Efficient Shooting Methods for the Second-Order Ordinary Differential Equations

In this paperwewill studythenumerical in- tegrations of second order boundary value problems un- der the imposed conditions at t = 0a ndt = T in a gen- eral setting. We can construct a compact space shoot- ing 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 impos- ing 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 solvingr 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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