2D Laplace's equation with mixed boundary conditions

[refind]

This isn't homework but could be labeled "textbook style" so I'm posting it here.

Homework Statement

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.

Homework Equations

The Attempt at a Solution

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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 $U(\omega,y)$.

 

[Asim Riaz]

The solution I think is to separate the domain into four regions. For example, Region 1: x \in [0,a], y \in [0,1] Region 2: x \in [-a,0], y \in [0,1] Region 3: x \in [-\infty,-a], y \in [0,1]Region 4: x \in [a,\infty], y \in [0,1] and then find the solution in each of these regions separately. For example, in Region 1, I could use the method of separation of variables and Fourier series to represent the solution as a sum of sines and cosines. The boundary conditions give the coefficients of the sine and cosine series. Similarly, for the other three regions, the boundary conditions give the coefficients of the solution in those regions. Then, I could stitch them together at x=\pm a at each of the boundaries (y=0,1) so that the solution is continuous across those boundaries. Finally, I could stitch the four regions together at x=0 so that the solution is continuous there too.Does this seem like the correct approach? Are there any other methods that could be used?