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Recent developments in the dual reciprocity method using compactly supported radial basis functions. (English) Zbl 1043.65133
Rashed, Y. F. (ed.) et al., Transformation of domain effects to the boundary. Southampton: WIT Press (ISBN 1-85312-896-1/hbk). Int. Ser. Adv. Bound. Elem. 14, 183-225 (2003).
Summary: We survey some recent developments of the application of compactly supported radial basis functions (CS-RBFs) in the context of the dual reciprocity method. Using the CS-RBFs as the main tool for approximating the right hand side of a given partial differential equation, we further introduce a number of numerical techniques so that a large class of partial differential equations can be solved numerically. Due to the virtue of compact support, the CS-RBFs are very promising for solving large scale and high dimensional problems.
In this survey paper, we summarize a collection of closed-form particular solutions for various differential operators. In particular, some of them are new and preliminary numerical results are also provided in this paper. We also point out a few minor errors in previous publications on CS-RBFs and offer the necessary corrections. A number of proposals for future research using CS-RBFs are also suggested.
For the entire collection see [Zbl 1031.65001].

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J40 Boundary value problems for higher-order elliptic equations
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35K05 Heat equation
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L05 Wave equation