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Topic: Numerical ODEs
Replies: 10   Last Post: May 16, 2013 4:22 PM

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Bart Goddard

Posts: 1,707
Registered: 12/6/04
Re: Numerical ODEs
Posted: May 15, 2013 4:15 PM
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Pretty much out of my field, (I'll do anything to avoid grading
final exams) but I notice that if you take the
derivative of both side of the DE and substitute, you discover
that f''(x)=2. As I understand, Runge-Kutta approximates the
second derivative and self-adjusts itself to account.

So it seems like we could cook up a numerical method which took
into account the information from the second derivative. This
would get you off the 0 solution. In this case, the second
derivative is constant, but in a more general case, we can always
find the value of the second derivative at x=0. Then assume that
the solution is the corresponding parabola to generate two more
points and then use a Runge-Kutta method from there.

José Carlos Santos <jcsantos@fc.up.pt> wrote in news:av9p7dFl84qU1

> Hi all,
> 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?
> Best regards,
> Jose Carlos Santos

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