DOI: https://doi.org/10.26089/NumMet.v17r319

Solution of a model inverse spectral problem for the Sturm-Liouville operator on a graph

Authors

  • N.F. Valeev
  • Yu.V. Martynova
  • Ya.T. Sultanaev

Keywords:

spectral theory of differential operators
geometric graph
Sturm-Liouville operator
spectral problems

Abstract

A model inverse spectral problem for the Sturm-Liouville operator on a geometric graph is studied. This problem consists in finding N parameters of the boundary conditions using its N known eigenvalues. It is shown that the problem under consideration possess the property of a monotonic dependence of its eigenvalues on the parameters of the boundary conditions. This problem is reduced to a multiparameter inverse spectral problem for the operator in a finite-dimensional space. A new algorithm for the numerical solution of this problem is proposed.

New computational technologies

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Published

2016-05-22

Issue

Vol. 17 (2016): Issue 3.

Section

Section 1. Numerical methods and applications

Author Biographies

N.F. Valeev

Institute of Mathematics with Computing Centre - Subdivision of the Ufa Federal Research Centre of RAS
• Senior Researcher

Yu.V. Martynova

RN-UfaNIPIneft, LLC,
ул. Бехтерева, 3, 450103, Уфа
• Leading Researcher

Ya.T. Sultanaev

Bashkir State Pedagogical University n. a. M. Akmulla
• Professor

References

  1. N. F. Valeev, “The Multiparameter Inverse Spectral Problems for Finite-Dimensional Operators,” Ufimsk. Mat. Zh. 2 (2), 3-19 (2010).
  2. N. F. Valeev, S. A. Rabtsevich, and E. R. Nugumanov, “About the Problem of Identification of the Sturm-Liouville Operator’s Boundary Conditions by the Spectrum,” Sistemy Upravl. Inform. Tekhnol., No. 3, 9-12 (2009).
  3. G. М. L. Gladwell, Inverse Problems in Vibration (Kluwer, Dordrecht, 2004; Inst. Komp’yut. Issled., Izhevsk, 2008).
  4. Yu. V. Martynova, “Model Inverse Spectral Problem for Sturm-Liouville Operator on Geometrical Graph,” Vestn. Bashkir Univ. 16 (1), 4-10 (2011).
  5. Yu. V. Pokornyi, O. M. Penkin, V. L. Pryadiev, A. V. Borovskikh, K. P. Lazarev, and S. A. Shabrov, Differential Equations on Geometric Graphs (Fizmatlit, Moscow, 2005) [in Russian].
  6. V. A. Sadovnichii, Ya. T. Sultanaev, and N. F. Valeev, “Multiparameter Inverse Spectral Problems and Their Applications,” Dokl. Akad. Nauk 426 (4), 457-460 (2009) [Dokl. Math. 79 (3), 390-393 (2009)].