High-Order Local Artificial Boundary Conditions for the Fractional Diffusion Equation on One-Dimensional Unbounded Domain

In this paper we consider the numerical solutions of the fractional diffusion equation on the unbounded spatial domain. With the application of Laplace transformation, we obtain one-way equations which absorb the wave touching on the artificial boundaries. By using Padé expansion to approximate the frequency in Laplace space and introducing auxiliary variables to reduce the order of the derivatives with respect to time t, we achieve a system of ODEs within the artificial boundaries. This system of ODEs, called high-order local absorbing boundary conditions (LABCs), reformulate the fractional diffusion problem on the unbounded domain to an initial-boundary-value (IBV) problem on a bounded computational domain. A fully discrete implicit difference scheme is constructed for the reduced problem. The stability and convergence rate are established for a finite difference scheme. Finally, numerical experiments are given to demonstrate the efficiency and accuracy of our approach.