During the past two decades, various techniques have been developed to overcome the difficulty of the domain integration in boundary element methods. One of the most popular methods is the so-called Dual Reciprocity Method (DRM) which was introduced by Brebbia and Nardini in 1983. The main feature of the DRM is the transference of domain integrals to the boundary by using a series of prescribed basis functions. In the original version of the DRM, 1+r was arbitrarily adopted as the basis function. In 1992, Partridge et al published the first book on the DRM which used 1+r almost exclusively as basis functions. Techniques based on this approach, typically developed between 1983-1992, are generally considered to comprise the first generation of the DRM.

After 1992, a number of researchers started to introduce radial basis functions (rbfs) as basis functions to the BEM community. Among them, Golberg and Chen established the connection between the mathematical foundations and engineering applications. Based on the rich theory of radial basis functions, the use of basis functions in the DRM was theoretically justified. We consider the period 1992-1998 of the DRM as the second generation.

The derivation of particular solutions using different basis functions is considered to be the most challenging task in the DRM process. In the past, the Laplacian was kept on the left hand side of the differential equation and the rest of the differential operator was moved to the right hand side and treated as a pseudo-forcing term. The accuracy and theoretical justification of this are still in great doubt. After 1998, the discovery of closed form particular solutions for Helmholtz-types operators using polyharmonic splines has opened the way to more effectively solving time-dependent problems. This new development has also indicated that it is possible to choose radial basis functions properly for a given differential operator so that the particular solutions can be obtained analytically.

Furthermore, the recent discovery of positive definite compactly supported radial basis functions has made it possible to produce sparse interpolation matrices in the DRM and this leads to the prospect of solving large scale industrial problems, particularly in 3D. We consider these post 1998 developments as the third generation of the DRM.

The ultimate goal in the development of the DRM is to be able to choose a basis function and derive a particular solution for a given differential operator. A new breed of basis functions has been under construction by mathematicians so that practitioners may have the flexibility to choose basis functions based on the problem at hand. We would consider these new developments as the fourth generation DRM.

In recognition of this progress, this special issue of Engineering Analysis with Boundary Elements is devoted to the presentation of a number of new theoretical developments and engineering applications of radial basis functions to the numerical solution of partial differential equations.

In honor of their invention of the DRM we have chosen to reprint the original article of Brebbia and Nardini as the first article in the issue. Following this is an in-depth study by Partridge of the behavior of various rbfs in the DRM. This is followed by the article of Cheng who presents a new derivation of some recent results of Muleshkov et al (Computational Mechanics, 23, pp. 411-419, 1999) concerning the particular solutions of Laplace and Helmholtz operators when the source functions are polyharmonics splines. In the next paper, Golberg et al show how to obtain closed form analytic particular solutions for 3D Helmholtz-type operators for Wendlandís compactly supported rbfs. Cheng et al then show how to use compactly supported rbfs to develop an efficient iterative DRBEM solver for Poissonís equations.

A continuing problem in applying rbfs to the DRM is that of obtaining efficient rbf approximations to the source term. One proposed solution to this is the use of compactly supported rbfs. An alternative approach is a multilevel approximation scheme given by Gaspar in the next paper.

Following this, Ramachandran and Balakrishnan survey some recent applications of rbfs to solve nonlinear Poisson-type partial differential equations while Wen et al discuss the accuracy of various rbfs to solve problems in plates and shells. KÖ gl and Gaul then present a new application of the DRM to solve problems in transient piezoelectricity. Hon and Wu use ideas from rbf interpolation theory to give a Treffts-type method to solve an inverse problem in potential theory. Finally, Jumarhon et al discuss the application of higher order splines to directly solve partial differential equation.

In closing, the editors would like to thank the authors for their contribution in putting together this special issue.

M.A. Golberg and C.S. Chen