I'm guessing that nothing changes in terms of the difficulty of literally solving these problems numerically, but that the difficulty here refers to the ease of getting efficient and stable/high quality integrations -- Maybe that would necessitate using more sophisticated approximations and mathematical restatements to make problems more amenable to numerical integration?
Is there any relationship between the Laplacian and the Laplace transform. I feel like I remember doing heat transfer problems with Laplace transforms, but I can't remember the specifics.
@zpb The difficulty is all about the difficulty of solving them numerically (stability, accuracy, etc.). These almost always translate directly to more computational effort. So it's not just a matter of "thinking harder"; it really does come down to more time steps, finer grids, more nonlinear equations to solve, etc.