# Heat Equation in semi infinite slab

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• Jun 15th 2008, 06:50 AM
Ehsan
Heat Equation in semi infinite slab
I have this PDE with following Boundary Conditions:

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

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

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

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

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

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

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 $\displaystyle t$ then based on $\displaystyle x$ and finally based on $\displaystyle y$, 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.
• Jul 9th 2008, 04:16 AM
Kiwi_Dave
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.