A Morgan‐Voyce Polynomial Framework for Solving Variable‐Order Atangana–Baleanu Fractional Differential Equations

This paper presents a novel and efficient spectral collocation framework for solving nonlinear variable‐order fractional differential equations (VO‐FDEs) involving the Atangana–Baleanu–Caputo (ABC) operator. Shifted Morgan‐Voyce polynomials (SMVPs) are employed as basic functions to construct a new operational matrix specifically adapted to the variable‐order ABC operator. This matrix substantially reduces computational complexity while maintaining high accuracy. The proposed approach converts VO‐FDEs into tractable nonlinear algebraic systems. In contrast to existing polynomial‐based techniques, the framework demonstrates enhanced robustness in handling nonlinearities and variable memory effects, delivering superior precision and stability. Numerical experiments are conducted on challenging nonlinear and multiterm VO‐FDEs. Comparative analyses confirm the advantages of the Morgan–Voyce polynomials (MVPs)‐based scheme over classical spectral methods. These findings establish the proposed method as a versatile and reliable tool for tackling complex VO‐FDEs with nonsingular memory kernels.