Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University549710.18287/2541-7525-2017-23-3-18-25UnknownNONLOCAL PROBLEM WITH DYNAMICAL BOUNDARY CONDITIONS FOR HYPERBOLIC EQUATIONDyuzhevaA. V.morenov.sv@ssau.ruSamara National Research University1511201723318251501201815012018Copyright © 2017, Dyuzheva A.V.2017<p>In this article, we consider a boundary-value problem with nonlocal dynamical conditions for hyperbolic equation. A feature of such conditions is the presence of both first and second order derivatives with respect to time-variable. Furthermore, boundary conditions are nonlocal to the extent that their representation is a relation between values of the derivatives on different parts of the boundary. The problem under consideration arise when we study vibration of a bar with damping and point masses. The existence and uniqueness of a generalized solution are proved. The proof is based on apriori estimates and Galerkin procedure.</p>нелокальная задача, динамические граничные условия, гиперболическое уравнение, обобщенное решение, априорные оценки, эффекта демпфирования, производная второго порядка, метод Галеркина.nonlocal problem, nonlocal dynamical conditions, hyperbolic equation, generalized solution, second order derivatives, bar with damping, apriori estimates, Galerkin procedure[[1] Beylin A.B., Pulkina L.S Zadacha o kolebaniiakh sterzhnia s neizvestnym usloviem ego zakrepleniia na chasim granitsy [A problem on vibration of a bar with unknown boundary condition on a part of the boundary]. Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia [Vestnik of Samara University. Natural science Series], 2017, №2(23), pp. 7–14 [in Russian].][[2] Gording L. Zadacha Koshi dlia giperbolicheskikh uravnenii [The Cauchy problem for hyperbolic equations]. M.: Izd-vo inostrannoi literatury, 1961, 120 p. [in Russian].][[3] Dyuzheva A.V. Zadacha s dinamicheskimi usloviiami dlia giperbolicheskogo uravneniia [Problem with timedependent boundary conditions for hyperbolic equation]. Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia [Vestnik of Samara University. Natural science Series], 2017, №2(23), pp. 7–14 [in Russian].][[4] Ladyzhenskaya O.A. Kraevye zadachi matematicheskoi fiziki [Boundary-value problems of mathematical physics]. M.: Nauka, 1973, 407 p. [in Russian].][[5] Lazhetich N.L. O klassicheskoi razreshimosti smeshannoi zadachi dlia odnomernogo giperbolicheskogo uravneniia vtorogo poriadka [On classical solvability of a mixed problem for one-dimensional hyperbolic equation of the second order]. Differents. uravneniia [Differential Equations], 2006, no. 42(8), pp. 1072–1077 [in Russian].][[6] Pulkina L.S., Dyuzheva A.V. Nelokal’naia zadacha s peremennymi po vremeni kraevymi usloviiami Steklova dlia giperbolicheskogo uravneniia [Nonlocal problem with Steklov’s time-varying boundary conditions for a hyperbolic equation]. Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia [Vestnik of Samara University. Natural science Series], 2010, no. 4(86), pp. 56–64 [in Russian].][[7] Steklov V.A. Osnovnye zadachi matematicheskoi fiziki [Basic problems of mathematical physics]. M.: Nauka, 1983 [in Russian].][[8] Tikhonov A.N., Samarskiy A.A. Uravneniia matematicheskoi fiziki [Equations of Mathematical Physics]. M.: Nauka, 2004, 798 p. [in Russian].]