Reducing Parabolic Partial Differential Equations to Canonical Form

Abstract

A simple method of reducing a parabolic partial differential equation to canonical form if it has only one term involving second derivatives is the following: find the general solution of the first-order equation obtained by ignoring that term and then seek a solution of the original equation which is a function of one more independent variable. Special cases of the method have been given before, but are not well known.

Applications occur in fluid mechanics and the theory of finance, where the Black-Scholes equation yields to the method, and where the variable corresponding to time appears to run backwards, but there is an information-theoretic reason why it should.