Solving partial differential equation.

Question

I want to solve this kind of equation: $${A∂f(x, t)\over ∂t} = {Bd^2f(x, t)\over \mathrm dx^2}$$ I tried to solve this equation in this way: $f(x, t) = a(x)b(t)$. But can I solve this more general?

Answer 1

If $A,B$ are just (nonzero) constants (i.e., this is just a heat equation), then yes it can be solved in other ways. For example, the Fourier transform would do the trick.

If $A,B$ aren't constants, then I'd need some more info on them in order to give you an answer.

Edit: If you're aim is to solve the BVP, then the approach you mention (separation of variables) is probably the way to go. In some domains (including an interval as in your example) you can use a Green function, but you'll probably have to use separation of variables to solve for it, so I'm not sure you're going to escape that approach in the long run.