A meshless, integration-free, and boundary-only RBF technique

TL;DR

This paper introduces a novel meshless, boundary-only RBF collocation method called the Boundary Knot Method (BKM) for solving PDEs, which is integration-free, symmetric, and does not require artificial boundaries.

Contribution

The paper presents the BKM, a new boundary-only RBF technique that uses non-singular solutions, eliminating the need for artificial boundaries and simplifying the numerical solution process.

Findings

BKM is meshless and boundary-only, reducing computational complexity.

The method produces symmetric system equations under certain conditions.

Numerical examples validate the efficiency and accuracy of BKM.

Abstract

Based on the radial basis function (RBF), non-singular general solution and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation techniques for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are very mathematically simple. In this study, the RBFs are employed to approximate the inhomogeneous terms via the DRM, while non-singular general solution leads to a boundary-only RBF formulation for homogenous 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 no 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 only use of non-singular part of complete fundamental solution is also discussed.