       Re: how use NDSolve with an ODE having parameters

• To: mathgroup at smc.vnet.net
• Subject: [mg39707] Re: [mg39691] how use NDSolve with an ODE having parameters
• From: Selwyn Hollis <selwynh at earthlink.net>
• Date: Mon, 3 Mar 2003 04:25:24 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Murray,

I can't get NonlinearRegress to work either, but using FindMinimum on
the least squares error seems to work. Here's the general idea:

model[x_, a_,  b_] := (y /. First@NDSolve[{ your DE, y == initval},
y, {t, 0, 2}])[x]

LSE[a_, b_] := Sum[(data[[i, 2]] - model[data[[i, 1]], a, b])^2, {i,
Length[data]}]

FindMinimum[LSE[a, b],  {a, a1, a2},  {b, b1, b2}]

---
Selwyn Hollis

On Saturday, March 1, 2003, at 10:05  PM, 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 == 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
>
>

```

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