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The Fu\[CHacek]\[IAcute]k spectrum is defined as the set of all couples \ ", StyleBox["(\[Mu],\[Nu]) \[Element]", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(R\^2\)\)\)], FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], " such that the problem has a nontrivial solution, and it can be figured by \ ", StyleBox["Mathematica", FontSlant->"Italic"], " as the zero-contour of a function ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["D", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["=", FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["D", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["(", FontColor->RGBColor[0, 0, 1]], StyleBox[\(\[Mu], \[Nu]\), FontColor->RGBColor[0, 0, 1]], StyleBox[")", FontColor->RGBColor[0, 0, 1]]}]}], TraditionalForm]]], ". Existence of a stable solution of an initial-boundary value problem that \ describes the oscillation of a suspension bridge is related to the shape of \ the Fu\[CHacek]\[IAcute]k spectrum of our problem." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Introduction", "SectionFirst"], Cell[CellGroupData[{ Cell["Mathematical model of a suspension bridge", "Subsection"], Cell[TextData[StyleBox["There are many mathematical models of suspension \ bridge, at different levels of simplification. 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The boundary conditions are called \ Dirichlet and mean that the ends of the bridge deck are clamped." }], "Text"], Cell[TextData[{ "The external force ", StyleBox["f", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " depends on the displacement ", StyleBox["u", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], ", as it is typical for the suspension bridge - if the bridge deck moves \ downwards (i.e. ", Cell[BoxData[ FormBox[ RowBox[{"u", ">", StyleBox["0", FontColor->RGBColor[0, 0, 1]]}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " - we consider the downwards direction to be positive), the cable stays \ pull it back, if it moves upwards (", Cell[BoxData[ FormBox[ RowBox[{"u", "<", StyleBox["0", FontColor->RGBColor[0, 0, 1]]}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], "), the cable stays do nothing." }], "Text"], Cell[TextData[{ "Let us assume that ", StyleBox["f", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " depends linearly on ", StyleBox["u", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], ", but with different constants for ", StyleBox["u", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " positive and ", StyleBox["u", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " negative, i.e. ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"f", "=", RowBox[{ SuperscriptBox[ StyleBox[ RowBox[{"\[Mu]", StyleBox["u", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]]}]], "+"], "-", SuperscriptBox[ StyleBox[ RowBox[{ StyleBox["\[Nu]", FontColor->RGBColor[0, 0, 1]], StyleBox["u", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]]}]], "-"]}]}], FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], ", where ", Cell[BoxData[ \(TraditionalForm\`\(u\^+\) = max {u, 0}\)], FontColor->RGBColor[0, 0, 1]], " is the positive and ", Cell[BoxData[ \(TraditionalForm\`\(u\^-\) = max {\(-u\), 0}\)], FontColor->RGBColor[0, 0, 1]], " the negative part of ", StyleBox["u", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " (hence, ", Cell[BoxData[ FormBox[ StyleBox[\(u = \(u\^+\) - \(u\^-\)\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], "). This is satisfied, if the cable stays behave like a spring with the \ modulus ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\)], FontColor->RGBColor[0, 0, 1]], " for ", StyleBox["u", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " positive and the modulus ", Cell[BoxData[ \(TraditionalForm\`\[Nu]\)], FontColor->RGBColor[0, 0, 1]], " for ", StyleBox["u", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " negative (", Cell[BoxData[ FormBox[ RowBox[{"\[Nu]", "=", StyleBox["0", FontColor->RGBColor[0, 0, 1]]}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " in our case). 