Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University514710.18287/2541-7525-2017-23-2-7-14UnknownA PROBLEM ON VIBRATION OF A BAR WITH UNKNOWN BOUNDARY CONDITION ON A PART OF THE BOUNDARYBeylinA. B.morenov@ssau.ruPulkinaL. S.morenov@ssau.ruSamara State Technical UniversitySamara National Research University210920172327142109201721092017Copyright © 1970, Beylin A.B., Pulkina L.S.1970<p>In this paper, we study an inverse problem for hyperbolic equation. This problem arises when we consider vibration of a nonhomogeneous bar if one endpoint is fixed by spring but behavior of the other is unknown and is the subject to find. Overdetermination is given in the form of integral with respect to spacial variable. Unique solvability of this problem is proved under some conditions on data. The proof is based on a priori estimates in Sobolev space.</p>гиперболическое уравнение, обратная задача, интегральное условие переопределения.hyperbolic equation, inverse problem, integral overdetermination.[[1] Babakov I.M. Teoriia kolebanii [Theory of vibrations]. M.: Nauka, 1968, 560 p. [in Russian].][[2] Birger I.A., Shorr B.F., Iosilevich G.B. Raschet na prochnost’ detalei mashin: Spravochnik [Valculation of stress of machine elements: Reference book]. M.: Mashinostroenie, 1993, 640 p. [in Russian].][[3] Khazanov Kh.S. Mekhanicheskie kolebaniia sistem s raspredelennymi parametrami: ucheb. posobie [Mechanical oscillations of systems with distributed parameters: textbook]. Samara, Samar. Gosud. Aerokosmich. Un-t, 2002, 80 p. [in Russian].][[4] Veitz V.L., Dondoshanskii V.K., Chiriaev V.I. Vynuzhdennye kolebaniia v metallorezhushchikh stankakh [Forced oscillations in cutting machines]. M-L.: Mashgiz, 1959, 288 p. [in Russian].][[5] Kumabe D. Vibratsionnoe rezanie [Vibration Cutting]. M.: Mashinostroenie, 1985, 424 p. [in Russian].][[6] Rao J.S. Advanced Theory of Vibration. N.Y.: Wiley, 1992, 431 p. [in Russian].][[7] Fedotov I.A., Polyanin A.D., Shatalov M.Yu. Teoriia svobodnykh i vynuzhdennykh kolebanii tverdogo sterzhnia, osnovannaia na modeli Releia [Theory of free and forced vibration of rigid rod based on Rayleigh model]. Doklady RAN [Dokladyi Akademii nauk], 2007, Vol. 417, no. 1, pp. 56–61 [in Russian].][[8] Beylin A.B., Pulkina L.S. Zadacha o prodol’nykh kolebaniiakh sterzhnia s dinamicheskimi granichnymi usloviiami [A Problem on Longitudinal Vibration in a Short Bar with Dynamical Boundary Conditions]. Vestnik SamGU. Estestvennonauchnaia seriia [Vestnik of Samara State University. Natural Science Series], 2014, no. 3(114), pp. 9–19.][[9] Beylin A.B. Zadacha o prodol’nykh kolebaniiakh uprugo zakreplennogo nagruzhennogo sterzhnia [The problem of longitudinal oscillations of an elastically fixed loaded rod]. Vestnik Samarskogo gos. Tekh. Un-ta. Seriia: Fiz.-mat. nauki [Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences], 2016, Vol. 20, no. 2, pp. 249–258 [in Russian].][[10] Prilepko A.I., Kostin A.B. Ob obratnykh zadachakh opredeleniia koeffitsienta v parabolicheskom uravnenii. II [On inverse problems of determining of a coefficient in a parabolic equation. II]. Sibirskii mat. zhurnal [Siberian Mathematical Journal], 1993, Vol. 34, no. 5 [in Russian].][[11] Kamynin V.L. Obratnaia zadacha opredeleniia mladshego koeffitsienta v parabolicheskom uravnenii pri uslovii integral’nogo nabliudeniia [The inverse problem of determining the lower-order coefficient in parabolic equation with integral observation]. Matem. zametki [Mathematical Notes], 2013, 94,(2), pp. 205–213 [in Russian].][[12] Cannon J.R., Lin Y. Determination of a parameter p(t) in some quasi-linear parabolic differential equations. Inverse Problems, 1988, no. 4, pp. 35–45 [in Russian].][[13] Denisov A.M. Obratnaia zadacha dlia giperbolicheskogo uravneniia s nelokal’nym kraevym usloviem, soderzhashchim zapazdyvaiushchii argument [The inverse problem for a hyperbolic equation with nonlocal boundary condition involving retarding argument]. Trudy instituta matematiki i mekhaniki UrO RAN [Trudy Instituta Matematiki i Mekhaniki UrO RAN], 2012, Vol. 18, No. 1 [in Russian].][[14] Ladyzhenskaya O.A. Kraevye zadachi matematicheskoi fiziki [The boundary value problems in mathematical physics]. M.: Nauka, 1973 [in Russian].][[15] Pulkina L.S. Kraevye zadachi dlia giperbolicheskogo uravneniia s nelokal’nymi usloviiami I i II roda [Boundary-value problems for a hyperbolic equation with nonlocal conditions of the I and II kind]. Izvestiia vuzov. Matematika [Russian Mathematics (Iz. VUZ)], 2012, no. 4, pp. 74–83 [in Russian].]