Numerical scheme for solving the time-fractional Huxley equation using shifted Dickson polynomials

Omar Mazen Alqubori; Shuja'a Ali Alsulami; Ahmed Gamal Atta; Amr Kamel Amin; Waleed Mohamed Abd-Elhameed; Omar Mazen Alqubori; Shuja'a Ali Alsulami; Ahmed Gamal Atta; Amr Kamel Amin; Waleed Mohamed Abd-Elhameed

doi:10.3934/math.2026566

AIMS Mathematics

2026, Volume 11, Issue 5: 13744-13766. doi: 10.3934/math.2026566

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Research article

Numerical scheme for solving the time-fractional Huxley equation using shifted Dickson polynomials

1.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
2.
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11341, Cairo, Egypt
3.
Department of Mathematics, Adham University College, Umm Al-Qura University, Makkah 28653, Saudi Arabia; akgadelrab@uqu.edu.sa
4.
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

Received: 24 February 2026 Revised: 28 April 2026 Accepted: 29 April 2026 Published: 15 May 2026
MSC : 11B83, 65M70, 35R11

This article introduces an efficient spectral collocation framework for numerically solving the time-fractional Huxley equation. New basis functions of shifted Dickson polynomials of the first kind are introduced and employed. To achieve this, new formulas for the shifted polynomials are derived, including a series representation, an inverse formula, and expressions for both integer and fractional derivatives, which together with the collocation method serve as the foundation of the proposed numerical algorithm for converting the equation with its governing conditions into a non-linear algebraic system. A convergence and error analysis of the proposed method is also provided. We present numerical results and compare them with existing methods to illustrate the high accuracy of the proposed algorithms and their applicability.
- time-fractional differential equations,
- Dickson polynomials,
- derivative expressions,
- collocation method,
- convergence analysis
Citation: Omar Mazen Alqubori, Shuja'a Ali Alsulami, Ahmed Gamal Atta, Amr Kamel Amin, Waleed Mohamed Abd-Elhameed. Numerical scheme for solving the time-fractional Huxley equation using shifted Dickson polynomials[J]. AIMS Mathematics, 2026, 11(5): 13744-13766. doi: 10.3934/math.2026566

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Abstract

This article introduces an efficient spectral collocation framework for numerically solving the time-fractional Huxley equation. New basis functions of shifted Dickson polynomials of the first kind are introduced and employed. To achieve this, new formulas for the shifted polynomials are derived, including a series representation, an inverse formula, and expressions for both integer and fractional derivatives, which together with the collocation method serve as the foundation of the proposed numerical algorithm for converting the equation with its governing conditions into a non-linear algebraic system. A convergence and error analysis of the proposed method is also provided. We present numerical results and compare them with existing methods to illustrate the high accuracy of the proposed algorithms and their applicability.

References

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