On the numerical solution of Fredholm integral equations utilizing the local radial basis function method

ABSTRACT

The current investigation describes a computational technique to solve one- and two-dimensional Fredholm integral equations of the second kind. The method estimates the solution using the discrete collocation method by combining locally supported radial basis functions (RBFs) constructed on a small set of nodes instead of all points over the analysed domain. In this work, we employ the Gauss–Legendre integration rule on the influence domains of shape functions to approximate the local integrals appearing in the method. In comparison with the globally supported RBFs for solving integral equations, the proposed method is stable and uses much less computer memory. The scheme does not require any cell structures, so it is meshless. We also obtain the error analysis of the proposed method and demonstrate that the convergence rate of the approach is high. Illustrative examples clearly show the reliability and efficiency of the new method.

KEYWORDS:

• Local radial basis function
• Fredholm integral equation
• discrete collocation method
• meshless method
• error analysis

2010 MSC SUBJECT CLASSIFICATIONS:

• 41A25
• 31B10
• 45B05
• 65R20

Acknowledgments

The authors are very grateful to three reviewers and the associate editor for their valuable comments and suggestions which have improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Pouria Assari  http://orcid.org/0000-0002-6541-7451

Mehdi Dehghan  http://orcid.org/0000-0002-2573-9755