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value generalized \ eigenvalue problem\ \>", "Text"], Cell[TextData[Cell[BoxData[ StyleBox[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ StyleBox["u", FontSlant->"Italic"], \((4)\)], "(", "t", ")"}], "=", RowBox[{ RowBox[{ SuperscriptBox[ StyleBox[ RowBox[{ StyleBox["\[Mu]", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["u", FontSlant->"Italic"]}]], "+"], "(", "t", ")"}], "-", RowBox[{ SuperscriptBox[ StyleBox[ RowBox[{"\[Nu]", StyleBox["u", FontSlant->"Italic"]}]], "-"], "(", "t", ")"}]}]}], ",", " ", \(t \[Element] \([0, 1]\)\), ",", "\[IndentingNewLine]", \(u( 0) = \(u' \((0)\) = \(u(1) = \(u' \((1)\) = 0. \)\)\)\)}], "TraditionalForm"], FontFamily->"Arial", FontSize->12, FontColor->RGBColor[0, 0, 1]]], CellMargins->{{171, 77}, {Inherited, Inherited}}, TextAlignment->Left, FontSize->10]], "Text", CellMargins->{{90, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "The Fu\[CHacek]\[IAcute]k (generalized) spectrum is defined as the set of \ all couples ", StyleBox["(\[Mu],\[Nu]) \[Element]", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox[ StyleBox["R", FontWeight->"Bold"], "2"]}], TraditionalForm]], FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], " such that the problem has a nontrivial solution (", StyleBox["eigenfunction", FontSlant->"Italic"], ")." }], "Text"], Cell[TextData[{ "If ", Cell[BoxData[ FormBox[ StyleBox["f", FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], " is at least ", StyleBox["asymptotically equal", FontSlant->"Italic"], " to ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SuperscriptBox[ StyleBox[ RowBox[{ StyleBox["\[Mu]", FontColor->RGBColor[0, 0, 1]], StyleBox["u", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]]}]], "+"], "-", SuperscriptBox[ StyleBox[ RowBox[{ StyleBox["\[Nu]", FontColor->RGBColor[0, 0, 1]], StyleBox["u", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]]}]], "-"]}], FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], ", i.e." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ StyleBox[\(lim\+\(s \[Rule] \(+\[Infinity]\)\)\ \(f(s)\)\/s = \[Mu]\), FontFamily->"Arial", FontSize->12, FontColor->RGBColor[0, 0, 1]], TraditionalForm]], CellMargins->{{171, 77}, {Inherited, Inherited}}, TextAlignment->Left, FontSize->10], " and ", Cell[BoxData[ FormBox[ StyleBox[\(lim\+\(s \[Rule] \(-\[Infinity]\)\)\ \(f(s)\)\/s = \[Nu]\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], "," }], "Text", CellMargins->{{90, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "and ", Cell[BoxData[ FormBox[ StyleBox[\((\[Mu], \[Nu])\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], " does not belong to the Fu\[CHacek]\[IAcute]k spectrum, then the \ topological degree theory can be used to prove that the solution is a priori \ bounded (otherwise there may exist an unbounded sequence of solutions). Hence \ it is important to know the Fu\[CHacek]\[IAcute]k spectrum well. Our aim is \ to ", StyleBox["figure the Fu\[CHacek]\[IAcute]k spectrum", FontWeight->"Bold"], " by ", StyleBox["Mathematica", FontSlant->"Italic"], " (we use the version 5.1.1.0)." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Shooting method", "Section"], Cell[CellGroupData[{ Cell["Initial value problem", "Subsection"], Cell[TextData[{ "Let us consider the fourth-order ", StyleBox["initial value problem", FontSlant->"Italic"], " (IVP)" }], "Text"], Cell[TextData[Cell[BoxData[ StyleBox[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ StyleBox["u", FontSlant->"Italic"], \((4)\)], "(", "t", ")"}], "=", RowBox[{ RowBox[{ SuperscriptBox[ StyleBox[ RowBox[{ StyleBox["\[Mu]", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["u", FontSlant->"Italic"]}]], "+"], "(", "t", ")"}], "-", RowBox[{ SuperscriptBox[ StyleBox[ RowBox[{"\[Nu]", StyleBox["u", FontSlant->"Italic"]}]], "-"], "(", "t", ")"}]}]}], ",", " ", \(t \[Element] \([0, 1]\)\), ",", "\[IndentingNewLine]", \(u(0) = \(u' \((0)\) = 