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Uniform convergence of compact and BDF methods for the space fractional semilinear delay reaction–diffusion equations

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  • Zhang, Qifeng
  • Ren, Yunzhu
  • Lin, Xiaoman
  • Xu, Yinghong

Abstract

In this article, two classes of finite difference methods are constructed to solve the space fractional semilinear delay reaction–diffusion equations. Firstly, fractional centered finite difference method in space and backward differential formula (BDF2) in time are employed to discrete the original equations. Secondly, the existence, uniqueness and convergence of the numerical method are analyzed at length by G-norm technique, and the convergence order is O(τ2+h2). Then the compact technique of finite difference method is used to further increase the accuracy in spatial direction, and a method with fourth accuracy in spatial dimension is obtained. The convergence orders are proved to be O(τ2+h4) in the sense of three classes of norms by GA-norm technique. Especially, we obtain the uniform convergence for the error estimation, which demonstrates competitive performance compared with the preceding work in the related references. Finally, two numerical examples are provided to verify our theoretical results and demonstrate the effectiveness of both methods when applied to simulate space fractional delay diffusive Nicholson’s blowflies equation.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:358:y:2019:i:c:p:91-110
    DOI: 10.1016/j.amc.2019.04.016
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    References listed on IDEAS

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    1. Wang, Dongling & Xiao, Aiguo & Yang, Wei, 2015. "Maximum-norm error analysis of a difference scheme for the space fractional CNLS," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 241-251.
    2. 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.
    3. 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.
    4. Zhang, Qifeng & Chen, Mengzhe & Xu, Yinghong & Xu, Dinghua, 2018. "Compact θ-method for the generalized delay diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 316(C), pages 357-369.
    5. 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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    Cited by:

    1. A. S. Hendy & R. H. De Staelen, 2020. "Theoretical Analysis (Convergence and Stability) of a Difference Approximation for Multiterm Time Fractional Convection Diffusion-Wave Equations with Delay," Mathematics, MDPI, vol. 8(10), pages 1-20, October.
    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. 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).
    4. Qin, Hongyu & Wu, Fengyan, 2019. "Several effective algorithms for nonlinear time fractional models," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.

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