Fun With Numerical Methods in LaPlace Space

We've all solved partial diffential equations (PDE's) by the LaPlace transform method -- right? You apply the LaPlace transform to the PDE and boundary conditions by looking them up on LaPlace transform tables. After applying the transformed initial condition, you're left with an ODE that can usually be solved analytically without too much difficulty. But then comes the moment of truth -- can you invert your solution back to real space using inverse LaPlace transform tables? In a graduate-level transport class you probably won't be given a PDE that can't by inverted back with the tables, but in real-life these situations can and do arise. Plus, in real-life you usually have some data that you want to fit with your LaPlace-space model. What do you do then? Well, if you really must know...