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Boundary shape function iterative method for nonlinear second-order boundary value problems with nonlinear boundary conditions

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  • Deng, Aimin
  • Lin, Ji
  • Liu, Chein-Shan

Abstract

Nonlinear boundary conditions are difficult to be fulfilled exactly, when one employs numerical methods to treat a highly nonlinear boundary value problem (NBVP). In this paper, a novel iterative algorithm to solve NBVP involved with two coupled nonlinear boundary conditions at two-end of a unit interval is developed, of which the solution can satisfy the nonlinear boundary conditions automatically. By letting the free function in the boundary shape function (BSF) be a new variable, an initial value problem (IVP) is created from the second-order NBVP. While the initial values of the new variable are given, the terminal values are viewed as unknown parameters to be determined iteratively. Therefore, a very accurate solution for the NBVP with nonlinear boundary conditions can be quickly determined through a few iterations. Some numerical examples confirm the efficiency and accuracy of the proposed iterative scheme, wherein the examples with multiple solutions and unique solution are worked out.

Suggested Citation

  • Deng, Aimin & Lin, Ji & Liu, Chein-Shan, 2022. "Boundary shape function iterative method for nonlinear second-order boundary value problems with nonlinear boundary conditions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 539-551.
    • Handle: RePEc:eee:matcom:v:194:y:2022:i:c:p:539-551
    DOI: 10.1016/j.matcom.2021.12.013
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    References listed on IDEAS

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    1. Dehghan, Mehdi, 2007. "The one-dimensional heat equation subject to a boundary integral specification," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 661-675.
    2. Zheyan Zhou & Jianhe Shen, 2010. "A Second-Order Boundary Value Problem with Nonlinear and Mixed Boundary Conditions: Existence, Uniqueness, and Approximation," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-20, August.
    3. Liu, Chein-Shan & El-Zahar, Essam R. & Chang, Chih-Wen, 2021. "A boundary shape function iterative method for solving nonlinear singular boundary value problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 614-629.
    4. Liu, Chein-Shan & Chang, Chih-Wen, 2022. "Modified asymptotic solutions for second-order nonlinear singularly perturbed boundary value problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 139-152.
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