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

• To: mathgroup at smc.vnet.net
• Subject: [mg75115] Re: [mg75096] how make function of solution by NDSolve depending on parameter?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 18 Apr 2007 04:51:36 -0400 (EDT)
• References: <200704170020.UAA08810@smc.vnet.net>

```
On 17 Apr 2007, at 09:20, Murray Eisenberg 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

The syntax you are trying to use for NMinimize is quite wrong; have
you looked at the documentation:

?NMinimize

You could do this:

f[k_?NumericQ] := Block[{y}, First[y /. NDSolve[{y'[t] == (k y[t] +
Exp[-t])/(1 + y[t]),y[0] == 1}, y, {t, 0, 1}]]]

In[2]:=
NMinimize[{f[k][0.5], 0 <= k <= 1}, k]

Out[2]=
{1.1879073663327735, {k -> 1.5474783566024743*^-8}}

Andrzej Kozlowski

```

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