Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique

TL;DR

The paper introduces the Boundary Knot Method (BKM), a meshless, boundary-only RBF technique with exponential convergence and no need for integration, suitable for solving various PDEs efficiently.

Contribution

It presents a novel boundary knot method that is meshless, boundary-only, and can solve nonlinear PDEs in a single step without iteration, differing from traditional methods.

Findings

BKM is meshless and boundary-only, simplifying implementation.

It achieves exponential convergence and does not require integration.

The method can solve nonlinear PDEs in one step without iteration.

Abstract

Based on the radial basis function (RBF), non-singular general solution and dual reciprocity principle (DRM), this paper presents an inheretnly meshless, exponential convergence, integration-free, boundary-only collocation techniques for numerical solution of general partial differential equation systems. The basic ideas behind this methodology are very mathematically simple and generally effective. The RBFs are used in this study to approximate the inhomogeneous terms of system equations in terms of the DRM, while non-singular general solution leads to a boundary-only RBF formulation. The present method is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of non-singular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does no need to introduce the artificial boundary and results in the symmetric system equations under certain conditions. It is also found that the BKM can solve nonlinear partial differential equations one-step without iteration if only boundary knots are used. The efficiency and utility of this new technique are validated through some typical numerical examples. Some promising developments of the BKM are also discussed.