Trapezoidal scheme for time–space fractional diffusion equation with Riesz derivative

In this paper, a finite difference scheme is proposed to solve time–space fractional diffusion equation which has second-order accuracy in both time and space direction. The time and space fractional derivatives are considered in the senses of Caputo and Riesz, respectively. First, the centered difference approach is used to approximate the Riesz fractional derivative in space. Then, the obtained fractional ordinary differential equations are transformed into equivalent Volterra integral equations. And then, the trapezoidal rule is utilized to approximate the Volterra integral equations. The stability and convergence of our scheme are proved via mathematical induction method. Finally, numerical experiments are performed to confirm the high accuracy and efficiency of our scheme. •An implicit numerical scheme for time–space fractional diffusion equation is proposed.•Centered difference approach is used to approximate the Riesz fractional derivative in space.•Trapezoidal formula is used to approximate equivalent integral equation.•Stability and convergence is investigated in detail.•Numerical simulation and error analysis is presented.