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Two novel meshless methods for solving general fourth-order PDEs in irregular domains: Applications to plate bending problems

Author

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  • Lin, Xubo
  • Reutskiy, Sergiy
  • Lin, Ji
  • Lu, Jun

Abstract

This paper presents two RBF-based meshless techniques for solving the fourth-order elliptic partial differential equations (PDEs) in complex geometries. The first method assumes splitting the given equation into a system of two Poisson ones. The second technique directly solves the given PDE of the fourth order if the splitting method does not work. The unifying idea of the both techniques is that the two problems (1) approximation of BC and (2) satisfaction of the governing PDE inside the solution domain are considered separately. Besides, the both techniques utilize series of the so-called modified RBFs (MRBFs) for approximation of the solution inside the solution domain. The direct approach yields a significantly higher accuracy and can be applied to a wide class of plate bending problems. The efficacy of the proposed methodologies is validated by illustrative test cases with different configuration and boundary conditions.

Suggested Citation

  • Lin, Xubo & Reutskiy, Sergiy & Lin, Ji & Lu, Jun, 2026. "Two novel meshless methods for solving general fourth-order PDEs in irregular domains: Applications to plate bending problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 248(C), pages 794-823.
    • Handle: RePEc:eee:matcom:v:248:y:2026:i:c:p:794-823
    DOI: 10.1016/j.matcom.2026.04.029
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    References listed on IDEAS

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    1. Karageorghis, Andreas, 2022. "A time–efficient variable shape parameter Kansa–radial basis function method for the solution of nonlinear boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    2. Qu, Wenzhen & Sun, Linlin & Li, Po-Wei, 2021. "Bending analysis of simply supported and clamped thin elastic plates by using a modified version of the LMFS," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 347-357.
    3. Chen, C.S. & Shen, Shu-Hui & Dou, Fangfang & Li, J., 2020. "The LMAPS for solving fourth-order PDEs with polynomial basis functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 500-515.
    4. Lin, Ji & Reutskiy, Sergiy, 2020. "A cubic B-spline semi-analytical algorithm for simulation of 3D steady-state convection-diffusion-reaction problems," Applied Mathematics and Computation, Elsevier, vol. 371(C).
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