution["\[InvisibleSpace]{\(-1\), 1}\[InvisibleSpace]",<>]"\), SequenceForm[ "SymbolicSolution[", {-1, 1}, ",<>]"], Editable->False]}], "}"}]], "Output"] }, Open ]], Cell["Here is the shape of the solution", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(jac[\([2]\)]\)[x] // Simplify\)], "Input"], Cell[BoxData[ \(\(-1.2413594482421528`\) - 2.907878090820242`\ x + 2.680340039062397`\ x\^2 + \((\(-2.067702246093674`\) + 6.325395761718564`\ x)\)\ Cos[ 2\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]] + \((\(\(5.183223173827969`\)\(\[InvisibleSpace]\)\) \ - 1.8634896484374277`\ x)\)\ Cos[ 3\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]] - 0.6988086181640432`\ Cos[ 4\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]] + 4.612136879882678`\ x\ Cos[ 4\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]] + 1.844854751953099`\ Cos[ 5\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]] - 1.687538997430238`*^-14\ x\ Cos[ 5\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]] + 8.171241461241152`*^-14\ Cos[ 6\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]] + 4.884981308350689`*^-14\ x\ Cos[ 6\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]] + 2.842170943040401`*^-14\ Cos[ 7\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]] + 1.438849039914203`*^-13\ x\ Cos[ 7\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]] + 5.5067062021407764`*^-14\ Cos[ 8\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]] + 1.532107773982716`*^-14\ x\ Cos[ 8\ ArcCos[\(\(0.`\)\(\[InvisibleSpace]\)\) - 1.`\ x]]\)], "Output"] }, Open ]], Cell["Here is a pure polynomial form of the solution", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(j0 = \(jac[\([2]\)]\)[x] // TrigExpand\)], "Input"], Cell[BoxData[ \(\(\(0.1275341796874514`\)\(\[InvisibleSpace]\)\) + 1.704258789062445`\ x - 1.4550644531240886`\ x\^2 - 8.082101171875394`\ x\^3 + 1.863489648433859`\ x\^4 + 7.379419007815134`\ x\^5 + 4.902744876744691`*^-12\ x\^6 - 4.177991286269389`*^-12\ x\^7 - 2.1600499167107046`*^-12\ x\^8 + 1.9610979506978765`*^-12\ x\^9\)], "Output"] }, Open ]], Cell["Here are the error function and its plot with the weight", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(j0 - JacobiP[n, a, b, x] // Expand\)], "Input"], Cell[BoxData[ \(\(-4.263256414560601`*^-14\) - 2.842170943040401`*^-14\ x + 9.663381206337363`*^-13\ x\^2 - 3.979039320256561`*^-13\ x\^3 - 3.645084234449314`*^-12\ x\^4 + 2.631672657571471`*^-12\ x\^5 + 4.902744876744691`*^-12\ x\^6 - 4.177991286269389`*^-12\ x\^7 - 2.1600499167107046`*^-12\ x\^8 + 1.9610979506978765`*^-12\ x\^9\)], "Output"] }, Open ]], Cell[CellGroupData[{ 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The third order equation with singular coefficients", \ "Subsubsection"], Cell["\<\ The general solution of the following considered equation has the form (see \ below)\ \>", "Text"], Cell[BoxData[ \(c[1] E\^x + x\ c[2] BesselJ[1, I\ x] + x\ c[3] BesselY[1, I\ x]\)], "Input"], Cell["We are interesting for the solution", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol[x_] = c[1] E\^x + x\ c[2] BesselJ[1, I\ x] + x\ c[3] BesselY[1, I\ x] /. {c[1] \[Rule] 0.5247162216356969`, c[2] \[Rule] \(-0.746574013239847`\) + 1.2605358525241253`\ \[ImaginaryI], c[3] \[Rule] \(-0.746574013239847`\)\ \[ImaginaryI]}\)], "Input"], Cell[BoxData[ \(0.5247162216356969`\ \[ExponentialE]\^x - \((\(\(0.746574013239847`\)\(\ \[InvisibleSpace]\)\) - 1.2605358525241253`\ \[ImaginaryI])\)\ x\ BesselJ[ 1, \[ImaginaryI]\ x] - 0.746574013239847`\ \[ImaginaryI]\ x\ BesselY[ 1, \[ImaginaryI]\ x]\)], "Output"] }, Open ]], Cell[TextData[{ "Here", StyleBox[" NLinearDSolve", FontWeight->"Bold"], " works" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(w = NLinearDSolve[{x\ \ \(y'''\)[x] - x\ \(y''\)[x] - \((x - 1)\) \(y'\)[x] + \((x - 1)\) y[x] \[Equal] 0, \[IndentingNewLine]y[0] \[Equal] \ 1, y[1] \[Equal] 1, y[2] \[Equal] 0\[InvisibleSpace]}, y, {x, 0, 2}, \[IndentingNewLine]PrecisionInformation \[Rule] True, \[IndentingNewLine]PrecisionGoal \[Rule] 4, \[IndentingNewLine]SymbolicSolution \[Rule] True\[IndentingNewLine]] // Timing\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"L2 norm (with weight->\"\>", "\[InvisibleSpace]", "1", "\[InvisibleSpace]", "\<\") of solution -> about \"\>", "\[InvisibleSpace]", TagBox[ InterpretationBox["\<\" 1.27\"\>", 1.2706310972370489, AutoDelete->True], (PaddedForm[ #, 3]&)]}], SequenceForm[ "L2 norm (with weight->", 1, ") of solution -> about ", PaddedForm[ 1.2706310972370489, 3]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Relative L2 error (with weight->\"\>", "\[InvisibleSpace]", "1", "\[InvisibleSpace]", "\<\")-> about \"\>", "\[InvisibleSpace]", TagBox[ InterpretationBox["\<\" 0.000208\"\>", .00020809514248933979, AutoDelete->True], (PaddedForm[ #, 3]&)]}], SequenceForm[ "Relative L2 error (with weight->", 1, ")-> about ", PaddedForm[ .00020809514248933979, 3]], Editable->False]], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{\(1.8120000000000118`\ Second\), ",", InterpretationBox[\("SymbolicSolution["\[InvisibleSpace]{0, 2}\[InvisibleSpace]",<>]"\), SequenceForm[ "SymbolicSolution[", {0, 2}, ",<>]"], Editable->False]}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "Here are the actual ", Cell[BoxData[ \(L\_2\)]], "error and ", Cell[BoxData[ \(L\_2\)]], "norm of the solution" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Sqrt[ NIntegrate[ Evaluate[Abs[\(w[\([2]\)]\)[x] - sol[x]]\^2], {x, 0, 2}]]\)], "Input"], Cell[BoxData[ \(0.00017996604028089366`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Sqrt[NIntegrate[Evaluate[Abs[sol[x]]\^2], {x, 0, 2}]]\)], "Input"], Cell[BoxData[ \(1.2705544329574903`\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["4.4. The same equation with integral conditions", "Subsubsection"], Cell[TextData[{ StyleBox["Let find the solution to the previous equation which is \ orthogonal to the subspace ", FontVariations->{"CompatibilityType"->0}], StyleBox["Span{Sin[\[Pi] x],Cos[\[Pi] x]} ", FontWeight->"Bold"], "and has integral value equals to one. 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