0\)\), ",", " ", \(u'' \((0)\) = 1\), ",", " ", \(u''' \((0)\) = \[Delta]\), ","}], "TraditionalForm"], FontFamily->"Arial", FontSize->12, FontColor->RGBColor[0, 0, 1]]], CellMargins->{{171, 77}, {Inherited, Inherited}}, TextAlignment->Left, FontSize->10]], "Text", CellMargins->{{90, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\[Mu], \[Nu] > 0\)], FontColor->RGBColor[0, 0, 1]], " and ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["\[Delta]", FontColor->RGBColor[0, 0, 1]], "\[Element]", StyleBox["R", FontWeight->"Bold"]}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". The solution can be proved to be unique, and so we may denote by ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["V", FontColor->RGBColor[0, 0, 1]], StyleBox["=", FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["V", FontColor->RGBColor[0, 0, 1]], StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Mu], \[Nu], \[Delta]\), ")"}]}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " the value of the solution at ", Cell[BoxData[ FormBox[ StyleBox[\(t = 1\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". Obviously, ", Cell[BoxData[ FormBox[ StyleBox["V", FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " is an increasing continuous function of ", Cell[BoxData[ FormBox[ StyleBox["\[Delta]", FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ", and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox[ StyleBox["lim", FontColor->RGBColor[0, 0, 1]], \(\[Delta] \[Rule] \(\[PlusMinus]\[Infinity]\)\)], \(V(\ \[Mu], \[Nu], \[Delta])\)}], "=", \(\[PlusMinus]\[Infinity]\)}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". Hence there exists exactly one constant ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_0 = \(\[Delta]\_0\)(\[Mu], \[Nu])\)], FontColor->RGBColor[0, 0, 1]], ", such that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["V", FontColor->RGBColor[0, 0, 1]], StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Mu], \[Nu], \(\[Delta]\_0\)(\[Mu], \[Nu])\), ")"}], "=", "0"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". Since clearly ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["V", FontColor->RGBColor[0, 0, 1]], StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Mu], \[Nu], \[Delta]\), ")"}], ">", "0"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " for any ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["\[Delta]", FontColor->RGBColor[0, 0, 1]], "\[GreaterEqual]", "0"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ", it must be ", Cell[BoxData[ FormBox[ RowBox[{\(\(\[Delta]\_0\)(\[Mu], \[Nu])\), StyleBox["<", FontColor->RGBColor[0, 0, 1]], StyleBox["0", FontColor->RGBColor[0, 0, 1]]}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " for all ", Cell[BoxData[ FormBox[ RowBox[{"\[Mu]", ",", RowBox[{"\[Nu]", ">", StyleBox["0", FontColor->RGBColor[0, 0, 1]]}]}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". Let ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["D", FontColor->RGBColor[0, 0, 1]], StyleBox["=", FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["D", FontColor->RGBColor[0, 0, 1]], StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Mu], \[Nu]\), ")"}]}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " denote the first derivative of the solution of the IVP with ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["\[Delta]", FontColor->RGBColor[0, 0, 1]], StyleBox["=", FontColor->RGBColor[0, 0, 1]], \(\(\[Delta]\_0\)(\[Mu], \[Nu])\)}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". If ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["D", FontColor->RGBColor[0, 0, 1]], StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Mu], \[Nu]\), ")"}], "=", "0"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ", then ", Cell[BoxData[ \(TraditionalForm\`\((\[Mu], \[Nu])\)\)], FontColor->RGBColor[0, 0, 1]], " belongs to the Fu\[CHacek]\[IAcute]k spectrum. Moreover, ", Cell[BoxData[ \(TraditionalForm\`\((\[Nu], \[Mu])\)\)], FontColor->RGBColor[0, 0, 1]], " belongs to the Fu\[CHacek]\[IAcute]k