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

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 λ > −¹₂ 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 L² norm was established, demonstrating spectral accuracy. Extensive numerical experiments confirmed that the proposed method outperforms the classical Legendre-based approach for optimal choices of λ, with errors reduced by several orders of magnitude. The superiority of optimized λ over the Legendre case (λ = ¹₂) was demonstrated through both numerical benchmarks and theoretical error bounds that explicitly depend on λ, 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.