Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis

A. H. Tedjani; A. Z. Amin; Abdel-Haleem Abdel-Aty; M. A. Abdelkawy; Mona Mahmoud; A. H. Tedjani; A. Z. Amin; Abdel-Haleem Abdel-Aty; M. A. Abdelkawy; Mona Mahmoud

doi:10.3934/math.2024388

AIMS Mathematics

2024, Volume 9, Issue 4: 7973-8000. doi: 10.3934/math.2024388

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Research article

Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
2.
Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
3.
Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
5.
Department of Physics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia

Received: 12 December 2024 Revised: 26 January 2024 Accepted: 06 February 2024 Published: 26 February 2024
MSC : 45Dxx, 65Mxx, 44-xx

The main purpose of this work was to develop a spectrally accurate collocation method for solving nonlinear fractional Fredholm integro-differential equations (non-FFIDEs). A proposed spectral collocation method is based on the Legendre-Gauss-Lobatto collocation (L-G-LC) method in which the main idea is to use Caputo derivatives and Legendre-Gauss interpolation for nonlinear FFIDEs. A rigorous convergence analysis is provided and confirmed by numerical tests. In addition, we provide some numerical test cases to demonstrate that the approach can preserve the non-smooth solution of the underlying problem.
- Legendre-Gauss-Lobatto,
- fractional Fredholm integro-differential equation,
- Caputo fractional derivative,
- convergence analysis,
- spectral method
Citation: A. H. Tedjani, A. Z. Amin, Abdel-Haleem Abdel-Aty, M. A. Abdelkawy, Mona Mahmoud. Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis[J]. AIMS Mathematics, 2024, 9(4): 7973-8000. doi: 10.3934/math.2024388

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Abstract

The main purpose of this work was to develop a spectrally accurate collocation method for solving nonlinear fractional Fredholm integro-differential equations (non-FFIDEs). A proposed spectral collocation method is based on the Legendre-Gauss-Lobatto collocation (L-G-LC) method in which the main idea is to use Caputo derivatives and Legendre-Gauss interpolation for nonlinear FFIDEs. A rigorous convergence analysis is provided and confirmed by numerical tests. In addition, we provide some numerical test cases to demonstrate that the approach can preserve the non-smooth solution of the underlying problem.

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