spectrum, too, and the corresponding \ eigenfunction is ", Cell[BoxData[ FormBox[ StyleBox[\(-u\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], "." }], "Text"], Cell[TextData[{ "On the other hand, let us assume that ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Mu], \[Nu]\), ")"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ", ", Cell[BoxData[ \(TraditionalForm\`\[Mu], \[Nu] > 0\)], FontColor->RGBColor[0, 0, 1]], ", belongs to the Fu\[CHacek]\[IAcute]k spectrum and let ", Cell[BoxData[ FormBox[ StyleBox["u", FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " be a corresponding eigenfunction. It can be proved that ", Cell[BoxData[ \(TraditionalForm\`u'' \((0)\) \[NotEqual] 0\)], FontColor->RGBColor[0, 0, 1]], ". Let us assume first ", Cell[BoxData[ FormBox[ StyleBox[\(u'' \((0)\) > 0\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". Since ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"\[Alpha]", StyleBox["u", FontSlant->"Italic"]}]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ", ", Cell[BoxData[ FormBox[ StyleBox[\(\[Alpha] > 0\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ", is an eigenfunction, too, we may suppose that ", Cell[BoxData[ FormBox[ StyleBox[\(u'' \((0)\) = 1\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". Then, clearly, ", Cell[BoxData[ FormBox[ StyleBox["u", FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " is the solution of the IVP with ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["\[Delta]", FontColor->RGBColor[0, 0, 1]], StyleBox["=", FontColor->RGBColor[0, 0, 1]], \(\(\[Delta]\_0\)(\[Mu], \[Nu])\)}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ", and so ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["D", FontColor->RGBColor[0, 0, 1]], StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Mu], \[Nu]\), ")"}], "=", "0"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". If ", Cell[BoxData[ FormBox[ StyleBox[\(u'' \((0)\) < 0\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ", then ", Cell[BoxData[ FormBox[ StyleBox[\(-u\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " is an eigenfunction, corresponding to ", Cell[BoxData[ \(TraditionalForm\`\((\[Nu], \[Mu])\)\)], FontColor->RGBColor[0, 0, 1]], " with ", Cell[BoxData[ FormBox[ StyleBox[\(\((\(-u\))\)'' \((0)\) > 0\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". Consequently, ", Cell[BoxData[ \(TraditionalForm\`D(\[Nu], \[Mu]) = 0\)], FontColor->RGBColor[0, 0, 1]], " in this case." }], "Text"], Cell[TextData[{ "We showed that ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Mu], \[Nu]\), ")"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ", ", Cell[BoxData[ \(TraditionalForm\`\[Mu], \[Nu] > 0\)], FontColor->RGBColor[0, 0, 1]], ", belongs to the Fu\[CHacek]\[IAcute]k spectrum, ", StyleBox["if and only if", FontWeight->"Bold"], " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"D", StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Mu], \[Nu]\), ")"}], "=", "0"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " or ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"D", StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Nu], \[Mu]\), ")"}], "=", "0"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". Hence it may be figured as a union of the zero contours of ", Cell[BoxData[ FormBox[ RowBox[{"D", StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Mu], \[Nu]\), ")"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " and ", Cell[BoxData[ FormBox[ RowBox[{"D", StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Nu], \[Mu]\), ")"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". Our aim now reduces to implementation of ", Cell[BoxData[ FormBox[ StyleBox["D", FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], "." