A collocation procedure for the numerical treatment of FitzHugh–Nagumo equation using a kind of Chebyshev polynomials

Waleed Mohamed Abd-Elhameed; Omar Mazen Alqubori; Ahmed Gamal Atta; Waleed Mohamed Abd-Elhameed; Omar Mazen Alqubori; Ahmed Gamal Atta

doi:10.3934/math.2025057

AIMS Mathematics

2025, Volume 10, Issue 1: 1201-1223. doi: 10.3934/math.2025057

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A collocation procedure for the numerical treatment of FitzHugh–Nagumo equation using a kind of Chebyshev polynomials

1.
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia
3.
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

Received: 05 September 2024 Revised: 28 December 2024 Accepted: 09 January 2025 Published: 20 January 2025
MSC : 365M60, 11B39, 40A05, 34A08

This manuscript aims to provide numerical solutions for the FitzHugh–Nagumo (FH–N) problem. The suggested approximate solutions are spectral and may be achieved using the standard collocation technique. We introduce and utilize specific polynomials of the generalized Gegenbauer polynomials. These introduced polynomials have connections with Chebyshev polynomials. The polynomials' series representation, orthogonality property, and derivative expressions are among the new formulas developed for these polynomials. We transform these formulas to obtain their counterparts for the shifted polynomials, which serve as basis functions for the suggested approximate solutions. The convergence of the expansion is thoroughly examined. We provide several numerical tests and comparisons to confirm the applicability and accuracy of our proposed numerical algorithm.
- FitzHugh–Nagumo equation,
- Chebyshev polynomials,
- collocation method,
- convergence analysis
Citation: Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori, Ahmed Gamal Atta. A collocation procedure for the numerical treatment of FitzHugh–Nagumo equation using a kind of Chebyshev polynomials[J]. AIMS Mathematics, 2025, 10(1): 1201-1223. doi: 10.3934/math.2025057

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Abstract

This manuscript aims to provide numerical solutions for the FitzHugh–Nagumo (FH–N) problem. The suggested approximate solutions are spectral and may be achieved using the standard collocation technique. We introduce and utilize specific polynomials of the generalized Gegenbauer polynomials. These introduced polynomials have connections with Chebyshev polynomials. The polynomials' series representation, orthogonality property, and derivative expressions are among the new formulas developed for these polynomials. We transform these formulas to obtain their counterparts for the shifted polynomials, which serve as basis functions for the suggested approximate solutions. The convergence of the expansion is thoroughly examined. We provide several numerical tests and comparisons to confirm the applicability and accuracy of our proposed numerical algorithm.

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