A nonuniform difference scheme for time-space fractional integro-differential equation with initial singularity

Purpose In this study, we want to develop a high-order numerical method to solve a one-dimensional time-space fractional integro-differential equation with singular kernels. Design/methodology/approach A nonuniform difference scheme is developed to solve a class of time-space fractional integro-differential equations with weak singularity. The Riemann–Liouville integral and the Caputo fractional derivative are approximated using the trapezoidal product integral method and the Alikhanov technique with graded meshes, respectively. After using the fractional central difference formula to approximate the Riesz derivative, the fully discrete difference scheme is obtained. Findings The stability and convergence of the proposed scheme are strictly analyzed with second-order accuracy in time and space. The theoretical analysis is verified by numerical results. Originality/value One of the main challenges in dealing with fractional partial differential equations is the initial singularity. Our presented numerical method can achieve the second-order accuracy in time and space and overcome the order reduction from initial singularity.