Recent developments in the dual receiprocity method using compactly supported radial basis functions

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 CSRBFs 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 virture 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.