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Implementation", "Section"], Cell[CellGroupData[{ Cell["The first problem: precision", "Subsection"], Cell[TextData[{ "From now on we write ", Cell[BoxData[ FormBox[ StyleBox[\(\[Mu]\^4\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " and ", Cell[BoxData[ FormBox[ StyleBox[\(\[Nu]\^4\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " instead of ", Cell[BoxData[ FormBox[ StyleBox["\[Mu]", FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " and ", Cell[BoxData[ FormBox[ StyleBox["\[Nu]", FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " in the differential equation. First we implement the function ", Cell[BoxData[ FormBox[ StyleBox["V", FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ". The simplest one might be" }], "Text"], Cell[BoxData[ \(\(ValueAtOne[\[Mu]_, \[Nu]_, \[Delta]_] := u[1] /. \(NDSolve[{\(\(Derivative[4]\)[u]\)[ t] \[Equal] \[Mu]^4*PosPart[u[t]] - \[Nu]^4* NegPart[u[t]], u[0] \[Equal] 0, \(\(Derivative[1]\)[u]\)[0] \[Equal] 0, \(\(Derivative[2]\)[u]\)[0] \[Equal] 1, \(\(Derivative[3]\)[u]\)[0] \[Equal] \[Delta]}, u, {t, 0, 1}]\)[\([1]\)];\)\)], "Input", CellLabel->"In[4]:="], Cell[TextData[{ "Now we look for ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[Delta]\_0\), FontColor->RGBColor[0, 0, 1]], "(", \(\[Mu], \[Nu]\), ")"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], " as a root of ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["V", FontColor->RGBColor[0, 0, 1]], StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(\[Mu], \[Nu], \[Delta]\), ")"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(p1 = Plot[ValueAtOne[40, 30, t], {t, \(-45\), \(-35\)}, DisplayFunction \[Rule] Identity];\)\), 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To find an optimal precision we can compare the values \ for different precisions.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[ Table[ValueAtOne[40, 30, \(-40.37441848177\), n, 2*n], {n, 15, 25}], TableHeadings \[Rule] {Table[ 2*n, {n, 15, 25}], {"\", "\"}}]\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ RowBox[{\(NDSolve::"mxst"\), \(\(:\)\(\ \)\), "\<\"Maximum number of \ \\!\\(10000\\) steps reached at the point \\!\\(t\\) == \ \\!\\(\\*TagBox[\\\"0.\ 1589158757426683234635010245507764804659237836450724607007599818701`48.\\\", \ HoldForm]\\). \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"NDSolve::mxst\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[12]:="], Cell[BoxData[ RowBox[{\(InterpolatingFunction::"dmval"\), \(\(:\)\(\ \)\), "\<\"Input \ value \\!\\({1}\\) lies outside the range of data in the interpolating \ function. Extrapolation will be used. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"InterpolatingFunction::dmval\\\"]\\)\ \"\>"}]], "Message", CellLabel->"From In[12]:="], Cell[BoxData[ InterpretationBox[GridBox[{ {"30", "4147.84400911262787903619421868233732168351`28.387216106321826"},\ {"32", "4147.85020676900216790826638636847029496421`30.387216106321866"},\ {"34", "4147.85071597914657631528590153414105173806`32.387216106321866"},\ {"36", "4147.850720342217179176002726697526783730857281252229`34.\ 387216106321866"}, {"38", "4147.850721037314656640768843634720493699231624791471`36.\ 387216106321866"}, {"40", "4147.850721173524050235211536551709108022012701454338`38.\ 38721610632187"}, {"42", "4147.850721189333104132293720867813757838250182067571`40.\ 387216106321866"}, {"44", "4147.85072119207624366125682054932031268129112143399543445322`42.\ 38721610632187"}, {"46", "4147.850721192632159324787421588165717445070076121773496215113`\ 44.38721610632186"}, {"48", "2.0138053157455014715430579`11.931616353720106*^26"}, {"50", "4147.850721192653970227069254744531259514847986678658854459168`\ 48.387216106321866"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { 4147.84400911262787903619421868233732168351`28.387216106321826, 4147.85020676900216790826638636847029496421`30.387216106321866, 4147.85071597914657631528590153414105173806`32.387216106321866, 4147.850720342217179176002726697526783730857281252229`34.\ 387216106321866, 4147.850721037314656640768843634720493699231624791471`36.\ 387216106321866, 4147.850721173524050235211536551709108022012701454338`38.\ 38721610632187, 4147.850721189333104132293720867813757838250182067571`40.