how use NDSolve with an ODE having parameters

• To: mathgroup at smc.vnet.net
• Subject: [mg39691] how use NDSolve with an ODE having parameters
• From: Murray Eisenberg <murraye at attbi.com>
• Date: Sat, 1 Mar 2003 22:05:27 -0500 (EST)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• Sender: owner-wri-mathgroup at wolfram.com

```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

```

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