Open Access

Solving an Inverse Sturm-Liouville Problem by a Lie-Group Method

Boundary Value Problems20082008:749865

https://doi.org/10.1155/2008/749865

Received: 8 September 2007

Accepted: 29 January 2008

Published: 29 January 2008

Abstract

Solving an inverse Sturm-Liouville problem requires a mathematical process to determine unknown function in the Sturm-Liouville operator from given data in addition to the boundary values. In this paper, we identify a Sturm-Liouville potential function by using the data of one eigenfunction and its corresponding eigenvalue, and identify a spatial-dependent unknown function of a Sturm-Liouville differential operator. The method we employ is to transform the inverse Sturm-Liouville problem into a parameter identification problem of a heat conduction equation. Then a Lie-group estimation method is developed to estimate the coefficients in a system of ordinary differential equations discretized from the heat conduction equation. Numerical tests confirm the accuracy and efficiency of present approach. Definite and random disturbances are also considered when comparing the present method with that by using a technique of numerical differentiation.

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Authors’ Affiliations

(1)
Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University
(2)
Department of Harbor and River Engineering, Taiwan Ocean University

Copyright

© Chein-Shan Liu. 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.