Legendre-Gauss-Lobatto collocation method for solving multi-dimensional systems of mixed Volterra-Fredholm integral equations

A. Z. Amin; M. A. Abdelkawy; Amr Kamel Amin; António M. Lopes; Abdulrahim A. Alluhaybi; I. Hashim; A. Z. Amin; M. A. Abdelkawy; Amr Kamel Amin; António M. Lopes; Abdulrahim A. Alluhaybi; I. Hashim

doi:10.3934/math.20231063

AIMS Mathematics

2023, Volume 8, Issue 9: 20871-20891. doi: 10.3934/math.20231063

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Legendre-Gauss-Lobatto collocation method for solving multi-dimensional systems of mixed Volterra-Fredholm integral equations

1.
Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan, Selangor, Malaysia
2.
Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
4.
Department of Basic Sciences, Adham University College, Umm AL-Qura University, Makkah, Saudi Arabia
5.
LAETA/INEGI, Faculty of Engineering, University of Porto, Porto, Portugal

Received: 22 February 2023 Revised: 15 June 2023 Accepted: 20 June 2023 Published: 30 June 2023
MSC : 45Dxx, 65Mxx, 44-XX

Integral equations play a crucial role in many scientific and engineering problems, though solving them is often challenging. This paper addresses the solution of multi-dimensional systems of mixed Volterra-Fredholm integral equations (SMVF-IEs) by means of a Legendre-Gauss-Lobatto collocation method. The one-dimensional case is addressed first. Afterwards, the method is extended to two-dimensional linear and nonlinear SMVF-IEs. Several numerical examples reveal the effectiveness of the approach and show its superiority in comparison to other alternative techniques for treating SMVF-IEs.
Citation: A. Z. Amin, M. A. Abdelkawy, Amr Kamel Amin, António M. Lopes, Abdulrahim A. Alluhaybi, I. Hashim. Legendre-Gauss-Lobatto collocation method for solving multi-dimensional systems of mixed Volterra-Fredholm integral equations[J]. AIMS Mathematics, 2023, 8(9): 20871-20891. doi: 10.3934/math.20231063

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Abstract

Integral equations play a crucial role in many scientific and engineering problems, though solving them is often challenging. This paper addresses the solution of multi-dimensional systems of mixed Volterra-Fredholm integral equations (SMVF-IEs) by means of a Legendre-Gauss-Lobatto collocation method. The one-dimensional case is addressed first. Afterwards, the method is extended to two-dimensional linear and nonlinear SMVF-IEs. Several numerical examples reveal the effectiveness of the approach and show its superiority in comparison to other alternative techniques for treating SMVF-IEs.

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