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Ufa Mathematical Journal, 2024, Volume 16, Issue 1, Pages 34–52
DOI: https://doi.org/10.13108/2024-16-1-34
(Mi ufa684)
 

Problem on string system vibrations on star-shaped graph with nonlinear condition at node

M. B. Zverevaab, M. I. Kamenskiiab

a Voronezh State University, Universitetskaya sq. 1, 394018, Voronezh, Russia
b Voronezh State Pedagogical University, Lenin str. 86, 394043, Voronezh, Russia
References:
Abstract: We consider a system of $n$ strings being in the equilibrium position along a geometrical star-graph. We suppose that the edges of the graph have the same lengths and the graph is oriented to the node. We study the case when the initial velocity of each string is zero. The initial shape of each string is defined by means of given functions on the edges. We assume that at the boundary vertices the strings are fixed. We study the oscillatory process for the case, when the node point of the string system is located inside the motion limiter. At the same time we suppose that the limiter can move in the direction perpendicular to the graph plane. While the limiter does not touch the node point of the string system, the transmission condition holds (the Kirchoff condition). Once the limiter touches the node, their joint motion begins and an additional restriction for the sign of the sum of derivatives at the node appears. Thus, at the node, a hysteresis type condition is satisfied.
In the work we obtain a representation for the solution and prove its existence. For a particular case we consider a case on periodic oscillations of the node point of the string system. We solve a problem on boundary control of the oscillatory process under the assumption that the oscillation time does not exceed the length of the strings.
Keywords: wave equation, oscillatory process, hysteresis, geometric graph.
Funding agency Grant number
Russian Science Foundation 22-71-10008
The research is financially supported by the Russian Science Foundation under the project no. 22-71-10008, \href{https://rscf.ru/project/22-71-10008/}{https://rscf.ru/project/22-71-10008/}.
Received: 10.04.2023
Document Type: Article
UDC: 517.954
MSC: 35L05, 35L20, 35R02
Language: English
Original paper language: Russian
Citation: M. B. Zvereva, M. I. Kamenskii, “Problem on string system vibrations on star-shaped graph with nonlinear condition at node”, Ufa Math. J., 16:1 (2024), 34–52
Citation in format AMSBIB
\Bibitem{ZveKam24}
\by M.~B.~Zvereva, M.~I.~Kamenskii
\paper Problem on string system vibrations on star-shaped graph with nonlinear condition at node
\jour Ufa Math. J.
\yr 2024
\vol 16
\issue 1
\pages 34--52
\mathnet{http://mi.mathnet.ru//eng/ufa684}
\crossref{https://doi.org/10.13108/2024-16-1-34}
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