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[0] == 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[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 > >