A Variable Step Third‐Order Adams–Bashforth Method for Caputo Fractional Differential Equations

ABSTRACT This study proposes a numerical framework for solving fractional differential equations with nonuniform step sizes using the Caputo derivative formulation. The suggested scheme, known as the Fractional Explicit Method (FEM), improves on the traditional third‐order Adams–Bashforth approach by integrating Lagrange interpolation to approximate the history term in fractional operators. The formulation allows for varied time increments, increasing flexibility for problems where uniform discretization is impractical. A theoretical examination of the method is performed, including derivation of its order of accuracy and convergence properties, and the method is compared numerically against an established fractional predictor–corrector method. As an illustrative example, the approach is applied to a fractional dynamical model of smoking habit, and the resulting solutions are studied using graphical representations. Conditional stability of the scheme is also discussed via the linear test equation, though a full characterization of the stability region for variable step sizes is left for future work. The results show that the suggested approach is a reliable and efficient tool for numerically treating fractional differential systems with nonuniform temporal discretization.