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

Author

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  • 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
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    References listed on IDEAS

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    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.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    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. 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.
    5. 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.

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