\ 387216106321866, 4147.85072119207624366125682054932031268129112143399543445322`42.\ 38721610632187, 4147.850721192632159324787421588165717445070076121773496215113`44.\ 38721610632186, 2.0138053157455014715430579`11.931616353720106*^26, 4147.850721192653970227069254744531259514847986678658854459168`48.\ 387216106321866}, TableHeadings -> {{30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50}, {"working", "result"}}]]], "Output", CellLabel->"Out[12]//TableForm="] }, Open ]], Cell[TextData[{ "Obviously, ", StyleBox["PrecisionGoal", "InlineInput", FontFamily->"Courier New", FontWeight->"Bold"], " 20 and ", StyleBox["WorkingPrecision", "InlineInput", FontFamily->"Courier New", FontWeight->"Bold"], " 40 is enough if we want the (absolute) precision of the result to be less \ than ", Cell[BoxData[ \(TraditionalForm\`10\^\(-6\)\)], FontColor->RGBColor[0, 0, 1]], ". But what happened when the ", StyleBox["WorkingPrecision", "InlineInput", FontFamily->"Courier New", FontWeight->"Bold"], " was 48!? Let us plot the solution for this case." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(sol = u /. \(NDSolve[{\(\(Derivative[4]\)[u]\)[t] \[Equal] SetPrecision[40^4, 48]*PosPart[u[t]] - SetPrecision[30^4, 48]*NegPart[u[t]], u[0] \[Equal] 0, \(\(Derivative[1]\)[u]\)[0] \[Equal] 0, \(\(Derivative[2]\)[u]\)[0] \[Equal] 1, \(\(Derivative[3]\)[u]\)[0] \[Equal] SetPrecision[\(-40.37441848177\), 48]}, u, {t, 0, 1}, PrecisionGoal \[Rule] 24, WorkingPrecision \[Rule] 48]\)[\([1]\)]\), "\n", \(\(Plot[sol[t], {t, 0, 1}, PlotRange \[Rule] {{0, 1}, {\(-0.01\), 0.01}}];\)\)}], "Input", CellLabel->"In[13]:="], Cell[BoxData[ RowBox[{\(NDSolve::"mxst"\), \(\(:\)\(\ \)\), "\<\"Maximum number of \ \\!\\(10000\\) steps reached at the point \\!\\(t\\) == \ \\!\\(\\*TagBox[\\\"0.\ 1589158757426683234635010245507764804659237836450724607007599818701`48.\\\", \ HoldForm]\\). \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"NDSolve::mxst\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[13]:="], Cell[BoxData[ TagBox[\(InterpolatingFunction[{{0, 0.1589158757426683234635010245507764804659237836450724607007599818\ 701`48. }}, "<>"]\), False, Editable->False]], "Output", CellLabel->"Out[13]="], Cell[BoxData[ RowBox[{\(InterpolatingFunction::"dmval"\), \(\(:\)\(\ \)\), "\<\"Input \ value \\!\\({0.1668924386666386`}\\) lies outside the range of data in the \ interpolating function. 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We can fix the problem using the ", StyleBox["StiffnessSwitching", "InlineInput", FontFamily->"Courier New", FontWeight->"Bold"], " method." }], "Text"], Cell[BoxData[ \(\(ValueAtOne[\[Mu]_, \[Nu]_, \[Delta]_, goal_, working_] := u[1] /. \(NDSolve[{\(\(Derivative[4]\)[u]\)[t] \[Equal] SetPrecision[\[Mu]^4, working]*PosPart[u[t]] - SetPrecision[\[Nu]^4, working]*NegPart[u[t]], u[0] \[Equal] 0, \(\(Derivative[1]\)[u]\)[0] \[Equal] 0, \(\(Derivative[2]\)[u]\)[0] \[Equal] 1, \(\(Derivative[3]\)[u]\)[0] \[Equal] SetPrecision[\[Delta], working]}, u, {t, 0, 1}, PrecisionGoal \[Rule] goal, WorkingPrecision \[Rule] working, Method \[Rule] {StiffnessSwitching}]\)[\([1]\)];\)\)], "Input", CellLabel->"In[15]:="], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[ Table[ValueAtOne[40, 30, \(-40.37441848177\), n, 2*n], {n, 15, 25}], TableHeadings \[Rule] {Table[ 2*n, {n, 15, 25}], {"\", "\"}}]\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ InterpretationBox[GridBox[{ {"30", "4147.8505391689472705880519726668579118069`28.38721610632187"}, {"32", "2354.62438059254047918615407099863768981411`30.38721608506277"}, {"34", "4073.37971376332793220102842337883033667399`32.38721610580085"}, {"36", "4147.850720494023771567460056185660633857629481469783`34.\ 387216106321866"}, {"38", "4147.833756630375885784607934016939141225055681419778`36.\ 38721610632176"}, {"40", "4147.850702963894458297945070097370343781294506129838`38.\ 38721610632187"}, {"42", "4144.690769953244960926007724027825195768205597388287`40.\ 38721610630028"}, {"44", "4147.61859372526944555661594669937064944536448098130905902749`42.