I need to solve the Laplace equation in two dimensions, $\nabla ^2 \phi(x,y) = 0$, $0<x<a, 0<y<b$ subject to $4$ boundary conditions:
$$\phi(0,y) = 0$$ $$\phi(a,y) = f(y)$$ $$\phi(x,b) = 0$$ $$\phi(x,0) = h(x)$$
Now, I have tried to really solve the differential equation, using separation of variable. Thinking that maybe I would need to use Fourier series. But the things got really messy, and so I have given up.
I am thinking that maybe the solutions have something to do with writing $\phi$ as some combination of the function $f(y)$ and $h(x)$, but yet I am not finding anything.
So, how to solve it?