A PROBLEM ON LONGITUDINAL VIBRATION IN A SHORT BAR WITH DYNAMICAL BOUNDARY CONDITIONS


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Abstract

In this paper, we consider an initial-boundary problem with dynamical nonlocal boundary condition for a pseudohyperbolic fourth-order equation in a rectangular. Dynamical nonlocal boundary condition represents a relation between values of a required solution, its derivatives with respect of spacial variables, second-order derivatives with respect of time-variables and an integral term. This problem may be used as a mathematical model of longitudinal vibration in a thick short bar and illustrates a nonlocal approach to such processes. The main result lies in justification of solvability of this problem. Existence and uniqueness of a generalized solution are proved. The proof is based on the a priori estimates obtained in this paper, Galerkin’s procedure and the properties of the Sobolev spaces.

About the authors

A. B. Beylin

Samara State Technical University

Author for correspondence.
Email: morenov.sv@ssau.ru
Russian Federation

L. S. Pulkina

Samara National Research University

Email: morenov.sv@ssau.ru
Russian Federation

References

  1. J.W.S.Rayleigh. Theory of sound. New York: Dover, 1945. (translated in Russian in 1955). M.: GITTL, 1955, Vol. I .
  2. Rao J.S. Advanced Theory of Vibration. N.Y.: Wiley, 1992 .
  3. Fedotov I.A., Polyanin A.D., Shatalov M.Yu. Teoriia svobodnykh i vynuzhdennykh kolebanii tverdogo sterzhnia, osnovannaia na modeli Releia . DAN , 2007, Vol. 417, pp. 56–61.
  4. Beilin A.B., Pulkina L.S. Zadacha o prodol’nykh kolebaniiakh sterzhnia s dinamicheskimi granichnymi usloviiami . Vestnik SamGU , 2014, no. 3(114), pp. 9–19 .
  5. Steklov V.A. Zadacha ob okhlazhdenii neodnorodnogo tverdogo tela Soobshch. Khar’kovskogo mat. o-va , 1896, Vol. 5, Issue 3-4, pp. 136–181 .
  6. Lazhetich N.L. O klassicheskoi razreshimosti smeshannoi zadachi dlia odnomernogo giperbolicheskogo uravneniia vtorogo poriadka . Differents. uravneniia , 2006, