I'm stuck solving $\partial^2 u/ \partial x^2 +\partial^2 u/ \partial y^2=0$ with these boundary conditions:
$$ u(x,0)=0, \ \ \ u(x,1)=1 \ \ for \ \ x\in [-a,a],\ a<<1$$ $$ \frac{\partial u}{\partial y} \biggr\rvert_{y=0, \ y=1}=0 \ \ for \ \ |x|>a $$ $$ \lim_{x\to\infty}\frac{\partial u}{\partial x} = 0, \ \ \frac{\partial u}{\partial x}\biggr\rvert_{x=0}=0 \ \ (symmetry)$$
the infinity condition can cause some problems so I am okay with changing it to $\frac{\partial u}{\partial x}\rvert_{x=L}=0$ for some large $L$.
Basically it's a rectangle of height 1 and infinite width and you apply a temperature of 1 in a small segment $x \in [-a,a]$ at the top $(y=1)$ and temperature 0 in the same segment $x \in [-a,a]$ at the bottom $(y=0)$, and no flux (insulated) everywhere else.
I have tried separation of variables but I cannot get the eigenfunctions to satisfy these boundary conditions. One of them must be exponential which fails to satisfy no-flux at the 2 end points. Laplace or Fourier transforms of the PDE also don't accept mixed conditions like I have. Does anyone have an idea?