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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  ]]
}
]