An enhanced ultraspherical collocation framework with Chebyshev nodes for the time-fractional FitzHugh–Nagumo equation

Youssri Hassan Youssri; Muhammad Mujtaba Shaikh; Iqbal M. Batiha; Nidal Anakira; Irianto Irianto; Tala Sasa; Youssri Hassan Youssri; Muhammad Mujtaba Shaikh; Iqbal M. Batiha; Nidal Anakira; Irianto Irianto; Tala Sasa

doi:10.3934/math.2026565

AIMS Mathematics

2026, Volume 11, Issue 5: 13710-13743. doi: 10.3934/math.2026565

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Research article

An enhanced ultraspherical collocation framework with Chebyshev nodes for the time-fractional FitzHugh–Nagumo equation

1.
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2.
Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan
3.
Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan
4.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, UAE
5.
Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman
6.
Department of General Education, Faculty of Resilience, Rabdan Academy, Abu Dhabi, United Arab Emirates
7.
Department of Mathematics, Faculty of Science, Private Applied Science University, Amman, Jordan

Received: 12 March 2026 Revised: 18 April 2026 Accepted: 24 April 2026 Published: 15 May 2026
MSC : 33C45, 65N35, 65L60, 35R11

This work introduces a novel spectral collocation scheme based on ultraspherical (Gegenbauer) polynomials evaluated at Chebyshev–Gauss–Lobatto nodes for the numerical treatment of the nonlinear inhomogeneous time-fractional FitzHugh–Nagumo differential problem. The proposed methodology exploits the flexibility of the ultraspherical parameter $ \lambda > -\frac{1}{2} $ to achieve enhanced accuracy and stability. We derived new operational matrices for both integer-order and Caputo fractional derivatives of the shifted ultraspherical basis, accompanied by rigorous proofs. A comprehensive convergence analysis in the $ L^2 $ norm was established, demonstrating spectral accuracy. Extensive numerical experiments confirmed that the proposed method outperforms the classical Legendre-based approach for optimal choices of $ \lambda $, with errors reduced by several orders of magnitude. The superiority of optimized $ \lambda $ over the Legendre case ($ \lambda = \frac{1}{2} $) was demonstrated through both numerical benchmarks and theoretical error bounds that explicitly depend on $ \lambda $, showing that the Legendre choice is not universally optimal for problems with boundary layers or specific regularity properties. An efficient algorithmic implementation was provided, and comparative tables illustrate the superiority of the ultraspherical framework across various fractional orders and parameter settings.
- ultraspherical polynomials,
- Chebyshev nodes,
- time-fractional FitzHugh–Nagumo equation,
- spectral collocation,
- operational matrices,
- $ L^2 $-convergence,
- error analysis
Citation: Youssri Hassan Youssri, Muhammad Mujtaba Shaikh, Iqbal M. Batiha, Nidal Anakira, Irianto Irianto, Tala Sasa. An enhanced ultraspherical collocation framework with Chebyshev nodes for the time-fractional FitzHugh–Nagumo equation[J]. AIMS Mathematics, 2026, 11(5): 13710-13743. doi: 10.3934/math.2026565

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Abstract

This work introduces a novel spectral collocation scheme based on ultraspherical (Gegenbauer) polynomials evaluated at Chebyshev–Gauss–Lobatto nodes for the numerical treatment of the nonlinear inhomogeneous time-fractional FitzHugh–Nagumo differential problem. The proposed methodology exploits the flexibility of the ultraspherical parameter $ \lambda > -\frac{1}{2} $ to achieve enhanced accuracy and stability. We derived new operational matrices for both integer-order and Caputo fractional derivatives of the shifted ultraspherical basis, accompanied by rigorous proofs. A comprehensive convergence analysis in the $ L^2 $ norm was established, demonstrating spectral accuracy. Extensive numerical experiments confirmed that the proposed method outperforms the classical Legendre-based approach for optimal choices of $ \lambda $, with errors reduced by several orders of magnitude. The superiority of optimized $ \lambda $ over the Legendre case ($ \lambda = \frac{1}{2} $) was demonstrated through both numerical benchmarks and theoretical error bounds that explicitly depend on $ \lambda $, showing that the Legendre choice is not universally optimal for problems with boundary layers or specific regularity properties. An efficient algorithmic implementation was provided, and comparative tables illustrate the superiority of the ultraspherical framework across various fractional orders and parameter settings.

References

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