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 Eurasian Math. J., 2017, Volume 8, Number 1, Pages 10–22 (Mi emj244)

Inverse problem for the diffusion operator with symmetric functions and general boundary conditions

A. M. Akhtyamovab, V. A. Sadovnichyc, Ya. T. Sultanaevb

a Bashkir State University, ul. Frunze 32, 450074 Ufa, Russia
b Mavlutov Institute of Mechanics, Russian Academy of Sciences, 71 Oktyabrya Pr, 450054 Ufa, Russia
c M.V. Lomonosov Moscow State University, Leninskie gory, 119992 Moscow, Russia

Abstract: For the inverse problem of reconstructing the nonself-adjoint diffusion operator with symmetric functions and general boundary conditions a uniqueness theorem is proved. As spectral data only one spectrum and six eigenvalues are used. Earlier this inverse problem was not considered. The inverse problem of reconstructing the self-adjoint diffusion operator with nonseparated boundary conditions was considered. To uniquely reconstruct this operator two spectra, some sequence of signs, and some complex number were used as spectral data. We show that in the symmetric case to uniquely reconstruct the self-adjoint diffusion operator one can use even less spectral data as compared with the reconstruction of a self-adjoint problem in earlier papers; more precisely, we need one spectrum and, in addition, five eigenvalues. The special cases of these general inverse problems are considered too. In these special cases less spectral data are used. Algorithms of reconstructing diffusion operator are given. Moreover, we show that results obtained in the present paper generalize the results for the inverse problem of reconstructing the diffusion operator with separated boundary conditions.

Keywords and phrases: inverse eigenvalue problem, diffusion operator, nonseparated boundary conditions.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-01095_a17-41-020195_r_a17-41-020230_r_a This work was supported by the Russian Foundation for Basic Research and by the Academy of Sciences of the Republic of Bashkortostan (projects No. 15-01-01095 a, 17-41-020195-r a, 17-41-020230-r a).

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Document Type: Article
MSC: 34A55, 34B05, 58C40
Language: English

Citation: A. M. Akhtyamov, V. A. Sadovnichy, Ya. T. Sultanaev, “Inverse problem for the diffusion operator with symmetric functions and general boundary conditions”, Eurasian Math. J., 8:1 (2017), 10–22

Citation in format AMSBIB
\Bibitem{AkhSadSul17} \by A.~M.~Akhtyamov, V.~A.~Sadovnichy, Ya.~T.~Sultanaev \paper Inverse problem for the diffusion operator with symmetric functions and general boundary conditions \jour Eurasian Math. J. \yr 2017 \vol 8 \issue 1 \pages 10--22 \mathnet{http://mi.mathnet.ru/emj244} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000411744800001} 

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This publication is cited in the following articles:
1. 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
2. Sadovnichii V.A., Sultanaev Ya.T., Akhtyamov A.M., “Inverse Problem For a Differential Operator With Nonseparated Boundary Conditions”, Dokl. Math., 97:2 (2018), 181–183
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