I have this PDE with following Boundary Conditions:

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 then based on and finally based on , 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.