Re: how make function of solution by NDSolve depending on parameter?

*To*: mathgroup at smc.vnet.net*Subject*: [mg75137] Re: how make function of solution by NDSolve depending on parameter?*From*: Szabolcs <szhorvat at gmail.com>*Date*: Wed, 18 Apr 2007 05:02:53 -0400 (EDT)*References*: <f014kb$94o$1@smc.vnet.net>

On Apr 17, 2:33 am, Murray Eisenberg <murray at math.umass.edu> wrote: > I have an initial-value problem that depends upon a parameter k. I want > to feed the result from NDSolve for it into a function of k and then > operate upon that function, say to find a minimum with respect to k. > > As a toy example: > > soln[k_]:=NDSolve[{y'[t] ==(k y[t]+Exp[-t])/(1+y[t]), > y[0]==1},y[t],{t,0,1}] > > I want to do something like this: > > NMinimize[Evaluate[y[t] /. soln[k]] /. t -> 0.5, {k, 0.1, 1}] > > That generates errors about non-numerical values. Yet I can get a > result from, for example: > > Table[Evaluate[y[t] /. soln[k]] /. t -> 0.5, {k, 0, 1}] > > So how can I create the function of k I want to feed into NMinimize? I > presume the issue is when NDSolve gets called, but I'm not sure how to > resolve this issue. > > -- > 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 fax 413 545-1801 > Amherst, MA 01003-9305 Hi! Unlike Table, NMinimize does not hold its argument until a numerical value is substituted for k. So you have to make sure that the function you pass to it is not evaluated for non-numerical arguments. In[1]:= soln[k_] := NDSolve[{y'[t] == (k y[t]+Exp[-t])/(1+y[t]), y[0]==1}, y, {t,0,1}] In[2]:= fun[k_?NumericQ] := y[0.5] /. First[soln[k]] In[3]:= NMinimize[fun[k], {k, 0.1, 1}] Out[3]= {1.2078762005934287*^-13, {k -> -5.023394217556404*^12}} In this particular example no minimum exists but the same approach should be usable for your real problem. Also note that I changed y[t] to y in NDSolve to avoid the double ReplaceAll. Szabolcs