S. P. Hu

This paper describes the applications of the method of fundamental solutions (MFS) as a mesh-free numerical method for the Stokes' first and second problems which prevail in the semi-infinite domain with constant and oscillatory velocity at the boundary in the fluid-mechanics benchmark problems. The time-dependent fundamental solutions for the semi-infinite problems are used directly to obtain the solution as a linear combination of the unsteady fundamental solution of the diffusion operator. The proposed numerical scheme is free from the conventional Laplace transform or the finite difference scheme to deal with the time derivative term of the governing equation. By properly placing the field points and the source points at a given time level, the solution is advanced in time until steady state solutions are reached. It is not necessary to locate and specify the condition at the infinite domain such as other numerical methods. Since the present method does not need mesh discretization and nodal connectivity, the computational effort and memory storage required are minimal as compared to the domain-oriented numerical schemes. Test results obtained for the Stokes' first and second problems show good comparisons with the analytical solutions. Thus the present numerical scheme has provided a promising mesh-free numerical tool to solve the unsteady semi-infinite problems with the space-time unification for the time-dependent fundamental solution.

The present work describes the application of the method of fundamental solutions (MFS) for the solution of cutoff wavelengths of elliptical waveguides. Since the MFS employs a formulation using boundary values only, the cutoff wavelengths are determined by applying the singular value decomposition (SVD) technique. The use of the MFS to solve the governing (Helmholtz) equation guarantees a solution without singularities, since it does not use discretized points to determine the solution at the interior of the computational domain. The combination of the MFS and SVD techniques has resulted in a simpler and efficient numerical solution procedure, as compared to other schemes. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 44: 552–558, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20695