A fast implicit difference scheme for a new class of time distributed-order and space fractional diffusion equations with variable coefficients

作者：Jian, Huan-Yan^*; Huang, Ting-Zhu; Zhao, Xi-Le; Zhao, Yong-Liang

来源：Advances in Difference Equations, 2018, 2018(1): 205.

DOI：10.1186/s13662-018-1655-2

• Multi-term time-space fractional diffusion
• Distributed-order fractional derivative
• Shifted Grunwald discretization
• Toeplitz matrix
• Circulant preconditioner
• Krylov subspace method

Recently, several problems in mathematics, physics, and engineering have been modeled via distributed-order fractional diffusion equations. In this paper, a new class of time distributed-order and space fractional diffusion equations with variable coefficients on bounded domains and Dirichlet boundary conditions is considered. By performing numerical integration we transform the time distributed-order fractional diffusion equations into multiterm time-space fractional diffusion equations. An implicit difference scheme for the multiterm time-space fractional diffusion equations is proposed along with a discussion about the unconditional stability and convergence. Then, the fast Krylov subspace methods with suitable circulant preconditioners are developed to solve the resultant linear system in light of their Toeplitz-like structures. The aforementioned methods are proved to acquire the capability to reduce the memory storage of the proposed implicit difference scheme from O(M-2) to O(M) and the computational cost from O(M-3) to O(MlogM) during iteration procedures, where M is the number of grid nodes. Finally, numerical experiments are employed to support the theoretical findings and show the efficiency of the proposed methods.

• 出版日期2018-6-13
• 单位电子科技大学