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Math Help - Heat Equation in semi infinite slab

  1. #1
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    Heat Equation in semi infinite slab

    I have this PDE with following Boundary Conditions:

     <br />
{\partial^2 \theta \over \partial x^2}+{\partial^2 \theta \over \partial y^2}={\partial \theta \over \partial t}<br />


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

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

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

     <br />
B.C. \ 3:\ {\partial \theta \over \partial y}=0 \ @ \ y=0<br />

     <br />
B.C. \ 4:\ {\partial \theta \over \partial y}=0 \ @ \ y=H<br />

    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.
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  2. #2
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    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.
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