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Efficient Shooting Methods for the Second-Order Ordinary Differential Equations

Chein-Shan Liu1

Department of Mechanicaland Mechatronic Engineering, Taiwan Ocean University, Keelung, Taiwan. E-mail: csliu@mail.ntou.edu.tw

Computer Modeling in Engineering & Sciences 2006, 15(2), 69-86. https://doi.org/10.3970/cmes.2006.015.069

Abstract

In this paper we will study the numerical integrations of second order boundary value problems under the imposed conditions at t=0 and t=T in a general setting. We can construct a compact space shooting method for finding the unknown initial conditions. The key point is based on the construction of a one-step Lie group element G(u0,uT) and the establishment of a mid-point Lie group element G(r). Then, by imposing G(u0,uT) = G(r) we can search the missing initial conditions through an iterative solution of the weighting factor r ∈ (0,1). Numerical examples were examined to convince that the new approach has high efficiency and accuracy with a fast convergence speed by solving r with a half-interval method. Even under a large span of the boundary coordinate, the new method is also applicable by requiring only a few iterations. The method is also extended to the BVP with general boundary conditions.

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Cite This Article

APA Style
Liu, C. (2006). Efficient shooting methods for the second-order ordinary differential equations. Computer Modeling in Engineering & Sciences, 15(2), 69-86. https://doi.org/10.3970/cmes.2006.015.069
Vancouver Style
Liu C. Efficient shooting methods for the second-order ordinary differential equations. Comput Model Eng Sci. 2006;15(2):69-86 https://doi.org/10.3970/cmes.2006.015.069
IEEE Style
C. Liu, "Efficient Shooting Methods for the Second-Order Ordinary Differential Equations," Comput. Model. Eng. Sci., vol. 15, no. 2, pp. 69-86. 2006. https://doi.org/10.3970/cmes.2006.015.069



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