A Legendre tau approach for high-order pantograph Volterra-Fredholm integro-differential equations

Mahmoud A. Zaky; Weam G. Alharbi; Marwa M. Alzubaidi; R.T. Matoog; Mahmoud A. Zaky; Weam G. Alharbi; Marwa M. Alzubaidi; R.T. Matoog

doi:10.3934/math.2025322

AIMS Mathematics

2025, Volume 10, Issue 3: 7067-7085. doi: 10.3934/math.2025322

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A Legendre tau approach for high-order pantograph Volterra-Fredholm integro-differential equations

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia
3.
Department of Mathematics, College of Duba, University of Tabuk, Tabuk, Saudi Arabia
4.
Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah, Saudi Arabia

Received: 16 January 2025 Revised: 17 March 2025 Accepted: 21 March 2025 Published: 27 March 2025
MSC : 65L05, 65R20, 65N35, 65L03

High-order pantograph Volterra-Fredholm integro-differential equations (P-VF-IDEs) arise in various scientific and engineering applications, where systems exhibit delay or scaling in their dynamics. This paper investigates a class of high-order P-VF-IDEs characterized by the presence of both Volterra and Fredholm integral terms as well as pantograph delay elements. We propose a spectral tau approach to approximate the solution P-VF-IDEs in both one- and two-dimensions. In particular, we employ the operational differentiation and integration matrices to approximate the integro-differential operator, transforming the continuous problem into a system of algebraic equations, and providing high accuracy with fewer computational modes. Numerical experiments illustrate the accuracy and convergence properties of the spectral Legendre tau method in solving high-order P-VF-IDEs and demonstrate its efficacy compared to other spectral approaches.
- pantograph Volterra-Fredholm,
- High-order integro-differential equations,
- spectral method,
- Legendre tau
Citation: Mahmoud A. Zaky, Weam G. Alharbi, Marwa M. Alzubaidi, R.T. Matoog. A Legendre tau approach for high-order pantograph Volterra-Fredholm integro-differential equations[J]. AIMS Mathematics, 2025, 10(3): 7067-7085. doi: 10.3934/math.2025322

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Abstract

High-order pantograph Volterra-Fredholm integro-differential equations (P-VF-IDEs) arise in various scientific and engineering applications, where systems exhibit delay or scaling in their dynamics. This paper investigates a class of high-order P-VF-IDEs characterized by the presence of both Volterra and Fredholm integral terms as well as pantograph delay elements. We propose a spectral tau approach to approximate the solution P-VF-IDEs in both one- and two-dimensions. In particular, we employ the operational differentiation and integration matrices to approximate the integro-differential operator, transforming the continuous problem into a system of algebraic equations, and providing high accuracy with fewer computational modes. Numerical experiments illustrate the accuracy and convergence properties of the spectral Legendre tau method in solving high-order P-VF-IDEs and demonstrate its efficacy compared to other spectral approaches.

References

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