Laplace Transform Methods for Transient Diffusion; or, Some Good Questions from Ralph White

Abstract

Motivated by years of correspondence with Prof. Ralph White, I discuss two unconventional ways to solve diffusion problems with Laplace transforms. A method to derive error-function series, alternatives to Fourier series that converge rapidly and avoid the Gibbs phenomenon at short times, is illustrated by example. It is shown how Mittag-Leffler partial-fractions expansions can facilitate derivations of Fourier-series solutions from the same starting point. Several basic problems pertinent to electrochemical transport are analyzed, culminating in the development of a modified Cottrell equation applicable to thin films of unsupported electrolytic solutions sandwiched between planar electrodes.