Numerical simulation of the fractional sub-diffusion equation with absorbing boundary conditions on a two-dimensional space unbounded domain

Anomalous diffusion is an important phenomenon in various physical processes. It is distinguished by the nonlinear correlation of the mean square displacement (MSD) with time. Traditional numerical methods , such as the direct truncation boundary conditions (DTBCs), often suffer from poor approximation results. To overcome these limitations, we employ the approach of constructing absorbing boundary conditions (ABCs) to separate the infinite domain into a finite computational region and an infinite outer region. Through the application of the Laplace transform, the precise ABCs are gained, enhancing computational efficiency while maintaining the influence of unbounded regions. Based on the L1 discretization scheme, the solution for the time-fractional order diffusion equation is presented, along with proofs of its stability and convergence properties. By approximating the convolution kernel of Caputo fractional derivative with exponential sum, a fast algorithm is proposed to save the computational cost. Furthermore, the accuracy of the discretization scheme, the rationality of the ABCs and the influence of different parameters on the anomalous diffusion are studied through several numerical examples. The results in this paper are helpful for solving fractional sub-diffusion problems numerically and provide a reliable method for dealing with complex physical diffusion processes over unbounded domains.