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

Chein-Shan Liu ()
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Chein-Shan Liu: National Taiwan University

Journal of Optimization Theory and Applications, 2012, vol. 152, issue 2, No 11, 468-495

Abstract: 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.

Keywords: Lie-group shooting method; Lie-group shooting equation; Multi-dimensional nonlinear boundary value problem; Multiple solutions; Optimal control (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10957-011-9913-4

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