Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time-space fractional reaction-diffusion equation

In this letter, we consider the numerical approximation of a two-dimensional distributed-order time-space fractional reaction-diffusion equation. The time- and space-fractional derivatives are considered in the senses of Caputo and Riesz, respectively. By using the composite mid-point quadrature, the original fractional problem is approximated by a multi-term time-space fractional differential equation. Then the multi-term Caputo fractional derivatives are discretized by the L2-1σ formula. We apply the Legendre-Galerkin spectral method for the spatial approximation. Two numerical experiments with smooth and non-smooth initial conditions, respectively, are performed to illustrate the robustness of the proposed method. The results show that: our scheme can arrive at the spectral accuracy (resp. algebraic accuracy) in space for the problem with smooth (resp. non-smooth) initial condition. For both of these two cases, our scheme can lead to the second-order accuracies in time. Additionally, the convergence rates in both spatial and temporal distributed-order variables are two.