2D Laplace's equation with mixed boundary conditions

1. Mar 30, 2016

refind

This isn't homework but could be labeled "textbook style" so I'm posting it here.
1. The problem statement, all variables and given/known data
I'm trying to solve

$$\frac{\partial^2 u} {\partial x^2} +\frac{\partial^2 u} {\partial y^2}=0$$

on the domain $x \in [-\infty,\infty], y\in[0,1]$ with the following mixed boundary conditions:

for $y=0: u(x,0)=0 \$if $x\in[-a,a]$ and $\partial u/ \partial y=0 \$if $|x|>a$

and very similarly for y=1:

$y=1: u(x,1)=1 \$if $x\in[-a,a]$ and $\partial u/ \partial y=0 \$if $|x|>a$

So I have Dirichlet and Neumann conditions for different segments of the boundary.
If the infinite domain $x \in [-\infty,\infty]$ poses a challenge, I am okay with changing the problem to be on a finite rectangle $x \in [-L,L]$ as long as $L>>1$.

The boundary conditions for x are simply that it's symmetric about x=0 (so $\partial u/ \partial x=0$ there) and also no-flux at infinity (or no flux at x=L if we switch to the finite domain).

The physical representation is a large plate which is insulated everywhere except for 2 small areas -a<x<a where I remove the insulation and apply temperatures.

2. Relevant equations

3. The attempt at a solution

Separation of variables doesn't work since the exponential solution cannot satisfy the boundary conditions of no flux. Also I tried Fourier transform in x and I cannot apply the mixed boundary conditions because it becomes a function $U(\omega,y)$.

2. Apr 4, 2016