New RBF collocation methods and kernel RBF with applications

TL;DR

This paper introduces new RBF collocation schemes and kernel RBF methods for solving PDEs, including boundary and domain approaches, with improved accuracy and reduced errors, applicable to various inhomogeneous problems.

Contribution

It develops novel RBF discretization schemes, including boundary and domain methods, and introduces kernel RBF variants, advancing numerical solutions for PDEs.

Findings

Developed symmetric boundary knot methods for PDEs.

Introduced the modified Kansa method with reduced boundary errors.

Presented five types of kernel RBF for enhanced flexibility.

Abstract

A few novel radial basis function (RBF) discretization schemes for partial differential equations are developed in this study. For boundary-type methods, we derive the indirect and direct symmetric boundary knot methods. Based on the multiple reciprocity principle, the boundary particle method is introduced for general inhomogeneous problems without using inner nodes. For domain-type schemes, by using the Green integral we develop a novel Hermite RBF scheme called the modified Kansa method, which significantly reduces calculation errors at close-to-boundary nodes. To avoid Gibbs phenomenon, we present the least square RBF collocation scheme. Finally, five types of the kernel RBF are also briefly presented.