Title: Improved composite methods using radial basis functions for solving nonlinear Volterra-Fredholm integral equations

Authors: Dalila Takouk; Rebiha Zeghdane; Belkacem Lakehali

Addresses: Faculty of Mathematics and Informatics, Department of Mathematics, University Mohamed El Bachir El Ibrahimi of Bordj-Bou-Arreridj, El-Anasser, 34030, Bordj-Bou-Arreridj, Algeria ' Faculty of Mathematics and Informatics, Department of Mathematics, University Mohamed El Bachir El Ibrahimi of Bordj-Bou-Arreridj, El-Anasser, 34030, Bordj-Bou-Arreridj, Algeria ' Faculty of Mathematics and Informatics, Department of Mathematics, University of M'sila, 28000, M'sila, Algeria

Abstract: In this paper, a hybrid radial basis function (HRBF) is developed for solving Volterra-Fredholm integral equations. It is based on combining between generalised multiquadric and gaussian radial basis functions (RBFs). The generalised multiquadric is one of the powerful RBFs for approximating solutions of Volterra-Fredholm integral equations because we benefit from the optimal choices of the exponent β for improving the accuracy of the composite technique. The interpolation scheme using RBFs has the advantage of being meshless and dimensionally independent because it takes Euclidean distance as input, which can be trivially computed in any dimension. This type of RBFs depends on a shape parameter, which needs to be defined by the user and controls the stability and accuracy of the RBFs approximation. We discuss the new method's effectiveness compared to the widely used traditional RBFs and also investigate the effect of parameters on the method's performance. Also, the adapted technique is compared with the use of Wendland's compactly supported radial basis functions (CSRBFs) in both condition number and accuracy by considering different numerical examples.

Keywords: generalised multiquadric radial basis functions; gaussian radial basis functions; CSRBFs; compactly supported radial basis functions; nonlinear Volterra-Fredholm integral equations; Legendre-Gauss-Lobatto nodes and weights.

DOI: 10.1504/IJCSM.2024.137260

International Journal of Computing Science and Mathematics, 2024 Vol.19 No.2, pp.93 - 106

Accepted: 05 May 2023
Published online: 08 Mar 2024 *

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