Re: how use NDSolve with an ODE having parameters
- To: mathgroup at smc.vnet.net
- Subject: [mg39708] Re: [mg39691] how use NDSolve with an ODE having parameters
- From: Reza Malek-Madani <research at usna.edu>
- Date: Mon, 3 Mar 2003 04:25:36 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Dear Murray: Is this along the lines of what you're looking for?
f[omega_, a_, b_] :=
NDSolve[{x'[t] == y[t], y'[t] == -Sin[x[t]] - 0.1*y[t] + Cos[omega
t], x[0] == a, y[0] == b}, {x, y}, {t, 0, 10}]
sol = f[6, 1, 1];
newx[t_] = First[x[t] /. sol[[1]]];
Print[NIntegrate[(newx[t])^2, {t, 0, 10}]];
g[omega_] := Block[{sol}, sol = f[omega, 1, 1]; newx[t_] = x[t] /. sol;
First[newx[3]]]
Plot[g[omega], {omega, 0.1, 3}]
This solves the usual pendulum equation with omega and initial conditions
as parameters, computes the L^2 norm of x for a specific set of
parameters, and plots the x value at t=3 as omega varies.
Rez
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On Sat, 1 Mar 2003, Murray Eisenberg wrote:
> This is a simplification of a question asked by a colleage. He wants to
> use as the model function argument to NonlinearRegress (from
> Statistics`NonlinearFit1) a solution of an initial-value problem for a
> differential equation, where the differential equation depends on a
> parameter.
>
> The catch is that the differential equation cannot be solved explicitly,
> so he has to resort to solving the initial-value problem by means of
> NDSolve. Of course, NDSolve will not do anything if the differential
> equation involves symbolic parameters. Thus the IDEA of what he wants
> to do is to use the "resulting function" from something like
>
> y[t]/.First@NDSolve[{y'[t] == a y[t] + b, y[0] == 1.}, y[t], {t, 0., 1.}]
>
> -- where two parameters a and b are involved -- as the model. Of course
> if NDSolve above is changed to DSolve, no difficulty. But in the ACTUAL
> problem at issue, with a much more complicated differential equation,
> DSolve does nothing.
>
> Is there some way to make this work?
>
> There are evidently two difficulties:
>
> (1) How to deal with NDSolve when the differential equation involves
> parameters (perhaps there's something regarding use of Hold that will
> help?); and
>
> (2) For each pair of particular values of the parameters, the result
> from NDSolve is an InterpolatingPolynomial object and NOT the sort of
> "expression in the variable" that seems to be required for the model
> argument to NonlinearRegress. How should the InterpolatingPolynomial
> object be massaged to allow it to be used as an ordinary expression in
> the variable?
>
> --
> Murray Eisenberg murray at math.umass.edu
> Mathematics & Statistics Dept.
> Lederle Graduate Research Tower phone 413 549-1020 (H)
> University of Massachusetts 413 545-2859 (W)
> 710 North Pleasant Street
> Amherst, MA 01375
>
>