Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

Abstract

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.

Keywords:

• Nonlinear boundary integral equation
• Laplace's equation
• thin plate spline
• discrete collocation method
• meshless method
• error analysis

2010 AMS Subject Classifications:

• 45G05
• 65G99
• 35J65
• 74S25

Acknowledgments

The authors are very grateful to both 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