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September 10, 2025Open Access

A Modified Collocation Technique for Addressing the Time-Fractional FitzHugh–Nagumo Differential Equation with Shifted Legendre Polynomials

Key Points

The method achieves accurate spectral approximations for the TFFNDP, outperforming other algorithms.
Comprehensive error studies validate the approach's effectiveness, demonstrating its reliability in solving complex equations.
The use of shifted Legendre polynomials facilitates orthogonality, enhancing convergence in numerical applications.
Benchmarking against analytical solutions highlights the proposed scheme's efficiency in resolving nonlinear differential problems.

Abstract

This study is devoted to solving the nonlinear inhomogeneous time-fractional FitzHugh–Nagumo differential problem (TFFNDP) using the spectral modified collocation method. The proposed algorithm uses non-symmetric polynomials, namely shifted Legendre polynomials (SLPs), which are orthogonal. The orthogonality property of SLPs and certain relations facilitate the acquisition of accurate spectral approximations. Comprehensive convergence and error studies are conducted to validate the accuracy of the suggested shifted Legendre expansion. Several numerical examples are presented to demonstrate the method’s effectiveness and accuracy. The proposed scheme is benchmarked against known analytical solutions and compared with other algorithms to ensure the applicability and efficiency of the proposed algorithm.

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