\ 387216106320274"}, {"46", "4146.327059251268925446636579313365351921577833676776145384502`\ 44.38721610631137"}, {"48", "4147.683873484625938621052746662330203035864920202035081552255`\ 46.38721610631956"}, {"50", "4147.850721192654054829697493157758874651140413936663102122787`\ 48.387216106321866"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { 4147.8505391689472705880519726668579118069`28.38721610632187, 2354.62438059254047918615407099863768981411`30.38721608506277, 4073.37971376332793220102842337883033667399`32.38721610580085, 4147.850720494023771567460056185660633857629481469783`34.\ 387216106321866, 4147.833756630375885784607934016939141225055681419778`36.\ 38721610632176, 4147.850702963894458297945070097370343781294506129838`38.\ 38721610632187, 4144.690769953244960926007724027825195768205597388287`40.\ 38721610630028, 4147.61859372526944555661594669937064944536448098130905902749`42.\ 387216106320274, 4146.327059251268925446636579313365351921577833676776145384502`44.\ 38721610631137, 4147.683873484625938621052746662330203035864920202035081552255`46.\ 38721610631956, 4147.850721192654054829697493157758874651140413936663102122787`48.\ 387216106321866}, TableHeadings -> {{30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50}, {"working", "result"}}]]], "Output", CellLabel->"Out[16]//TableForm="] }, Open ]], Cell[TextData[{ "The equation was always solved on the whole ", Cell[BoxData[ FormBox[ StyleBox[\([0, 1]\), FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], ". However, the results are obviously not consistent." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The second problem: jumping nonlinearity", "Subsection"], Cell[TextData[{ "The reason why we got such nonconsistent results is that our ", Cell[BoxData[ FormBox[ StyleBox["f", FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], " (the right-hand side) is the jumping nonlinearity which is not \ differentiable at ", Cell[BoxData[ FormBox[ StyleBox["0", FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], ". It brings unpredictable errors into the solution as it changes its sign. \ Consequently, we must stop the integration whenever the solution attains the \ zero value, and restart the integration. This is easily accomplished with the \ ", StyleBox["EventLocator", "InlineInput", Background->GrayLevel[1]], " method, new in ", StyleBox["Mathematica", FontSlant->"Italic"], " 5.1." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(state = First[NDSolve`ProcessEquations[{\(\(Derivative[4]\)[u]\)[t] \[Equal] SetPrecision[40^4, 130]*PosPart[u[t]] - SetPrecision[30^4, 130]*NegPart[u[t]], u[0] \[Equal] 0, \(\(Derivative[1]\)[u]\)[0] \[Equal] 0, \(\(Derivative[2]\)[u]\)[0] \[Equal] 1, \(\(Derivative[3]\)[u]\)[0] \[Equal] SetPrecision[\(-40.37441848177\), 130]}, u, t, Method \[Rule] {EventLocator, "\" \[Rule] u[t], Method \[Rule] StiffnessSwitching}, WorkingPrecision \[Rule] 130, AccuracyGoal \[Rule] 30, PrecisionGoal \[Rule] 30, MaxSteps \[Rule] Infinity]];\)\), "\n", \(\(result = Reap[While[Sow[state["\"["\"]]] < 1, state = NDSolve`ReinitializeVector[state, state["\"["\"]], state["\"["\"]]]; \ \[IndentingNewLine]NDSolve`Iterate[state, 1];]; \[IndentingNewLine]state["\"["\\ "]]];\)\), "\n", \(SetPrecision[\(result[\([1]\)]\)[\([1]\)], 30]\)}], "Input", CellLabel->"In[17]:="], Cell[BoxData[ \(4147.85072119265408024709603968172472572635`30. \)], "Output", CellLabel->"Out[19]="] }, Open ]], Cell[TextData[{ "The used precision appears to be sufficient for any ", Cell[BoxData[ \(TraditionalForm\`\[Mu], \[Nu] \[Element] \([0, 50]\)\)], FontColor->RGBColor[0, 0, 1]], ". Using the ", StyleBox["EventLocator", "InlineInput", Background->GrayLevel[1]], " is not appropriate when ", Cell[BoxData[ \(TraditionalForm\`\[Mu] = \[Nu]\)], FontColor->RGBColor[0, 0, 1]], ", and so we finally define" }], "Text"], Cell[BoxData[ \(\(ValueAtOne[order_, \[Mu]_, \[Nu]_, \[Delta]_, goal_, working_] := If[Abs[\[Mu] - \[Nu]] < 10^\((\(-10\))\), SetPrecision[\(\(Derivative[order]\)[u]\)[ 