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A meshless, integration-free, and boundary-only RBF technique. (English) Zbl 0999.65142
Summary: Based on the radial basis function (RBF), nonsingular general solution, and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation technique for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are mathematically very simple.
In this study, the RBFs are employed to approximate the inhomogeneous terms via the DRM, while nonsingular general solution leads to a boundary-only RBF formulation for homogeneous solution. The present scheme is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of nonsingular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions.
The efficiency and utility of this new technique are validated through a number of typical numerical examples. Completeness concern of the BKM due to the sole use of the nonsingular part of complete fundamental solution is also discussed.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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