Time-dependent fundamental solutions for homogeneous diffusion problems. (English) Zbl 1098.76622

Summary: This paper describes applications of the method of fundamental solutions (MFS) to 1-, 2- and 3-D diffusion equations. The time-dependent fundamental solutions for diffusion equations are used directly to obtain the solution as a linear combination of fundamental solutions of diffusion operator. The proposed scheme is free from conventionally used Laplace transform or from finite difference schemes to deal with the time derivative of the governing equation. By properly placing the field points and source points at a given time level, the solution is advanced in time until steady state solutions are reached. Test results obtained for 1-, 2- and 3-D diffusion problems show good comparisons with analytical solutions and MFS-based modified Helmholtz fundamental solutions. Thus the presented MFS with space-time unification has been demonstrated as a promising mesh-free numerical tool to solve homogeneous diffusion problem.


76R50 Diffusion
76M25 Other numerical methods (fluid mechanics) (MSC2010)
Full Text: DOI


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