Analysis and fast approximation of a steady-state spatially-dependent distributed-order space-fractional diffusion equation

We prove the wellposedness of a distributed-order space-fractional diffusion equation with variably distribution and its support, which could adequately model the challenging phenomena such as the anomalous diffusion in multiscale heterogeneous porous media, and smoothing properties of its solutions. We develop and analyze a collocation scheme for the proposed model based on the proved smoothing properties of the solutions. Furthermore, we approximately expand the stiffness matrix by a sum of Toeplitz matrices multiplied by diagonal matrices, which can be employed to develop the fast solver for the approximated system. We prove that it suffices to apply O(log N) terms of expansion to retain the accuracy of the numerical discretization of degree N, which reduces the storage of the stiffness matrix from O(N 2) to O(N log N), and the computational cost of matrix-vector multiplication from O(N 2) to O(N log2 N). Numerical results are presented to verify the effectiveness and the efficiency of the fast method.