An operational approach for one- and two-dimension high-order multi-pantograph Volterra integro-differential equation

Ali H. Tedjani; Sharifah E. Alhazmi; Samer S. Ezz-Eldien; Ali H. Tedjani; Sharifah E. Alhazmi; Samer S. Ezz-Eldien

doi:10.3934/math.2025426

AIMS Mathematics

2025, Volume 10, Issue 4: 9274-9294. doi: 10.3934/math.2025426

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An operational approach for one- and two-dimension high-order multi-pantograph Volterra integro-differential equation

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
2.
Mathematics Department, Al-Qunfudah University College, Umm Al-Qura University, Mecca, KSA
3.
Department of Mathematics, Faculty of Science, New Valley University, El-Kharga 72511, Egypt

Received: 06 February 2025 Revised: 21 March 2025 Accepted: 28 March 2025 Published: 23 April 2025
MSC : 65L05, 65R20, 65N35, 65L03

High-order Volterra integro-differential equations are of great interest to many authors because of their important applications in physics and engineering, especially if they contain delay or pantograph terms that enable them to describe the memory effect. Providing an efficient numerical scheme for high-order Volterra integro-differential equations helps to explain many problems in mathematical biology and quantum mechanics. In this manuscript, we use shifted Jacobi polynomials as the basis for a spectral collocation approach to solve high-order one- and two-dimensional Volterra integro-differential equations with variable coefficients. A pantograph operational matrix, based on shifted Jacobi polynomials, is used for the first time, together with the Gauss-Jacobi quadrature rule, to reduce the problem to the problem of solving a system of algebraic equations. To ensure the validity of the proposed approach, we compare the numerical results with those of other numerical schemes in the literature.
- pantograph equation,
- Volterra integro-differential equation,
- spectral method,
- Jacobi polynomial,
- operational matrix
Citation: Ali H. Tedjani, Sharifah E. Alhazmi, Samer S. Ezz-Eldien. An operational approach for one- and two-dimension high-order multi-pantograph Volterra integro-differential equation[J]. AIMS Mathematics, 2025, 10(4): 9274-9294. doi: 10.3934/math.2025426

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Abstract

High-order Volterra integro-differential equations are of great interest to many authors because of their important applications in physics and engineering, especially if they contain delay or pantograph terms that enable them to describe the memory effect. Providing an efficient numerical scheme for high-order Volterra integro-differential equations helps to explain many problems in mathematical biology and quantum mechanics. In this manuscript, we use shifted Jacobi polynomials as the basis for a spectral collocation approach to solve high-order one- and two-dimensional Volterra integro-differential equations with variable coefficients. A pantograph operational matrix, based on shifted Jacobi polynomials, is used for the first time, together with the Gauss-Jacobi quadrature rule, to reduce the problem to the problem of solving a system of algebraic equations. To ensure the validity of the proposed approach, we compare the numerical results with those of other numerical schemes in the literature.

References

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