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A fast second-order implicit scheme for non-linear time-space fractional diffusion equation with time delay and drift term

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  • Zhao, Yong-Liang
  • Zhu, Pei-Yong
  • Luo, Wei-Hua

Abstract

In this paper, a second-order accurate implicit scheme based on the L2–1σ formula for temporal variable and the fractional centered difference formula for spatial discretization is established to solve a class of time-space fractional diffusion equations with time drift term and non-linear delayed source function. The stability of this scheme is proved rigorously by the discrete energy method under several auxiliary assumptions, then we theoretically and numerically show that the proposed scheme converges in the L2-norm with the order O((Δt)2+h2) with time step Δt and mesh size h. Moreover, it finds that the discreted linear systems are symmetric Toeplitz systems. In order to solve these systems efficiently, the conjugate gradient method with suitable circulant preconditioners is designed. In each iterative step, the storage requirements and the computational complexity of the resulting equations are O(N) and O(NlogN) respectively, where N is the number of grid nodes. Numerical experiments are carried out to demonstrate the effectiveness of our proposed circulant preconditioners.

Suggested Citation

  • Zhao, Yong-Liang & Zhu, Pei-Yong & Luo, Wei-Hua, 2018. "A fast second-order implicit scheme for non-linear time-space fractional diffusion equation with time delay and drift term," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 231-248.
    • Handle: RePEc:eee:apmaco:v:336:y:2018:i:c:p:231-248
    DOI: 10.1016/j.amc.2018.05.004
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    1. Sun, Hong & Sun, Zhi-zhong & Gao, Guang-hua, 2016. "Some high order difference schemes for the space and time fractional Bloch–Torrey equations," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 356-380.
    2. Luo, Wei-Hua & Huang, Ting-Zhu & Wu, Guo-Cheng & Gu, Xian-Ming, 2016. "Quadratic spline collocation method for the time fractional subdiffusion equation," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 252-265.
    3. Hao, Zhaopeng & Fan, Kai & Cao, Wanrong & Sun, Zhizhong, 2016. "A finite difference scheme for semilinear space-fractional diffusion equations with time delay," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 238-254.
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    3. Wang, Furong & Yang, Xuehua & Zhang, Haixiang & Wu, Lijiao, 2022. "A time two-grid algorithm for the two dimensional nonlinear fractional PIDE with a weakly singular kernel," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 38-59.

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