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 the di usion equation. For non- harmonic initial conditions this gives rise to a non-homogeneous modi ed Helmholtz equation which we solve by the method of fundamental solutions. To do this a particular solution must be obtained which we nd through a method suggested by Atkinson.17 To avoid costly 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 numerically using Stehfestʹs algorithm.13 Two numerical examples are given to illustrate the simplicity and e ectiveness of our approach to solving di usion equations in 2-D and 3-D.