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\title{Основные задачи математики}
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\author{Иванов}{П.И.}{Уфимский университет науки и технологий}{Иванов Петр Иванович, д.ф.-м.н., профессор,
УУНиТ (Уфа, Россия); Petr Ivanov (Ufa University of Science and Technology, Ufa, Russia)}, %
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\author{Петров}{А.Б.}{Национальный исследовательский Нижегородский государственный университет им. Н.И. Лобачевского}
{Петров Андрей Борисович, к.ф.-м.н., доцент, ННГУ имени Н.И.Лобачевского (Нижний Новгород, Россия); Andrey Petrov
(Lobachevsky State University of Nizhny Novgorod, Russia)}\\%
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\abstractRUS{Краткая аннотация. Краткая аннотация. Краткая аннотация. Краткая аннотация. Краткая аннотация. Краткая аннотация.
Краткая аннотация. Краткая аннотация. Краткая аннотация. Краткая аннотация. Краткая аннотация. Краткая аннотация (3-10 строк).}
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\keywordsRUS{математика, дифференциальные уравнения, спектральная теория.}
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\titleENG{The main tasks of mathematics}
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\abstractENG{Short abstract. Short abstract. Short abstract. Short abstract. Short abstract. Short abstract. Short abstract. Short abstract.
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\keywordsENG{mathematics, differential equations, spectral theory.}
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При ссылках номера лемм (теорем, формул и~т.\,д.) должны указываться явно:  теорема~1, используя лемму~5, согласно формуле~(2).

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\textbf{Теорема~1.} {\it Пусть $p,q,r\in\mathbb{R}$. Тогда
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Отметим также, что
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e^{x}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\\
+\frac{x^8}{8!}+\frac{x^9}{9!}+\frac{x^{10}}{10!}+\frac{x^{11}}{11!}+\frac{x^{12}}{12!}+o(x^{12}),\ x\to 0.
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TeX-файл должен компилироваться без ошибок и переполнений (``over\-full'').\par
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\references{Литература}
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\reference{1. \emph{Левитан Б.М.} Сопряженные операторы обобщенного сдвига // Докл. АН СССР, \textbf{123}:3 (1958), 401-404}
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\reference{2. \emph{Lax P.D., Phillips R.S.} Scattering theory // Петровский И.Г. (ред.) Труды Межд. конгресса математиков.~---  Москва: Мир, 1968.~--- 542-545.}
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\reference{3. \emph{Kelley J.L.} General Topology.~--- Princeton: Noostrand, 1964.}
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MATHEMATICAL MODELING OF STOCHASTIC OSCILLATIONS OF
A VISCOELASTIC ROPE WITH A MOVING BOUNDARY
K. V. Litvinova
kristinalitvinova900@rambler.ru
MSC: 74H45, 74K05
The study of viscoelasticity includes the analysis of the stochastic stability of stochastic
viscoelastic systems, their reliability, etc. The paper considers stochastic linear longitudinal
oscillations of a viscoelastic beam with moving boundaries, taking into account the influence of
damping forces. The case of a difference kernel makes it possible to reduce the problem of
analyzing a system of stochastic integro-differential equations to the study of a system of
stochastic differential equations. To estimate the expansion coefficients, it is proposed to apply
the statistical numerical Monte Carlo method.
Keywords: stochastic longitudinal oscillations, vibrations of a viscoelastic rope, moving
boundaries
Currently, reliability issues in the design of machines and mechanisms require more and
more complete consideration of the dynamic phenomena occurring in the objects being designed.
The widespread use in technology of mechanical objects with moving boundaries necessitates the
development of methods for their calculation. The problem of oscillations of systems with moving
boundaries is associated with obtaining solutions of integro-differential equations and partial
differential equations in time-varying domains [1-10]. Such problems are currently insufficiently
studied. Their peculiarity is the difficulty of using known methods of mathematical physics
suitable for problems with fixed boundaries. The complexity of the solutions obtained is explained
by the fact that there is still no sufficiently general approach to analyzing the features of the
dynamics of such systems. In connection with the danger of resonance, the study of forced
oscillations is of great importance here. Attempts have been made to study this process, but the
results obtained are limited mainly to a qualitative description of dynamic phenomena [1–4]. In
addition, it is recognized that deterministic modeling of systems cannot be adequate for some types
of problems, so it is necessary to move on to probabilistic-statistical modeling, where random
variables and stochastic fluctuations are present. When solving here, predominantly approximate
methods are used [5–9], since obtaining exact solutions is possible only in the simplest cases [10].
If the damping of transverse vibrations occurs primarily due to the action of external
damping forces, then in the case of longitudinal vibrations, the damping is mainly affected by
elastic imperfections in the material of the vibrating object [5-10]. The study of viscoelasticity
includes an analysis of the stochastic stability of stochastic viscoelastic systems, their reliability,
etc. The work examines stochastic linear vibrations of a viscoelastic rope with moving boundaries.
The case of a difference kernel allows us to reduce the problem of analyzing a system of stochastic
integrodifferential equations to studying a system of stochastic differential equations. To estimate
the expansion coefficients, it is proposed to use the statistical numerical Monte Carlo method [11].
References
?? Savin G.N., Goroshko O.A. Dynamics of a variable length thread // Nauk. dumka, Kiev, 1962,
332 p.
?? Samarin Yu.P. On a nonlinear problem for the wave equation in one-dimensional space //
Applied Mathematics and Mechanics. - 1964. - T. 26, V. 3. - P. 77–80.
?? Vesnitsky A.I. Waves in systems with moving boundaries and loads // Fizmatlit, Moscow, 2001,
320 p.
?? Lezhneva A.A. Bending vibrations of a beam of variable length // Izv. Academy of Sciences of
the USSR. Rigid Body Mechanics. - 1970. - No. 1. - P. 159-161.
?? Litvinov V.L. Solution of boundary value problems with moving boundaries using an
approximate method for constructing solutions of integro-differential equations // Tr. Institute
of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences. 2020.Vol.
26, No. 2. P. 188-199.
?? Anisimov V.N., Litvinov V.L. Mathematical models of longitudinal-transverse vibrations of
objects with moving boundaries // Vestn. Himself. tech. un-t. Ser. Phys and mat. science, 2015.
Vol. 19, No. 2. P. 382-397.
?? Anisimov V.N., Litvinov V.L. Mathematical modeling and study of the resonance properties
of mechanical objects with a changing boundary: monograph / V. L. Litvinov, V. N. Anisimov
- Samara: Samar. state tech. un – t, 2020. - 100 p.
?? Litvinov V.L., Anisimov V.N. Application of the Kantorovich - Galerkin method for solving
boundary value problems with conditions on moving boundaries // Bulletin of the Russian
Academy of Sciences. Rigid Body Mechanics. 2018. No. 2. P. 70–77.
?? Litvinov V.L., Anisimov V.N. Transverse vibrations of a rope moving in a longitudinal
direction // Bulletin of the Samara Scientific Center of the Russian Academy of Sciences. 2017.
T. 19. No. 4. – P.161–165.
??? Litvinov V.L., Anisimov V.N. Mathematical modeling and research of oscillations of one
–dimensional mechanical systems with moving boundaries: monograph / V. L. Litvinov, V. N.
Anisimov – Samara: Samar. state tech. un–t, 2017.– 149 p.
??? Elepov B. S., Kronberg A. A., Mikhailov G. A. and Sabelfeld K. K. Solution of boundary
value problems by the Monte Carlo method. - Novosibirsk: Nauka, 1980. - 174 p.
_______________________
Litvinova Kristina Vladislavovna – student, Moscow State University (Moscow, Russia)
