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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2018, Volume 11, Issue 3, Pages 5–17 (Mi vyuru440)  

Mathematical Modelling

Boundary inverse problem for star-shaped graph with different densities strings-edges

A. M. Akhtyamova, Kh. R. Mamedovb, E. N. Yilmazoglub

a Bashkir State University, Ufa, Russian Federation
b Mersin University, Mersin, Turkey

Abstract: The paper is devoted to the mathematical modelling of star-shaped geometric graphs with n rib-strings of different density and the solution of the boundary inverse spectral problem for Sturm–Liouville differential operators on these graphs. Earlier it was shown that if strings have the same length and densities, then the stiffness coefficients of springs at the ends of graph strings are not uniquely recovered from natural frequencies. They are found up to permutations of their places. We show, that if the strings have different densities, then the stiffness coefficients of springs on the ends of graph strings are uniquely recovered from all natural frequencies. Counterexamples are shown that for the unique recovery of the stiffness coefficients of springs on n dead ends of the graph, it is not enough to use $n$ natural frequencies. Examples are also given showing that it is sufficient to use $n + 1$ natural frequencies for the uniqueness of the recovery of the stiffness coefficients of springs at the n ends of the strings. Those, the uniqueness or non-uniqueness of the restoration of the stiffness coefficients of springs at the ends of the strings depends on whether the string densities are identical or different.

Keywords: natural frequencies; star-shaped graph; inverse problems; strings; densities; boundary conditions.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-51-06002-Aš_a
18-01-00250_a
17-41-020230-­_a
Science Development Foundation under the President of the Republic of Azerbaijan EIF-BGM-4-RFTF-1/2017
The reported research was funded by Russian Foundation for Basic Research, the government of the region of the Republic of Bashkortostan (projects 18-51-06002-Az_a 18-01-00250_a, 17-41-020230-r_a), and the Science Development Fund under the President of the Republic of Azerbaijan (project on the 1st Azerbaijan-Russian International Grant Competition (EIF-BGM-4-RFTF-1/2017)).


DOI: https://doi.org/10.14529/mmp180301

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Document Type: Article
UDC: 517.984.54
MSC: 45C05, 35G16
Received: 04.07.2018
Language: English

Citation: A. M. Akhtyamov, Kh. R. Mamedov, E. N. Yilmazoglu, “Boundary inverse problem for star-shaped graph with different densities strings-edges”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 11:3 (2018), 5–17

Citation in format AMSBIB
\Bibitem{AkhMamYil18}
\by A.~M.~Akhtyamov, Kh.~R.~Mamedov, E.~N.~Yilmazoglu
\paper Boundary inverse problem for star-shaped graph with different densities strings-edges
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2018
\vol 11
\issue 3
\pages 5--17
\mathnet{http://mi.mathnet.ru/vyuru440}
\crossref{https://doi.org/10.14529/mmp180301}
\elib{http://elibrary.ru/item.asp?id=32711843}


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