I am looking for a general solution so you may assume F(x,y) be any two-dimentional function. Does the solution rely very much on the form of F(x,y)? Is there a general way of solving such problem?
But you may assume F(x,y)=x^2+y^2
Thanks very much for your help!
If F(x, y) had the seperable form F(x, y) = a(x) b(y), for example , then you could assume a solution of the form w = X(x) Y(y) and it's blue sky.
For things are less straightforward.
I haven't really had a close look - a suggestion that might work is to use a transform technique eg Fourier or Laplace. Although the success of this might depend on the given boundary conditions (which have not been mentioned so far).
If I have the time, I will try to take a closer look if no-one else does.