A finite difference scheme for semilinear space-fractional diffusion equations with time delay

A linearized quasi-compact finite difference scheme is proposed for semilinear space-fractional diffusion equations with a fixed time delay. The nonlinear source term is discretized and linearized by Taylor’s expansion to obtain a second-order discretization in time. The space-fractional derivatives are approximated by a weighted shifted Grünwald–Letnikov formula, which is of fourth order approximation under some smoothness assumptions of the exact solution. Under the local Lipschitz conditions, the solvability and convergence of the scheme are proved in the discrete maximum norm by the energy method. Numerical examples verify the theoretical predictions and illustrate the validity of the proposed scheme.