IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v166y2019icp298-314.html
   My bibliography  Save this article

An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations

Author

Listed:
  • Qiu, Wenlin
  • Chen, Hongbin
  • Zheng, Xuan

Abstract

An implicit difference scheme with the truncation of order 2−α(0<α<1) for time and order 2 for space is considered for the one-dimensional time-fractional Burgers equations. The L1-discretization formula of the fractional derivative in the Caputo sense is employed. The second-order spatial derivative is approximated by means of the three-point centered formula and the nonlinear convection term is discretized by the Galerkin method based on piecewise linear test functions. The stability and convergence in the L∞ norm are proved by the energy method. Meanwhile, a novel iterative algorithm is proposed and implemented to solve the nonlinear systems. Numerical experiment shows that the results are consistent with our theoretical analysis, and the comparison between the proposed iterative algorithm and the existing methods shows the efficiency of our method.

Suggested Citation

  • Qiu, Wenlin & Chen, Hongbin & Zheng, Xuan, 2019. "An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 298-314.
    • Handle: RePEc:eee:matcom:v:166:y:2019:i:c:p:298-314
    DOI: 10.1016/j.matcom.2019.05.017
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475419301934
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2019.05.017?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. Momani, Shaher, 2006. "Non-perturbative analytical solutions of the space- and time-fractional Burgers equations," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 930-937.
    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. Chen, Hao & Nikan, Omid & Qiu, Wenlin & Avazzadeh, Zakieh, 2023. "Two-grid finite difference method for 1D fourth-order Sobolev-type equation with Burgers’ type nonlinearity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 209(C), pages 248-266.
    2. Qiao, Leijie & Qiu, Wenlin & Xu, Da, 2023. "Error analysis of fast L1 ADI finite difference/compact difference schemes for the fractional telegraph equation in three dimensions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 205-231.
    3. Qiu, Wenlin & Xu, Da & Guo, Jing, 2021. "Numerical solution of the fourth-order partial integro-differential equation with multi-term kernels by the Sinc-collocation method based on the double exponential transformation," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    4. Peng, Xiangyi & Xu, Da & Qiu, Wenlin, 2023. "Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers’ equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 702-726.
    5. 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. Eriqat, Tareq & El-Ajou, Ahmad & Oqielat, Moa'ath N. & Al-Zhour, Zeyad & Momani, Shaher, 2020. "A New Attractive Analytic Approach for Solutions of Linear and Nonlinear Neutral Fractional Pantograph Equations," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    2. Odibat, Zaid M., 2009. "Computational algorithms for computing the fractional derivatives of functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(7), pages 2013-2020.
    3. Yadav, Swati & Pandey, Rajesh K., 2020. "Numerical approximation of fractional burgers equation with Atangana–Baleanu derivative in Caputo sense," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    4. Memarbashi, Reza, 2008. "Numerical solution of the Laplace equation in annulus by Adomian decomposition method," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 138-143.
    5. Yu, Yongguang & Li, Han-Xiong, 2009. "Application of the multistage homotopy-perturbation method to solve a class of hyperchaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2330-2337.
    6. Rashid Nawaz & Laiq Zada & Abraiz Khattak & Muhammad Jibran & Adam Khan, 2019. "Optimum Solutions of Fractional Order Zakharov–Kuznetsov Equations," Complexity, Hindawi, vol. 2019, pages 1-9, December.
    7. Yu, Yongguang & Li, Han-Xiong, 2008. "The synchronization of fractional-order Rössler hyperchaotic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(5), pages 1393-1403.
    8. Safyan Mukhtar & Salah Abuasad & Ishak Hashim & Samsul Ariffin Abdul Karim, 2020. "Effective Method for Solving Different Types of Nonlinear Fractional Burgers’ Equations," Mathematics, MDPI, vol. 8(5), pages 1-19, May.

    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:matcom:v:166:y:2019:i:c:p:298-314. 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: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    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.