Thread: Heat Equation in semi infinite slab

1. Heat Equation in semi infinite slab

I have this PDE with following Boundary Conditions:

$
{\partial^2 \theta \over \partial x^2}+{\partial^2 \theta \over \partial y^2}={\partial \theta \over \partial t}
$

$I.C.:\ \theta=0 \ @ \ t=0$

$B.C. \ 1:\ \theta=f(y) \ @ \ x=0$

$B.C. \ 2:\ \theta=0 \ @ \ {x \to \infty}$

$
B.C. \ 3:\ {\partial \theta \over \partial y}=0 \ @ \ y=0
$

$
B.C. \ 4:\ {\partial \theta \over \partial y}=0 \ @ \ y=H
$

I could not solve it by separation of varaibles because it has a non-homogeneous boundary condition. I tried to solve it by laplace transform. First, I took the transform based on $t$ then based on $x$ and finally based on $y$, but I have problems with the inverse transform. I do not even know if this method is right or wrong.

Can anybody help me to solve this PDE problem? It is a part of my university project and I really need the solution.

2. First I am no expert. On the surface it looks to me like B.C.1 and I.C. are only consistent for the special case f(y) = 0? And that seems to lead to the trivial solution.