Re: Delay Differential Equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg8823] Re: Delay Differential Equations*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Mon, 29 Sep 1997 02:39:48 -0400*Organization*: University of Western Australia*Sender*: owner-wri-mathgroup at wolfram.com

Matthias Weiss wrote: > Can I solve DDE of the form > > x'(t) = F(t, x(t), x(t-a)) > > with NDSolve? It appears it is not able to refer back to its own > solution while it is still in progress. See The Mathematica Journal 5(3): 25 and 6(1): 25-26. I have appended a Notebook below. Cheers, Paul ____________________________________________________________________ Paul Abbott Phone: +61-8-9380-2734 Department of Physics Fax: +61-8-9380-1014 The University of Western Australia Nedlands WA 6907 mailto:paul at physics.uwa.edu.au AUSTRALIA http://www.pd.uwa.edu.au/~paul God IS a weakly left-handed dice player ____________________________________________________________________ Notebook[{ Cell[CellGroupData[{ Cell["Delay Differential Equation", "Section"], Cell[TextData[{ "Differential equations involving delay such as ", Cell[BoxData[ \(TraditionalForm \`\(q\^\[Prime]\)[t] \[Equal] \(-q[t]\) + q[t - 2]\)]], ", with the initial conditions ", Cell[BoxData[ \(TraditionalForm\`q[0] \[Equal] 1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`q[t_\ /; \ t\ < \ 0] = 0\)]], ", can be solved iteratively in an elegant way using ", Cell[BoxData[ FormBox[ StyleBox["FixedPoint", "Input"], TraditionalForm]]], ". After entering" }], "Text", CellTags->{"FixedPoint", "delay differential equation"}], Cell[BoxData[ \(TraditionalForm \`Off(NDSolve::"\<ndnum\>", InterpolatingFunction::"\<dmval\>")\)], "Input", CellTags->"Off"], Cell[TextData[{ "to suppress the warning messages that arising during intermediate \ computations prior to reaching the fixed point, we use ", StyleBox["FixedPoint", "Input"], " to solve the equation iteratively:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"\[ScriptQ]", "=", RowBox[{"FixedPoint", "(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"NDSolve", "(", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["q", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "==", \(If(t < 2, 0, #1 (\(t - 2\))) - q(t)\)}], ",", \(q(0) == 1\)}], "}"}], ",", "q", ",", \({t, 0, 7}\)}], ")"}], ")"}], "\[LeftDoubleBracket]", \(1, 1, 2\), "\[RightDoubleBracket]"}], "&"}], ",", "0"}], ")"}]}], TraditionalForm]], "Input"], Cell[BoxData[ FormBox[ TagBox[ RowBox[{"InterpolatingFunction", "(", RowBox[{ RowBox[{"(", GridBox[{ {"0.`", "7.`"} }, ColumnAlignments->{Decimal}], ")"}], ",", "\<\"<>\"\>"}], ")"}], False, Editable->False], TraditionalForm]], "Output"] }, Open ]], Cell[BoxData[ \(TraditionalForm \`\(Plot(\[ScriptQ](t), {t, 0, 7}, PlotRange \[Rule] {All, {0, 1}}); \)\)], "Input"], Cell[TextData[{ "Here we use ", Cell[BoxData[ FormBox[ StyleBox[\(PlotRange \[Rule] {All, {0, 1}}\), "Input"], TraditionalForm]]], " to control the plot range in the ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " directions separately." }], "Text", CellTags->"PlotRange"], Cell["It is easy to check how well the equation is satisfied:", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Plot", "(", RowBox[{ RowBox[{"Evaluate", "(", RowBox[{ \(-\(If(t < 2, 0, \[ScriptQ](t - 2))\)\), "+", \(\[ScriptQ](t)\), "+", RowBox[{ SuperscriptBox["\[ScriptQ]", "\[Prime]", MultilineFunction->None], "(", "t", ")"}]}], ")"}], ",", \({t, 0, 7}\), ",", \(PlotRange \[Rule] {\(-0.001\), 0.001}\)}], ")"}], ";"}], TraditionalForm]], "Input"], Cell[TextData[{ "We can use ", Cell[BoxData[ FormBox[ StyleBox["Nest", "Input"], TraditionalForm]]], " instead of ", Cell[BoxData[ FormBox[ StyleBox["FixedPoint", "Input"], TraditionalForm]]], " if we are sure of the number of iterations needed." }], "Text"] }, Open ]] } ]