Date: May 13, 2013 11:30 AM
Author: Jose Carlos Santos
Subject: Re: Numerical ODEs
On 13/05/2013 14:53, dullrich@sprynet.com wrote:

>> This question is perhaps too vague to have a meaningful answer, but

>> here it goes.

>>

>> In what follows, I am only interested in functions defined in some

>> interval of the type [0,a], with a > 0.

>>

>> Suppose that I want to solve numerically the ODE f'(x) = 2*sqrt(f(x)),

>> under the condition f(0) = 0. Of course, the null function is a

>> solution of this ODE. The problem is that I am not interested in that

>> solution; the solution that I am after is f(x) = x^2.

>>

>> For my purposes, numerical solutions are enough, but if I try to solve

>> numerically an ODE of the type f'(x) = g(f(x)) (with g(0) = 0) and

>> f(0) = 0, what I get is the null function. So far, my way of dealing

>> with this has been to solve numerically the ODE f'(x) = g(f(x)) and

>> f(0) = k, where _k_ is positive but very small and to hope that the

>> solution that I get is very close to the solution of the ODE that I am

>> interested in (that is, the one with k = 0). Do you know a better way

>> of dealing with this problem?

>

> I suspect you may never get the sort of answer you want here.

> There are those numerical methods out there. The proof that

> they work typically depends on certain hypotheses, which

> hypotheses typical entail the existence and uniqueness of

> the solution. You have a DE that does not satisfy those

> hypotheses, so you shouldn't expect those methods to

> give the solution you want.

You are probably right. :(

Best regards,

Jose Carlos Santos