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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2015, Volume 8, Issue 3, Pages 116–126 (Mi vyuru279)  

This article is cited in 5 scientific papers (total in 5 papers)

Mathematical Modelling

A numerical method for inverse spectral problems

S. I. Kadchenkoa, G. A. Zakirovab

a Magnitogorsk State Technical University named after G. I. Nosov, Magnitogorsk, Russian Federation
b South Ural State University, Chelyabinsk, Russian Federation

Abstract: Basing on the Galerkin methods, we develop a new numerical method for solving the inverse spectral problems generated by discrete lower semibounded operators. The restrictions on the perturbing operator are relaxed in comparison with the method based on the theory of regular traces. A Fredholm integral equation of the first kind enables us to recover the values of the perturbing operator at the discretization nodes. We tested the method on spectral problems for the Sturm–Liouville operator, and the results of numerous simulations demonstrate its computational efficiency.
We found simple formulas for the eigenvalues of a discrete lower semibounded operator avoiding the roots of the corresponding secular equations. The calculation of eigenvalues of these operators can start at an arbitrary index independently of the (un)availability of the eigenvalues with smaller indices. For perturbed selfadjoint operators we can calculate eigenvalues with large indices when the Galerkin method becomes difficult to apply.

Keywords: inverse spectral problem; discrete selfadjoint operators; eigenvalues; eigenfunctions; ill-posed problems.

DOI: https://doi.org/10.14529/mmp150307

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Bibliographic databases:

UDC: 519.642.8
MSC: 47A75
Received: 09.02.2015
Language:

Citation: S. I. Kadchenko, G. A. Zakirova, “A numerical method for inverse spectral problems”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8:3 (2015), 116–126

Citation in format AMSBIB
\Bibitem{KadZak15}
\by S.~I.~Kadchenko, G.~A.~Zakirova
\paper A numerical method for inverse spectral problems
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2015
\vol 8
\issue 3
\pages 116--126
\mathnet{http://mi.mathnet.ru/vyuru279}
\crossref{https://doi.org/10.14529/mmp150307}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000422201600007}
\elib{http://elibrary.ru/item.asp?id=24078399}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. E. V. Bychkov, Ya. O. Al'-Ani, “A linearized model of vibrations in the DNA molecule in the quasi-Banach spaces”, J. Comp. Eng. Math., 3:1 (2016), 20–26  mathnet  crossref  mathscinet  zmath  elib
    2. S. I. Kadchenko, G. A. Zakirova, “Calculation of eigenvalues of discrete semibounded differential operators”, J. Comp. Eng. Math., 4:1 (2017), 38–47  mathnet  crossref  mathscinet  elib
    3. I. S. Strepetova, L. M. Fatkullina, G. A. Zakirova, “Spectral problems for one mathematical model of hydrodynamics”, J. Comp. Eng. Math., 4:1 (2017), 48–56  mathnet  crossref  mathscinet  elib
    4. A. M. Akhtyamov, Kh. R. Mamedov, E. N. Yilmazoglu, “Boundary inverse problem for star-shaped graph with different densities strings-edges”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 11:3 (2018), 5–17  mathnet  crossref  elib
    5. S. I. Kadchenko, G. A. Zakirova, L. S. Ryazanova, O. A. Torshina, “Calculation of discrete semi-bounded operators' eigenvalues with large numbers”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 11:1 (2019), 10–15  mathnet  crossref  elib
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