A Stable and Convergent Implicit Finite Difference Scheme for Variable-Order Time-Fractional Convection–Diffusion Equations

Abstract

This study develops and investigates an implicit finite difference approach (FDA) for solving a class of linear variable order (VO) time-fractional partial differential equations (PDEs) that involve both convection and diffusion effects. The scheme is constructed by approximating the VO time-fractional derivative through a finite difference formulation and applying central difference operators for the spatial derivatives. A comprehensive theoretical analysis is carried out. By means of Fourier analysis, the method is shown to be unconditionally stable and convergent. In addition, the unique solvability of the resulting discrete system is demonstrated. To support the theoretical findings, several numerical experiments are pre sented, which confirm the accuracy, efficiency, and robustness of the proposed method.