Open Access

ARTICLE

A Lie-Group Shooting Method for Computing Eigenvalues and Eigenfunctions of Sturm-Liouville Problems

Chein-Shan Liu¹

1 Department of Mechanical & Mechatronic Engineering, Department of Harbor & River Engineering, Taiwan Ocean University, Keelung, Taiwan. E-mail: csliu@mail.ntou.edu.tw

Computer Modeling in Engineering & Sciences 2008, 26(3), 157-168. https://doi.org/10.3970/cmes.2008.026.157

Abstract

For the Sturm-Liouville eigenvalues problem we construct a very effective Lie-group shooting method (LGSM) to search the eigenvalues, and when eigenvalue is determined we can also search a missing left-boundary condition of the slope through a weighting factor r ∈ (0,1). Hence, the eigenvalues and eigenfunctions can be calculated with a better accuracy. Because a closed-form formula is derived to calculate unknown slope in terms of λ for the estimation of eigenvalues, the present method is easy to implement and has a low computational cost. Similarly by applying the LGSM to find a corresponding eigenfunction in terms of λ is easily carried out in a finer range of r. Numerical examples were examined to show that the Lie-group shooting method has a significantly improved accuracy than before.

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Keywords

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