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The method of fundamental solutions and quasi‐Monte‐Carlo method for diffusion equations
Author(s) -
Chen C. S.,
Golberg M. A.,
Hon Y. C.
Publication year - 1998
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19981230)43:8<1421::aid-nme476>3.0.co;2-v
Subject(s) - mathematics , helmholtz equation , laplace transform , monte carlo method , singularity , mathematical analysis , boundary value problem , statistics
The Laplace transform is applied to remove the time‐dependent variable in the diffusion equation. For non‐harmonic initial conditions this gives rise to a non‐homogeneous modified Helmholtz equation which we solve by the method of fundamental solutions. To do this a particular solution must be obtained which we find through a method suggested by Atkinson. 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. Two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2‐D and 3‐D. © 1998 John Wiley & Sons, Ltd.