Accurate Boundary Treatment for Riesz Space Fractional Diffusion Equations

We consider numerical boundary treatment for solving the Cauchy problems of the Riesz space fractional diffusion equation with compact initial data in one and two space dimension(s). First, the Riesz space fractional equation is semi-discretized into a lattice system. Then we derive an equivalent decoupled form for its dynamics using kernel functions. Series expansions and path integration are devised to numerically evaluate the kernel functions with high accuracy. For the first time, this allows an accurate numerical boundary treatment for the Riesz space fractional diffusion equation. Numerical results demonstrate the effectiveness of the method. The methodology may be extended to treat other fractional partial differential equations.