@Article{cmes.2008.036.261,
AUTHOR = {Chein-Shan  Liu, Satya N.  Atluri},
TITLE = {A Novel Fictitious Time Integration Method for Solving the Discretized Inverse Sturm-Liouville Problems, For Specified Eigenvalues},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {36},
YEAR = {2008},
NUMBER = {3},
PAGES = {261--286},
URL = {http://www.techscience.com/CMES/v36n3/26741},
ISSN = {1526-1506},
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 <i>SL</i>(2,R) to construct nonlinear algebraic equations (NAEs), when discrete eigenvalues are specified. Our purpose here is to use these NAEs to solve the unknown function in the Sturm-Liouville operator. Then, we use a fictitious time integration method (FTIM) developed by Liu and Atluri (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.},
DOI = {10.3970/cmes.2008.036.261}
}