An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order

In this paper, we study a time-fractional diffusion equation with a time-invariant type variable fractional order. We propose an implicit finite difference scheme to approximate the variable-order Caputo fractional derivative, while the central difference method is employed to discretize the spatial differential operator. A novel decomposition of the temporal discretization coefficients is adopted to overcome their loss of monotonicity due to the impact of the variable order and thus to support the proof of the convergence and unconditionally stability of the numerical scheme. Numerical examples are presented to verify the effectiveness of the proposed method.