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From: Daniel Lichtblau on 21 May 2010 06:44
susy wrote:
> Hallo,
>
> I need to solve the following differential equation:
>
> (1/4 r^2 + 2 Sin[F[r]]^2) F''[r] + 1/2 r F'[r] +
> Sin[2 F[r]] (F'[r])^2 - 1/4 Sin[2 F[r]] - (Sin[F[r]]^2 Sin[2 F[r]])/r^2 == 0
>
> with boundary values:
> F[0]==Pi, F[Infinity]==0.
>
> I tried NDSolve, but failed to get a solution.
> How can I solve that equation?
>
> Best regards,
> susy
>

I had some luck by changing to a finite interval via r->1/(1+r). But I
still needed to scoot in a bit from the origin (corresponding to
infinity in the original coordinate).

new = (1/4 r^2 + 2 Sin[ff[r]]^2) D[ff[r], {r, 2}] +
1/2 r D[ff[r], {r, 1}] + Sin[2 ff[r]] (D[ff[r], {r, 1}])^2 -
1/4 Sin[2 ff[r]] - (Sin[ff[r]]^2 Sin[2 ff[r]])/r^2 /.
Derivative[j_][ff][r] :> Derivative[j][ff][r]*D[1/(1 + r), {r, j}];

In[125]:= eps = .0001;

In[126]:= gg =
ff[r] /. First[
NDSolve[{new == 0, ff[1 - eps] == Pi, ff[eps] == 0},
ff[r], {r, 1 - eps, eps}]];

During evaluation of In[126]:= FindRoot::sszero: The step size in the
search has become less than the tolerance prescribed by the
PrecisionGoal option, but the function value is still greater than the
tolerance prescribed by the AccuracyGoal option. >>

During evaluation of In[126]:= NDSolve::berr: There are significant
errors {-7.77603*10^-8,0.000328812} in the boundary value residuals.
Returning the best solution found. >>

(*Ignoring the warning messages, the plot seems reasonable. *)

hh[s_?NumberQ] := gg /. r -> 1/(s + 1)

Plot[hh[s], {s, 1/(1 - eps) - 1, 1/eps - 1}]

Daniel Lichtblau
Wolfram Research

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