TT-M FE method for a 2D nonlinear time distributed-order and space fractional diffusion equation

2021 ◽  
Vol 181 ◽  
pp. 117-137
Author(s):  
Xinghua Gao ◽  
Baoli Yin ◽  
Hong Li ◽  
Yang Liu
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Fe Method ◽  
Space Fractional Diffusion Equation ◽  
Distributed Order

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

2021 ◽  
Vol 24 (5) ◽  
pp. 1477-1506
Author(s):  
Jinhong Jia ◽  
Xiangcheng Zheng ◽  
Hong Wang
Keyword(s):  
Diffusion Equation ◽  
Stiffness Matrix ◽  
Toeplitz Matrices ◽  
Computational Cost ◽  
Fractional Diffusion ◽  
Heterogeneous Porous Media ◽  
Fractional Diffusion Equation ◽  
Fast Method ◽  
Space Fractional Diffusion Equation ◽  
Distributed Order

Abstract 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.

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