# Homework Help: 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