An enhanced shifted Jacobi collocation framework for the time-fractional FitzHugh–Nagumo equation

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 [Formula: see text] and [Formula: see text] (both greater than [Formula: see text]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 [Formula: see text]-norm shows the method achieves spectral accuracy. Theoretical results also show that, with the optimal choice of [Formula: see text] and [Formula: see text], 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.