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Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 2019, Volume 11, Issue 1, Pages 10–15 (Mi vyurm397)  

Mathematics

Calculation of discrete semi-bounded operators’ eigenvalues with large numbers

S. I. Kadchenkoa, G. A. Zakirovab, L. S. Ryazanovaa, O. A. Torshinaa

a Magnitogorsk State Technical University of G.I. Nosova, Magnitogorsk, Russian Federation
b South Ural State University, Chelyabinsk, Russian Federation

Abstract: In previous works of the article’s authors on development of the Galerkin method, linear formulas for calculating the approximate eigenvalues of discrete lower semi-bounded operators have been obtained. The formulas allow calculating the eigenvalues of the specified operators of any number, regardless of whether the eigenvalues of the previous numbers are known or not. At that, it is possible to calculate the eigenvalues with large numbers when application of the Galerkin method is becoming difficult. It is shown that eigenvalues of small numbers of various boundary-value problems, generated by discrete lower semi-bounded operators and calculated by linear formulas and by the Galerkin method, are in a good conformity.
In this paper we use linear formulas to calculate approximate eigenvalues with large numbers of discrete lower semi-bounded operators. Results of calculation of eigenvalues by linear formulas and by known asymptotic formulas for two spectral problems are given. Comparison of the results of calculations of the approximate eigenvalues shows that they almost coincide for sufficiently large numbers. This proves the fact that linear formulas can be used for the considered spectral problems and sufficiently large numbers of eigenvalues.

Keywords: spectral problem, discrete operators, semi-bounded operators, eigenvalues and eigenfunctions of an operator, Galerkin method.

DOI: https://doi.org/10.14529/mmph190102

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UDC: 519.642.8
MSC: 47A10
Received: 10.12.2018
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Citation: S. I. Kadchenko, G. A. Zakirova, L. S. Ryazanova, O. A. Torshina, “Calculation of discrete semi-bounded operators’ eigenvalues with large numbers”, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 11:1 (2019), 10–15

Citation in format AMSBIB
\Bibitem{KadZakRya19}
\by S.~I.~Kadchenko, G.~A.~Zakirova, L.~S.~Ryazanova, O.~A.~Torshina
\paper Calculation of discrete semi-bounded operators’ eigenvalues with large numbers
\jour Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz.
\yr 2019
\vol 11
\issue 1
\pages 10--15
\mathnet{http://mi.mathnet.ru/vyurm397}
\crossref{https://doi.org/10.14529/mmph190102}
\elib{https://elibrary.ru/item.asp?id=36816017}


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