Hermite series solutions of differential equations

Abstract

A standard approach both for the numerical solution as well as the deduction of certain theoretical properties of partial differential equations is their replacement by appropriate difference equations. Another standard approach is the use of Fourier transforms. These two approaches could be combined if the transformed equation could be replaced by a difference equation which could then be solved. IJnfortunately, this combination is usually of little use for numerical considerations since the Fourier transform of a good approximation to the solution of an equation is not gcncrally a good approximation to the Fourier transform of the solution and vise versa However, there is one type of approximating function which is well qualified to give a good approximation both to the solution and to its Fourier transform, and that is a partial sum of a Hermite Series. This is true because the Hermite functions are eigenfunctions of the Fourier transform corresponding to eigenvalues of modules 1. Thus solutions in terms of Hermite series of the transformed equation may be interesting from a numerical point of view. The transformed equation of a partial differential equation with constant coefficients is just an equation of the form