Re: Help solving PDE system.

Thanks to both of you, Peter and Olof. I was quite sure I can solve this 
as two integrals, but I need a confirmation. Anyway, I have a further 
question. Using the fact that f,k->0 when x->INF, I can write:

int_x^INF f_x dx = f(INF) - f(x) = -f(x)

Using this trivial result my integral equations become:

f = int_x^INF g*h_x dx

k = y int_x^INF (g_x*h_y - g_y*h_x) dx - f

As far as I can understand the two methods are analitically equivalent, 
but I can't say if the numerical solutions are more or less stable and 
accurate.

In order to improve stability I've thought to re-map the open interval 
0<=x<INF into a closed interval (e.g. x = tan(u/2) with 0<= u <= Pi). 
The integration over the new variable should be:

f = int_u^Pi |J| g*h_u du

k = y int_u^Pi |J|(g_u*h_y - g_y*h_u) du - f

where |J| is the trasformation jacobian.

Do you think I can gain any accuracy and stability in this way? And 
where can I find any good reference to numerical integration over a grid (
Numerical Recipes, and other text books I read unfit my needs)??

Thanks in advance,
        Andrea.

In <[EMAIL PROTECTED]> Olof wrote:
> Hi,
> 
> 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