Boundary Value Problems
Volume 2008 (2008), Article ID 749865, 18 pages
doi:10.1155/2008/749865
Research Article
Solving an Inverse Sturm-Liouville Problem by a Lie-Group Method
1Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung 20224, Taiwan
2Department of Harbor and River Engineering, Taiwan Ocean University, Keelung 20224, Taiwan
Received 8 September 2007; Revised 21 December 2007; Accepted 29 January 2008
Academic Editor: Colin Rogers
Copyright © 2008 Chein-Shan Liu. 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.
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.