Second order finite difference approximations for the two-dimensional time-space Caputo-Riesz fractional diffusion equation

In this paper, we discuss the time-space Caputo-Riesz fractional diffusion equation with variable coefficients on a finite domain. The finite difference schemes for this equation are provided. We theoretically prove and numerically verify that the implicit finite difference scheme is unconditionally stable (the explicit scheme is conditionally stable with the stability condition $\frac{\tau^{\gamma}}{(\Delta x)^{\alpha}}+\frac{\tau^{\gamma}}{(\Delta y)^{\beta}} <C$) and 2nd order convergent in space direction, and $(2-\gamma)$-th order convergent in time direction, where $\gamma \in(0,1]$.