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A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel

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Abstract

In this study, we propose a new nonlinear compact difference scheme (NCDS) for a fourth-order nonlinear Burgers type equation (FO-NBTE) with a weakly singular kernel. The Crank–Nicolson method and product-integration rule are utilized for discretizing the time derivative and Riemann–Liouville (R–L) time fractional integral term on the graded meshes, respectively. A nonlinear operator is constructed to discretize the nonlinear convective term, while the double reduced-order method is employed to handle fourth-order spatial derivative, then the FO-NBTE is transformed into three coupled nonlinear equations. The main advantage of the proposed NCDS is that it achieves second-order convergence in time and simultaneously fourth-order convergence in space, addressing the limitation observed in Tian et al. (Comput Appl Math 41:328, 2022) where the order of spatial convergence was restricted to second-order. In addition, a series of theoretical proofs for the proposed NCDS are presented, including the existence, stability, convergence and uniqueness. Finally, two numerical examples are given which consistent with the theoretical analysis.

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Authors and Affiliations

  1. School of Science, Hunan University of Technology, Zhuzhou, 412007, China

    Caojie Li, Haixiang Zhang & Xuehua Yang

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  1. Caojie Li
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  2. Haixiang Zhang
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  3. Xuehua Yang
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Correspondence to Haixiang Zhang.

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The work was supported by National Natural Science Foundation of China Mathematics Tianyuan Foundation (12226337, 12226340), Scientific Research Fund of Hunan Provincial Education Department (21B0550), Hunan Provincial Natural Science Foundation of China (2022JJ50083).

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Li, C., Zhang, H. & Yang, X. A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02039-x

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  • DOI: https://doi.org/10.1007/s12190-024-02039-x

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