Matrix transfer technique for anomalous diffusion equation involving fractional Laplacian

The fractional Laplacian, (−△)^s, s∈(0,1), appears in a wide range of physical systems, including Lévy flights, some stochastic interfaces, and theoretical physics in connection to the problem of stability of the matter. In this paper, a matrix transfer technique (MTT) is employed combining with spectral/element method to solve fractional diffusion equations involving the fractional Laplacian. The convergence of the MTT method is analyzed by the abstract operator theory. Our method can be applied to solve various fractional equation involving fractional Laplacian on some complex domains. Numerical results indicate exponential convergence in the spatial discretization which is in good agreement with the theoretical analysis.