A Novel Fictitious Time Integration Method for Solving the Discretized Inverse Sturm-Liouville Problems, For Specified Eigenvalues

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