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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:

Computer Modeling in Engineering & Sciences 2006, 15(2), 69-86.


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.


Cite This Article

Liu, C. (2006). Efficient Shooting Methods for the Second-Order Ordinary Differential Equations. CMES-Computer Modeling in Engineering & Sciences, 15(2), 69–86.

cc This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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