A finite difference scheme for the nonlinear time-fractional partial integro-differential equation

In this paper, a finite difference scheme is proposed for solving the nonlinear time-fractional integro-differential equation. This model involves two nonlocal terms in time, ie, a Caputo time-fractional derivative and an integral term with memory. The existence of numerical solutions is shown by the Leray-Schauder theorem. And we obtain the discrete L-2 stability and convergence with second order in time and space by the discrete energy method. Then the uniqueness of numerical solutions is derived. Moreover, an iterative algorithm is designed for solving the derived nonlinear system. Numerical examples are presented to validate the theoretical findings and the efficiency of the proposed algorithm.