Advanced Numerical Treatment for One- and Two-Dimensional Time-Fractional Coupled FitzHugh–Nagumo Models

Singular behavior near the initial times, arising from the presence of fractional-order derivatives in fractional differential equations, often leads to the deterioration of solutions obtained using spectral methods when applied based on classical orthogonal polynomials. Consequently, instead of relying on smooth polynomials as the basis for spectral approaches, significant attention has been devoted to developing appropriate nonsmooth functions that can form the basis for various spectral methods to address the limitation imposed by the fractional-order derivatives. In this study, we develop a numerical framework for solving one- and two-dimensional time-fractional coupled FitzHugh–Nagumo (FHN) models. We construct a novel, time-nonsmooth but spatially smooth function, called the orthogonal shifted Chebyshev function, in both one- and two-dimensional dimensions, that serves as the basis of the spectral collocation approach. Furthermore, we derive novel operational matrices of second-order and fractional-order derivatives, based on the new basis function in the spatial and time directions, respectively. These matrices are then used in conjunction with the spectral collocation technique in both spatial and time directions to reduce the problem to a system of algebraic equations. The numerical results demonstrate the accuracy of the presented numerical scheme and confirm the superiority of the new basis over the classical shifted Chebyshev polynomials when applied to time-fractional models.