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

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Roland Franzius

Posts: 455
Registered: 12/7/04
Re: Numerical ODEs
Posted: May 13, 2013 11:54 PM
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Am 13.05.2013 17:28, schrieb José Carlos Santos:
> On 12/05/2013 21:32, Peter Percival 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

>>
>> What method are you using? I'm just being curious, I don't know a
>> solution to your problem.

>
> I use Mathematica and, more specifically, the NDSolve command. I suppose
> that the default method that it uses is the Euler method.


Mathematica methods can be discussed in the moderated group

comp.soft-sys.math.mathematica


Mathematica yields general solutions of

DSolve[ f'[x]==2 Sqrt[f[x]], f[x], x]

You can use general solutions (of an approximation to the original
equation ) at branch points or non-Lipschitz points of ( x0, f(x0) ) to
get a first guess of a nonsingular start value (x0+h, f(x0+h) ) as data
for NDSolve. The final numerical solution should always be checked by a
Plot of {f'[x]- 2 Sqrt[f[x]]}.

Mathematica is able to DSolve or NDSolve using the substitution method I
suggested to remove the branch point

NDSolve[{ D[g[x]^2],x]==2 Sqrt[g[x]^2], g[x]==0},g,{x,0,1}]

--

Roland Franzius



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