A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems

Chih-Yu Liu and Cheng-Yu Ku
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Chih-Yu Liu: Graduate Institute of Applied Geology, National Central University, Taoyuan 320317, Taiwan
Cheng-Yu Ku: School of Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan

Mathematics, 2022, vol. 10, issue 10, 1-23

Abstract: In this article, we propose a simplified radial basis function (RBF) method with exterior fictitious sources for solving elliptic boundary value problems (BVPs). Three simplified RBFs, including Gaussian, multiquadric (MQ), and inverse multiquadric (IMQ) without the shape parameter, are adopted in this study. With the consideration of many exterior fictitious sources outside the domain, the radial distance of the RBF is always greater than zero, such that we can remove the shape parameter from RBFs. Additionally, simplified Gaussian, MQ, and IMQ RBFs and their derivatives in the governing equation are always smooth and nonsingular. Comparative analysis is conducted for three different collocation types, including conventional uniform centers, randomly fictitious centers, and exterior fictitious sources. Numerical examples of elliptic BVPs in two and three dimensions are carried out. The results demonstrate that the proposed simplified RBFs with exterior fictitious sources can significantly improve the accuracy, especially for the Laplace equation. Furthermore, the proposed simplified RBFs exhibit the simplicity of solving elliptic BVPs without finding the optimum shape parameter.

Keywords: radial basis function; the shape parameter; multiquadric; inverse multiquadric; Gaussian (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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