Re: Help solving PDE system.

Hi,

[EMAIL PROTECTED] wrote in message news:<[EMAIL PROTECTED]>...
> Hi,
> I need a little help solving 2 systems of PDE numerically. The first 
> system is the following:
> 
> Df/Dx = - g(x,y) * Dh/Dx
> 
> Df/Dy - (k(x,y) - f(x,y)) / y = - g(x,y) * Dh/Dy
> 
> where x,y > 0, D/Dx and D/Dy are partial derivative, Dh/Dx, Dh/Dy and g(
> x,y) are known on a
> precomputed grid, f(x,y) and k(x,y) are the unknown functions and I want 
> to know their value on
> the grid. Boundary conditions tell me that f,k = 0 when x,y->Infinity.

This is in fact just two integrals, not a PDE (assuming smooth
solutions, i.e. smooth g, h). If you use the fact that f_xy=f_yx under
these assumptions, then you can rewrite the system as:

f_x = - g*h_x,

k_x = y*(g_x*h_y - g_y*h_x) - g*h_x.

The right hand side contains no f or g, so you
just have

f = f0(y) - int_0^x g*h_x dx,

k = k0(y) + int_0^x y*(g_x*h_y - g_y*h_x) - g*h_x dx.

Use a numerical integration scheme such as
the trapezoidal rule to compute the integrals
for each y, to a rather large x. Since 
f,k->0 as x->oo, the value of the integrals for
large x give you f0 and k0.

Olof
 
> I solve the system in quite naive way, integrating over x the first 
> equation with a trapezoidal
> rule and computing Df/Dy numerically as described in Numerical Recipes (
> par. 5.7). But the
> results are not enthusiastic. I've looked at many books about PDE but 
> I've not found any hint to
> solve my problem. Do you have any suggestions?? Any references?
> 
> Thanks in advance,
> Andrea.