Abstract
This paper addresses the numerical study of variable-order fractional differential equation based on finite-difference method. We utilize the implicit numerical scheme to find out the solution of two-dimensional variable-order fractional modified sub-diffusion equation. The discretized form of the variable-order Riemann-Liouville differential operator is used for the fractional variable-order differential operator. The theoretical analysis including for stability and convergence is made by the von Neumann method. The analysis confirmed that the proposed scheme is unconditionally stable and convergent. Numerical simulation results are given to validate the theoretical analysis as well as demonstrate the accuracy and efficiency of the implicit scheme.