The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations

The Laplace transform is applied to remove the time-dependent variable in t he diffusion equation. For nonharmonic initial conditions this gives rise t o a non-homogeneous modified Helmholtz equation which we solve by the metho d of fundamental solutions. To do this a particular solution must be obtain ed which we find through a method suggested by Atkinson.(17) To avoid costl y Gaussian quadratures, we approximate the particular solution using quasi- Monte-Carlo integration which has the advantage of ignoring the singularity in the integrand. The approximate transformed solution is then inverted nu merically using Stehfest's algorithm.(13) Two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving d iffusion equations in 2-D and 3-D. (C) 1998 John Wiley & Sons, Ltd.