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This is my first post...

I am trying to make a code to numerically solve a problem, which is a heat conduction problem (temperature) in a moving slab (in y-z plane) with a source term in it:

A(dT/dy)=(d2T/dy2 + d2T/dz2) + B

dT/dy=0 at y=0, T=given at y=0, boundary conditions along y at z=0 and W are described by some simple functions. Say the width of the slab is W and the length is L.

I think this is called "convection (or advection)-diffusion" problem.

Because the way boundary conditions are given, to me (I don't know much about numerical methods), it looks like a "shooting method" can handle this problem. Or maybe the Crank-Nicolson method can be used, while taking y in this problem like t in a conventional C-N problem. Well, it seems like I cannot numerically solve this problem using these methods. The solution blows up.

Maybe I should use a method to handle a typical elliptic equation problem. But in this problem, one side is open so that I cannot specify the boundary condition at one end of y=L (I can specify both T and dT/dy at y=0, instead).

So here I am. I am stuck. Does anybody can give me some advices? In summary, my questions are:

(1) can a shooting method (like Runge-Kutta or whatever) be used to solve this kind of PDE problems?

(2) if I have to apply a method for a boundary value problem, how can I do to this problem that does not have a specified boundary condition at one end?

Thank you very much!