Re: NDSolve with delayed terms in equations
- To: mathgroup@smc.vnet.net
- Subject: [mg11854] Re: NDSolve with delayed terms in equations
- From: Paul Abbott <paul@physics.uwa.edu.au>
- Date: Fri, 3 Apr 1998 03:45:18 -0500
- Organization: University of Western Australia
- References: <6fsj92$gi8@smc.vnet.net>
Alban P Tsui wrote: > I have a set of differential equations. Usually I can solve it with > NDSolve with no problem. However, I do not know how to solve something > like this, say > > x'[t]==y[t]^2+x[t-a] > y'[t]==y[t]x[t] > > with the usuaual intial conditions and a is real and positive. > > x[t-a] is a delayed term. > > What is the best way to tackle this with NDSolve? See The Mathematica Journal 5(3): 25 and 6(1): 25-26. I have appended a Notebook below. In addition, Scott Herod <Scott.Herod@Colorado.EDU> has developed a delay differential equation package which I have attached (hopefully attachments sent to comp.soft-sys.math.mathematica now work ok). 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@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[{ "See The ", StyleBox["Mathematica", FontSlant->"Italic"], " Journal ", StyleBox["5", FontWeight->"Bold"], "(3): 25 and ", StyleBox["6", FontWeight->"Bold"], "(1): 25-26." }], "Text"], Cell[TextData[{ "Differential equations involving delay such as ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["q", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "==", \(q(t - 2) - q(t)\)}], TraditionalForm]]], ", with the initial conditions ", Cell[BoxData[ \(TraditionalForm\`q(0) == 1\)]], " 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 ", Cell[BoxData[ FormBox[ StyleBox["FixedPoint", "Input"], TraditionalForm]]], " to solve the equation iteratively:" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"r", "=", RowBox[{"FixedPoint", "[", 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", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(Plot[r(x), {x, 0, 7}, PlotRange \[Rule] {All, {0, 1}}]; \)\)], "Input", AspectRatioFixed->True], 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[{\(-\(If(t < 2, 0, r(t - 2))\)\), "+", \(r(t)\), "+", RowBox[{ SuperscriptBox["r", "\[Prime]", MultilineFunction->None], "(", "t", ")"}]}], ")"}], ",", \({t, 0, 7}\), ",", \(PlotRange \[Rule] {\(-0.0005\), 0.0005}\)}], ")"}], ";"}], 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 ]] } ]
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