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

Authors

  • N.F. Valeev Institute of Mathematics with Computing Centre - Subdivision of the Ufa Federal Research Centre of RAS
  • Yu.V. Martynova RN-UfaNIPIneft
  • Ya.T. Sultanaev Bashkir State Pedagogical University n. a. M. Akmulla

DOI:

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

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.

Author Biographies

N.F. Valeev

Yu.V. Martynova

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

Ya.T. Sultanaev

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)].

Published

22-05-2016

How to Cite

Валеев Н.Ф., Мартынова Ю.В., Султанаев Я.Т. Solution of a Model Inverse Spectral Problem for the Sturm-Liouville Operator on a Graph // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2016. 17. 204-211. doi 10.26089/NumMet.v17r319

Issue

Section

Section 1. Numerical methods and applications

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