High-Order Local Artificial Boundary Conditions for the Fractional Diffusion Equation on One-Dimensional Unbounded Domain (Journal of Mathematical Study, 2017)
In this paper we consider the numerical solutions of the fractional diffusionequation on the unbounded spatial domain. With the application of Laplace transformation,we obtain one-way equations which absorb the wave touching on the artificialboundaries. By using Padé expansion to approximate the frequency in Laplace spaceand introducing auxiliary variables to reduce the order of the derivatives with respectto time $t,$ we achieve a system of ODEs within the artificial boundaries. This systemof ODEs, called high-order local absorbing boundary conditions (LABCs), reformulatethe fractional diffusion problem on the unbounded domain to an initial-boundary-value(IBV) problem on a bounded computational domain. A fully discrete implicitdifference scheme is constructed for the reduced problem. The stability and convergencerate are established for a finite difference scheme. Finally, numerical experimentsare given to demonstrate the efficiency and accuracy of our approach.