Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations

ABSTRACT

The present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.

KEYWORDS:

• Non-linear integral equation
• thin-plate spline
• weakly singular kernel
• discrete Galerkin method
• error analysis

AMS SUBJECT CLASSIFICATIONS:

• 41A25
• 45B05
• 45E99
• 45G05

Acknowledgements

The author is very grateful to the reviewers for their valuable comments and suggestions which have improved the paper.

Notes

No potential conflict of interest was reported by the author.