It is worth noting that it is impossible to solve the initial value problem for the standard heat equation in the real analytic category. Here, there are asymptotic expansions available but no Taylor series.
ADDED: It should also be noted that the directionality of time, as exhibited in the heat and diffusion processes, is a phenomenon lives outside the real analytic category. I believe that any PDE in the real analytic category that is well-posed as an initial value problem can be solved in both positive and time directions. That's not true for the heat equation in the smooth category. So the need for going outside the real analytic category appears already in fundamental classical physics.
This can be avoided, I suppose, by working purely with discrete models, but that for some of us is a cure worse than the original "problem".