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Robust Implicit Difference Approach For The Time-Fractional Kuramoto–Sivashinsky Equation With The Non-Smooth Solution

Author

Listed:
  • XIANG-LIN HAN

    (Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China)

  • TAO GUO

    (Key Laboratory of Computing and Stochastic, Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China)

  • OMID NIKAN

    (School of Mathematics and Computer Science, Iran University of Science and Technology, Narmak, Tehran, Iran)

  • ZAKIEH AVAZZADEH

    (Department of Mathematical Sciences, University of South Africa, Florida, South Africa)

Abstract

This paper formulates the L1 implicit difference scheme (L1IDS) for the time-fractional Kuramoto–Sivashinsky equation (TFKSE) with non-smooth solution. The TFKSE is one of useful descriptions for modeling flame-propagation, viscous flow problems, and reaction–diffusion systems. The proposed method approximates the unknown solution by using two main stages. At the first stage, the L1 method with nonuniform meshes and the general centered difference method is adopted to discretize the Caputo fractional derivative and the spatial derivative, respectively. In the second stage, the fully-discrete L1IDS is established with the help of the Galerkin scheme based on piecewise linear test functions. Meanwhile, an iterative algorithm is adopted to solve the nonlinear systems. Furthermore, the convergence and stability of the proposed method are both demonstrated and confirmed numerically. Finally, three numerical examples highlight the accuracy and efficiency of the proposed strategy.

Suggested Citation

  • Xiang-Lin Han & Tao Guo & Omid Nikan & Zakieh Avazzadeh, 2023. "Robust Implicit Difference Approach For The Time-Fractional Kuramoto–Sivashinsky Equation With The Non-Smooth Solution," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(04), pages 1-12.
    • Handle: RePEc:wsi:fracta:v:31:y:2023:i:04:n:s0218348x23400613
    DOI: 10.1142/S0218348X23400613
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    Keywords

    Fractional Kuramoto–Sivashinsky Equation; Caputo Fractional Derivative; L1 Formula; Graded Meshes; Stability and Convergence;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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