IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v336y2018icp231-248.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300318304077
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2018.05.004?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Omran, A.K. & Zaky, M.A. & Hendy, A.S. & Pimenov, V.G., 2022. "An easy to implement linearized numerical scheme for fractional reaction–diffusion equations with a prehistorical nonlinear source function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 218-239.
    2. Guo, Lin & Zhao, Xi-Le & Gu, Xian-Ming & Zhao, Yong-Liang & Zheng, Yu-Bang & Huang, Ting-Zhu, 2021. "Three-dimensional fractional total variation regularized tensor optimized model for image deblurring," Applied Mathematics and Computation, Elsevier, vol. 404(C).
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Zhang, Qifeng & Ren, Yunzhu & Lin, Xiaoman & Xu, Yinghong, 2019. "Uniform convergence of compact and BDF methods for the space fractional semilinear delay reaction–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 91-110.
    2. Omran, A.K. & Zaky, M.A. & Hendy, A.S. & Pimenov, V.G., 2022. "An easy to implement linearized numerical scheme for fractional reaction–diffusion equations with a prehistorical nonlinear source function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 218-239.
    3. Zhang, Xue & Gu, Xian-Ming & Zhao, Yong-Liang & Li, Hu & Gu, Chuan-Yun, 2024. "Two fast and unconditionally stable finite difference methods for Riesz fractional diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 462(C).
    4. Iyiola, O.S. & Tasbozan, O. & Kurt, A. & Çenesiz, Y., 2017. "On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 94(C), pages 1-7.
    5. Yu, Hao & Wu, Boying & Zhang, Dazhi, 2018. "A generalized Laguerre spectral Petrov–Galerkin method for the time-fractional subdiffusion equation on the semi-infinite domain," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 96-111.
    6. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    7. Wang, Jinfeng & Yin, Baoli & Liu, Yang & Li, Hong & Fang, Zhichao, 2021. "Mixed finite element algorithm for a nonlinear time fractional wave model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 60-76.
    8. Yan, Xiong-bin & Zhang, Zheng-qiang & Wei, Ting, 2022. "Simultaneous inversion of a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    9. Kashfi Sadabad, Mahnaz & Jodayree Akbarfam, Aliasghar, 2021. "An efficient numerical method for estimating eigenvalues and eigenfunctions of fractional Sturm–Liouville problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 547-569.
    10. Bu, Weiping & Zhao, Yanmin & Shen, Chen, 2021. "Fast and efficient finite difference/finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equation," Applied Mathematics and Computation, Elsevier, vol. 398(C).
    11. Allaberen Ashyralyev & Deniz Agirseven, 2019. "Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations," Mathematics, MDPI, vol. 7(12), pages 1-38, December.
    12. Sarita Gajbhiye Meshram & Vijay P. Singh & Ozgur Kisi & Chandrashekhar Meshram, 2021. "Soil erosion modeling of watershed using cubic, quadratic and quintic splines," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 108(3), pages 2701-2719, September.
    13. Sowa, Marcin, 2018. "Application of SubIval in solving initial value problems with fractional derivatives," Applied Mathematics and Computation, Elsevier, vol. 319(C), pages 86-103.
    14. Pezza, L. & Pitolli, F., 2018. "A multiscale collocation method for fractional differential problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 147(C), pages 210-219.
    15. Li, Lili & Zhou, Boya & Chen, Xiaoli & Wang, Zhiyong, 2018. "Convergence and stability of compact finite difference method for nonlinear time fractional reaction–diffusion equations with delay," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 144-152.
    16. Wang, Yihong & Cao, Jianxiong, 2021. "A tailored finite point method for subdiffusion equation with anisotropic and discontinuous diffusivity," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    17. Luo, Wei-Hua & Gu, Xian-Ming & Carpentieri, Bruno, 2022. "A hybrid triangulation method for banded linear systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 97-108.
    18. Jian, Huan-Yan & Huang, Ting-Zhu & Ostermann, Alexander & Gu, Xian-Ming & Zhao, Yong-Liang, 2021. "Fast numerical schemes for nonlinear space-fractional multidelay reaction-diffusion equations by implicit integration factor methods," Applied Mathematics and Computation, Elsevier, vol. 408(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:336:y:2018:i:c:p:231-248. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.