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A boundary shape function iterative method for solving nonlinear singular boundary value problems

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  • Liu, Chein-Shan
  • El-Zahar, Essam R.
  • Chang, Chih-Wen

Abstract

In this paper, a novel iterative algorithm is developed to solve second-order nonlinear singular boundary value problem, whose solution exactly satisfies the Robin boundary conditions specified on the boundaries of a unit interval. The boundary shape function is designed such that the boundary conditions can be fulfilled automatically, which renders a new algorithm with the solution playing the role of a boundary shape function. When the free function is viewed as a new variable, the original singular boundary value problem can be properly transformed to an initial value problem. For the new variable the initial values are given, whereas two unknown terminal values are determined iteratively by integrating the transformed ordinary differential equation to obtain the new terminal values until they are convergent. As a consequence, very accurate solutions for the nonlinear singular boundary value problems can be obtained through a few iterations. The present method is different from the traditional shooting method, which needs to guess initial values and solve nonlinear algebraic equations to approximate the missing initial values. As practical applications of the present method, we solve the Blasius equation for describing the boundary layer behavior of fluid flow over a flat plate, where the Crocco transformation is employed to transform the third-order differential equation to a second-order singular differential equation. We also solve a nonlinear singular differential equation of a pressurized spherical membrane with a strong singularity.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:187:y:2021:i:c:p:614-629
    DOI: 10.1016/j.matcom.2021.03.030
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    References listed on IDEAS

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    Cited by:

    1. 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.
    2. Chein-Shan Liu & Essam R. El-Zahar & Chih-Wen Chang, 2022. "Higher-Order Asymptotic Numerical Solutions for Singularly Perturbed Problems with Variable Coefficients," Mathematics, MDPI, vol. 10(15), pages 1-20, August.
    3. Ramos, Higinio & Rufai, Mufutau Ajani, 2022. "An adaptive pair of one-step hybrid block Nyström methods for singular initial-value problems of Lane–Emden–Fowler type," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 497-508.

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