1) Can this Laplace equation, with its non trivial boundaries (on a rectangular domain), be solved analytically?
$$\frac{d^2U}{dx^2}+\frac{d^2U}{dy^2}=0$$ $$U_x(0,y)=0\quad,\quad U_x(a,y)=f(y)$$ $$U_y(x,0)=0\quad,\quad U(x,b)=0$$
$f(y)$ is a function described as a vector (discrete values) which I cant describe as a continuous function.
2) If the answer to (1) is no, is there an easy procedure to solve it numerically (I don't have much experience with numerics or time to study it deeply)
The difficulty arises because $b>>>a$ $$b=0.1\quad,\quad a=1e-5$$
Thanks.