Numerical approximation of ABC-type fractional equations using a Vieta-Lucas polynomial

Abstract

This paper presents a spectral collocation method based on shifted VietaLucas polynomials (SVLPs) for the numerical approximation of multi-term variable-order fractional differential equations (MT-VOFDEs) involving the Atangana-Baleanu-Caputo (ABC) fractional derivative in variable-order form. A novel operational matrix of the ABC derivative is derived in the SVLP framework, enabling an efficient transformation of the fractional system into an algebraic system. The stability and convergence of the method are theoretically analyzed. The proposed SVLP-based approach is then applied to several test problems, including systems of MT-VOFDEs with known exact polynomial solutions. Numerical results demonstrate exponential convergence, high accuracy, and reduced computational cost compared to other polynomial-based collocation methods. The orthogonality and recursive structure of the SVLPs contribute to computational efficiency and robustness. These findings highlight the effectiveness of the proposed method for solving complex fractional models governed by non-singular kernel operators.