### Abstract

The inverse Sturm-Liouville problem finds its applications in the identification of mechanical properties and/or geometrical configurations of a vibrating continuous medium; however, this problem is hard to solve, either theoretically or numerically. Previously, Liu (2008a) has constructed a Lie-group shooting method to determine the eigenvalues, and the corresponding eigenfunctions, for the direct Sturm-Liouville problem. In this study, we are concerned with solving the inverse Sturm-Liouville problem, by developing a Lie-group of SL(2,R) to construct non-linear algebraic equations (NAEs), when discrete eigenvalues are specified. Our purpose here is to use these NAEs to solve the unknown function in the SturmLiouville operator. Then, we use a fictitious time integration method (FTIM) developed by Liu and Atiuri (2008), to find the potential function, impedance function or weighting function, in a discretized manner. Numerical examples are presented to show that the Lie-group and FTIM methods have a significantly improved accuracy, along with ease of numerical implementation. The numerical examples also include the inverse problem of determining the material properties and cross-sectional area of a tapered rod undergoing axial vibrations, when the eigen-frequencies are specified.

Original language | English |
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Pages (from-to) | 261-285 |

Number of pages | 25 |

Journal | CMES - Computer Modeling in Engineering and Sciences |

Volume | 36 |

Issue number | 3 |

State | Published - 2008 |

### Keywords

- Eigenfunctions
- Eigenvalues
- Fictitious time integration method (FTIM)
- Inverse Sturm-Liouville problem
- Inverse problem of a vibrating rod for specified frequencies
- Lie-group method
- Lie-group shooting method (LGSM)