This work presents a new spectral collocation method using shifted Jacobi polynomials to solve the nonlinear inhomogeneous time-fractional FitzHugh–Nagumo differential equation numerically. By adjusting the Jacobi parameters 𝛼 and 𝛽 (both greater than -1), the method improves accuracy and stability. We develop new operational matrices for both integer-order and Caputo fractional derivatives based on the shifted Jacobi basis, with detailed proofs. A thorough convergence analysis in the weighted L 2 -norm shows the method achieves spectral accuracy. Theoretical results also show that, with the optimal choice of 𝛼 and 𝛽, this approach performs better than traditional Legendre-based methods, significantly reducing errors. The method is efficiently implemented and can handle different fractional orders and parameter settings.
Youssri et al. (Thu,) studied this question.