Hermite, Higher order Hermite, Laguerre type polynomials and Burgers like equations

Abstract

The multivariable version of ordinary and generalized Hermite polynomials are the natural solutions of the classical heat equation and of its higher order versions. We derive the associated Burgers equations and show that analogous non-linear partial differential equations can be derived for Laguerre polynomials. The starting point of this extension is the Laguerre diffusive equation, whose nonlinear extension reveals interesting implications involving families of mixed polynomials. In this way we have a general scheme to obtain new exact explicit solutions for nonlinear PDEs by using Laguerre, Hermite and other families of polynomials of Appèl and non-Appèl type.