A Unified Framework Using Orthogonal Hybrid Functions for Solving Linear and Nonlinear Fractional Differential Systems

This paper presents a novel and computationally efficient numerical method for solving systems of fractional-order differential equations using orthogonal hybrid functions (HFs). The proposed HFs are constructed by combining piecewise constant orthogonal sample-and-hold functions with piecewise linear orthogonal right-handed triangular functions, resulting in a flexible and accurate approximation basis. A central innovation of the method is the derivation of generalized one-shot operational matrices that approximate the Riemann–Liouville fractional integral, enabling direct integration of differential operators of arbitrary order. These matrices act as unified integrators for both integer and non-integer orders, enhancing the method’s applicability and scalability. A rigorous convergence analysis is provided, establishing theoretical guarantees for the accuracy of the numerical solution. The effectiveness and robustness of the approach are demonstrated through several benchmark problems, including fractional-order models related to smoking dynamics, lung cancer progression, and Hepatitis B infection. Comparative results highlight the method’s superior performance in terms of accuracy, numerical stability, and computational efficiency when applied to complex, nonlinear, and high-dimensional fractional-order systems.