A comparison of two equivalent solutions of the diffusion equation

Abstract

Plane radial flow of water in a closed aquifer towards a circular well or heat flow through a homogeneous conducting solid from a circular hole to infinity are well known problems that were solved long ago. The solution is expressed in terms of an integral of ordinary Bessel functions. A new solution, which is expressed in terms of an integral of modified Bessel functions, is derived by means of Laplace transformation instead of integration of Green's function. One particular choice of the Bromwich contour in the invers transformation, which appears unnecessarily complicated, gives a solution the integral of which converges much more rapidly than the corresponding integral that is given in the literature. In contrast with the classical solution, the new one gives an explicit expression for the transient part. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 657–671, 1999