Particular solutions of Laplace and bi-harmonic operators using Matérn radial basis functions

ABSTRACT

In this paper, we derive closed-form particular solutions of Matérn radial basis functions for the Laplace and biharmonic operator in 2D and Laplace operator in 3D. These derived particular solutions are essential for the implementation of the method of particular solutions for solving various types of partial differential equations. Four numerical examples in 2D and 3D are given to demonstrate the effectiveness of the derived particular solutions.

KEYWORDS:

• Radial basis functions
• Matérn function
• the method of particular solutions

2010 AMS SUBJECT CLASSIFICATIONS:

• 65Y40
• 68W05

Acknowledgments

The authors would like to thank Professor Andreas Karaageorghis for his help to proof read the final version of this paper. The authors also thanks referees for their constructive comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.