1] /. \(\(NDSolve[{\(\(Derivative[4]\)[u]\)[t] \[Equal] SetPrecision[\[Mu]^4, working]*PosPart[u[t]] - SetPrecision[\[Nu]^4, working]*NegPart[u[t]], u[0] \[Equal] 0, \(\(Derivative[1]\)[u]\)[0] \[Equal] 0, \(\(Derivative[2]\)[u]\)[0] \[Equal] 1, \(\(Derivative[3]\)[u]\)[0] \[Equal] SetPrecision[\[Delta], working]}, u, {t, 0, 1}, Method \[Rule] StiffnessSwitching, WorkingPrecision \[Rule] working, AccuracyGoal \[Rule] goal, PrecisionGoal \[Rule] goal, MaxSteps \[Rule] Infinity]\)[\([1]\)]\)[\([1]\)], goal], Module[{state, result}, state = First[ NDSolve`ProcessEquations[{\(\(Derivative[4]\)[u]\)[ t] \[Equal] SetPrecision[\[Mu]^4, working]*PosPart[u[t]] - SetPrecision[\[Nu]^4, working]*NegPart[u[t]], u[0] \[Equal] 0, \(\(Derivative[1]\)[u]\)[0] \[Equal] 0, \(\(Derivative[2]\)[u]\)[0] \[Equal] 1, \(\(Derivative[3]\)[u]\)[0] \[Equal] SetPrecision[\[Delta], working]}, u, t, Method \[Rule] {EventLocator, "\" \[Rule] u[t], Method \[Rule] StiffnessSwitching}, WorkingPrecision \[Rule] working, AccuracyGoal \[Rule] goal, PrecisionGoal \[Rule] goal, MaxSteps \[Rule] Infinity]]; \[IndentingNewLine]result = Reap[While[Sow[state["\"["\"]]] < 1, state = NDSolve`ReinitializeVector[state, state["\"["\"]], state["\"["\"]]]; \ \[IndentingNewLine]NDSolve`Iterate[state, 1];]; \[IndentingNewLine]state["\"["\<\ Forward\>"]]]; \[IndentingNewLine]SetPrecision[\(result[\([1]\)]\)[\([order + 1]\)], goal]]];\)\)], "Input", CellLabel->"In[20]:="], Cell[CellGroupData[{ Cell[BoxData[ \(ValueAtOne[0, 40, 30, \(-40.37441848177\), 30, 130] // Timing\)], "Input", CellLabel->"In[21]:="], Cell[BoxData[ \({2.7639999999999816`\ Second, 4147.85072119265408024709603968172472572635`30. }\)], "Output", CellLabel->"Out[21]="] }, Open ]], Cell[TextData[{ "Having the function ", Cell[BoxData[ \(TraditionalForm\`V\)], FontColor->RGBColor[0, 0, 1]], " implemented, we can easily compute ", Cell[BoxData[ \(TraditionalForm\`\(\[Delta]\_0\)(\[Mu], \[Nu])\)], FontColor->RGBColor[0, 0, 1]], " and ", Cell[BoxData[ \(TraditionalForm\`D(\[Mu], \[Nu])\)], FontColor->RGBColor[0, 0, 1]], "." }], "Text"], Cell[BoxData[{ \(\(LookForNeg[fction_] := NestWhile[#1*2 &, \(-1\), fction[#1] > 0 &];\)\), "\n", \(\(InitInt[x_, prec_] := SetPrecision[If[x \[Equal] \(-1\), {\(-1\), 0}, {x, x/2}], prec];\)\), "\n", \(\(Halving[{l_, r_, f_}] := If[f[\((l + r)\)/2] < 0, {\((l + r)\)/2, r, f}, {l, \((l + r)\)/2, f}];\)\), "\n", \(\(MyFindRoot[fction_, goal_, working_] := \(NestWhile[Halving, Append[InitInt[LookForNeg[fction], working], fction], #1[\([2]\)] - #1[\([1]\)] > 10^\((\(-goal\))\) &]\)[\([1]\)];\)\), "\n", \(\(DerZero[\[Mu]_, \[Nu]_] := ValueAtOne[1, \[Mu], \[Nu], MyFindRoot[ValueAtOne[0, \[Mu], \[Nu], #1, 30, 130] &, 25, 35], 30, 130];\)\)}], "Input", CellLabel->"In[22]:="], Cell[CellGroupData[{ Cell[BoxData[{ \(MyFindRoot[ValueAtOne[0, 40, 30, #, 30, 130] &, 25, 35]\), "\n", \(DerZero[40, 30] // Timing\)}], "Input", CellLabel->"In[27]:="], Cell[BoxData[ \(\(-40.374418580317266981581822140568217210308517642336`35.\ 00000000000001\)\)], "Output", CellLabel->"Out[27]="], Cell[BoxData[ \({297.337`\ Second, \ \(-0.021774824527892359583448133522141301091159936739`30. \)}\)], "Output", CellLabel->"Out[28]="] }, Open ]], Cell[TextData[{ "Finally, the Fu\[CHacek]\[IAcute]k spectrum may be figured by the ", StyleBox["ContourPlot", "InlineInput"], " function as follows\n", StyleBox["ContourPlot[DerZero[\[Mu], \[Nu]]*DerZero[\[Mu], \[Nu]],\n \ {\[Mu], 0, 50}, {\[Nu], 0, 50}, ContourShading -> None, \n Contours -> {0}, \ PlotPoints -> 200]", FontFamily->"Courier New", FontWeight->"Bold"], "\nBut it would take months..." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Optimization", "Section"], Cell[TextData[{ "The ", StyleBox["ContourPlot", "InlineInput", FontFamily->"Courier New", FontWeight->"Bold"], " method evaluates the function uniformly at all ", Cell[BoxData[ \(TraditionalForm\`n\[Times]n\)], FontColor->RGBColor[0, 0, 1]], " points. We develop an adaptive iterative algorithm for figuring the zero \ contour of a function. 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