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The Lie-Group Shooting Method for Solving Multi-dimensional Nonlinear Boundary Value Problems

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  • Chein-Shan Liu

    (National Taiwan University)

Abstract

This paper presents a Lie-group shooting method for the numerical solutions of multi-dimensional nonlinear boundary-value problems, which may exhibit multiple solutions. The Lie-group shooting method is a powerful technique to search unknown initial conditions through a single parameter, which is determined by matching the multiple targets through a minimum of an appropriately defined measure of the mis-matching error to target equations. Several numerical examples are examined to show that the novel approach is highly efficient and accurate. The number of solutions can be identified in advance, and all possible solutions can be numerically integrated by using the fourth-order Runge–Kutta method. We also apply the Lie-group shooting method to a numerical solution of an optimal control problem of the Duffing oscillator.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:152:y:2012:i:2:d:10.1007_s10957-011-9913-4
    DOI: 10.1007/s10957-011-9913-4
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    References listed on IDEAS

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    1. ,, 2000. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 16(2), pages 287-299, April.
    2. C. L. Chen & Y. C. Liu, 1998. "Solution of Two-Point Boundary-Value Problems Using the Differential Transformation Method," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 23-35, October.
    3. Attili, Basem S. & Syam, Muhammed I., 2008. "Efficient shooting method for solving two point boundary value problems," Chaos, Solitons & Fractals, Elsevier, vol. 35(5), pages 895-903.
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    Cited by:

    1. 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.
    2. 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.

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