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

Abstract

The Laplace transform is applied to remove the time-dependent variable in the di usion equation. For nonharmonic initial conditions this gives rise to a non-homogeneous modiÿed Helmholtz equation which we solve by the method of fundamental solutions. To do this a particular solution must be obtained which we ÿnd 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 Stehfest's algorithm.

The method of fundamental solutions for a biharmonic inverse boundary determination problem

“…Since then, the MFS has been successfully used in numerical solutions of the Poisson equation [3,8], diffusion equation [4,27], Helmholtz equation [11,28], biharmonic equation [12][13][14]22], and Stokes equations [29][30][31]. The MFS has also been effectively used for solving inverse problems in which some of the ingredients necessary to solve a direct problem are missing, see, e.g.…”

Α Study on Radial Basis Function and Quasi-Monte Carlo Methods

“…As was pointed out by Chen et al [3], the dual reciprocity method (DRM) and QMC are two general techniques evaluating particular solution. What follows is a brief description of these two strategies with emphasis on the analysis of the RBF and MQC.…”

“…(4), the Quasi-Monte Carlo scheme [3] directly evaluates domain integral of Eq. (11), which is in fact equivalent to the particular solution.…”

Mathematical analysis and treatment for the true and spurious eigenequations of circular plates by the meshless method using radial basis function

“…This method was attributed to Kupradze in 1964. The method of fundamental solutions can be applied to potential (Fairweather and Karageorghis, 1998), Helmholtz (Karageorghis, 2001), diffusion (Chen et al, 1998), biharmonic (Poullikkas et al, 1998) and elasticity problems (Kupradze, 1964). The method of fundamental solutions can be seen as one kind of meshless method.…”