A fast numerical method for two-dimensional Riesz space fractional diffusion equations on a convex bounded region

Fractional differential equations have attracted considerable attention because of their many applications in physics, geology, biology, chemistry, and finance. In this paper, a two-dimensional Riesz space fractional diffusion equation on a convex bounded region (2D-RSFDE-CBR) is considered. These regions are more general than rectangular or circular domains. A novel alternating direction implicit method for the 2D-RSFDE-CBR with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the method are discussed. The resulting linear systems are Toeplitz-like and are solved by the preconditioned conjugate gradient method with a suitable circulant preconditioner. By the fast Fourier transform, the method only requires a computational cost of O(nlog⁡n) per time step. These numerical techniques are used for simulating a two-dimensional Riesz space fractional FitzHugh-Nagumo model. The numerical results demonstrate the effectiveness of the method. These techniques can be extended to three spatial dimensions, which will be the topic of our future research.