Analytical Expansions for Parabolic Equations

Abstract

We consider the Cauchy problem associated with a general parabolic partial differential equation in d dimensions. We find a family of closed-form asymptotic approximations for the unique classical solution of this equation as well as rigorous short-time error estimates. Using a boot-strapping technique, we also provide convergence results for arbitrarily large time intervals.

Keywords: parabolic PDE, asymptotic expansion, singular perturbation, analytical approximation

JEL Classification: C00