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.