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A novel semi-implicit finite difference approach for the Sobolev equation with generalized Burgers-type nonlinear term

Author

Listed:
  • Li, Kexin
  • Zhang, Hao
  • Nikan, Omid
  • Qiu, Wenlin

Abstract

This paper investigates the approximate solution of the Sobolev equation with generalized Burgers-type nonlinear term. For this purpose, a backward Euler (BE) semi-implicit difference approach is proposed, whose primary advantage is that, unlike general implicit finite difference (FD) schemes, it circumvents the necessity of iterative methods in computation and greatly reduces computing costs. A rigorous numerical analysis of the proposed strategy is then conducted based on the energy method. Specifically, the existence, uniqueness, and boundedness of the approximate solution are established via the Leray–Schauder theorem and discrete Sobolev’s inequality. Furthermore, the convergence and perturbation stability of the proposed strategy are derived using the discrete Gronwall inequality. Finally, the theoretical findings are corroborated through several numerical examples.

Suggested Citation

  • Li, Kexin & Zhang, Hao & Nikan, Omid & Qiu, Wenlin, 2026. "A novel semi-implicit finite difference approach for the Sobolev equation with generalized Burgers-type nonlinear term," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 244(C), pages 181-195.
    • Handle: RePEc:eee:matcom:v:244:y:2026:i:c:p:181-195
    DOI: 10.1016/j.matcom.2025.12.024
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    References listed on IDEAS

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    1. Singh, Ajeet & Cheng, Hanz Martin & Kumar, Naresh & Jiwari, Ram, 2025. "A high order numerical method for analysis and simulation of 2D semilinear Sobolev model on polygonal meshes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 227(C), pages 241-262.
    2. 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.
    3. 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.
    4. Zhao, Zhihui & Li, Hong & Wang, Jing, 2021. "The study of a continuous Galerkin method for Sobolev equation with space-time variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    5. Pany, Ambit K. & Bajpai, Saumya & Mishra, Soumyarani, 2020. "Finite element Galerkin method for 2D Sobolev equations with Burgers’ type nonlinearity," Applied Mathematics and Computation, Elsevier, vol. 387(C).
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