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

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Bart Goddard

Posts: 1,565
Registered: 12/6/04
Re: Numerical ODEs
Posted: May 16, 2013 4:22 PM
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José Carlos Santos <jcsantos@fc.up.pt> wrote in news:avkirmF3aj0U1
@mid.individual.net:

> On 15-05-2013 21:15, Bart Goddard wrote:
>

>> 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.

>
> This works indeed for the ODE f'(x) = 2 sqrt(f(x)), but, more
> generally, I am interested in ODEs of the type f'(x) = g(f(x)), under
> the initial condition f(0) = 0 and where g(0) = 0. Then, all I get is:
>
> f''(x) = g'(f(x))*g(f(x))
>
> and I don't see how to proceed from here.


Me either, really, but my thought was you could plug in 0 to
get f''(0) = g'(0)*g(0) = a. (Where a might be a limit here.)
If we're lucky and a is not 0, then
we assume y = (a/2)x^2 is a reasonable approximation to f
around x=0. Then run the numerical method, using stepsize h,
and assume that the first three points on the solution are (0,0),
(h, ah^2/2) and (2h, a4h^2/2), which should be enough to seed
a RK method.

If we're unlucky and a = 0, then take another derivative, plug in 0
and if the 3rd derivative is not 0, approximate by
bx^3/6 and so on. Some derivative has to be nonzero, and that should
give us a bump off of the f(x)=0 solution.(?)

B.



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