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 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 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.