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Modified asymptotic solutions for second-order nonlinear singularly perturbed boundary value problems

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  • Liu, Chein-Shan
  • Chang, Chih-Wen

Abstract

We introduce a coordinate transformation of independent variable, such that the second-order nonlinear singularly perturbed boundary value problem (SPBVP) in the transformed coordinate is less stiff within the boundary layer. An initial value problem for a new dependent variable can be derived easily through the variable transformation. While the zero initial values are given, an unknown terminal value of the new variable at the right end is determined iteratively. We propose the modifications of the asymptotic solution and the uniform approximate solution of the SPBVP; hence, the modified analytic solutions can exactly satisfy both the boundary conditions at two ends. Some examples confirm that the novel methods can achieve better analytic and numerical solutions of the nonlinear SPBVP.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:193:y:2022:i:c:p:139-152
    DOI: 10.1016/j.matcom.2021.10.005
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    References listed on IDEAS

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    1. Liu, Chein-Shan, 2018. "Solving singularly perturbed problems by a weak-form integral equation with exponential trial functions," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 154-174.
    2. Nurettin Doğan & Vedat Suat Ertürk & Ömer Akın, 2012. "Numerical Treatment of Singularly Perturbed Two-Point Boundary Value Problems by Using Differential Transformation Method," Discrete Dynamics in Nature and Society, Hindawi, vol. 2012, pages 1-10, April.
    3. Chein-Shan Liu, 2012. "The Lie-Group Shooting Method for Solving Multi-dimensional Nonlinear Boundary Value Problems," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 468-495, February.
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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.

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