Non-Singular Generalized RBF Solution and Weaker Singularity MFS: Laplace Equation and Anisotropic Laplace Equation

Chein-Shan Liu and Chung-Lun Kuo ()
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Chein-Shan Liu: Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
Chung-Lun Kuo: Department of Applied Artificial Intelligence, Ming Chuan University, Taoyuan 333321, Taiwan

Mathematics, 2025, vol. 13, issue 22, 1-19

Abstract: This paper introduces a singular distance function r s in terms of a symmetric non-negative metric tensor S . If S satisfies a quadratic matrix equation involving a parameter β then for the Laplace equation r s β is a non-singular generalized radial basis function solution if 2 > β > 0, and a weaker singularity fundamental solution if − 1 < β < 0. With a unit vector as a medium to express S , we can derive the metric tensor in closed form and prove that S is a singular projection operator. For the anisotropic Laplace equation, the corresponding closed-form representation of S is also derived. The concept of non-singular generalized radial basis function solution for the Laplace-type equations is novel and useful, which has not yet appeared in the literature. In addition, a logarithmic type method of fundamental solutions is developed for the anisotropic Laplace equation. Owing to non-singularity and weaker singularity of the bases of solutions, numerical experiments verify the accuracy and efficiency of the proposed methods.

Keywords: Laplace equation; anisotropic Laplace equation; radial basis function (RBF); non-singular generalized RBF solution; weaker singularity MFS; singular projection operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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