A Laguerre expansion of the Cauchy problem for the diffusion equation

Abstract

The non-characteristic Cauchy problem for the diffusion equation in plane geometry is re-examined and an analytic solution is derived. There is a well known series solution to this classic problem which involves derivatives of all orders of the Cauchy data. In this work, we present a new series solution to the problem which does not require higher derivatives of the Cauchy data. The solution is expressed in a series of Laguerre polynomials in the time variable with coefficients involving new specialized functions of the space variable. A complete analysis of these specialized functions is presented and their properties exploited to great effect in a number of numerical examples. All computations are performed using an efficient numerical algorithm based on a truncated form of the series solution.0266-5611/94/051165+19%19.50 0 1994 1OP Publishing LtdOnce again, substituting series (5.1) into the above equation and collecting like powers

On polynomials of Sheffer type arising from a Cauchy problem

“…This is usually dealt with by some form of regularization, a process that relegates the Cauchy data to a well-posedness class of functions, but which nevertheless leaves an infinite series of derivatives of the mollified data. In Pettigrew and Meredith [11], the authors overspecified the problem with the imposition of the initial condition u(x, 0) = f (x). In this way, a series solution that did not require derivatives of the Cauchy data was derived.…”