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  • Research Article
  • Open Access

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

Boundary Value Problems20082008:749865

  • Received: 8 September 2007
  • Accepted: 29 January 2008
  • Published:


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.


  • Partial Differential Equation
  • Ordinary Differential Equation
  • Parameter Identification
  • Functional Equation
  • Differential Operator

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Authors’ Affiliations

Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung, 20224, Taiwan
Department of Harbor and River Engineering, Taiwan Ocean University, Keelung, 20224, Taiwan


© 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.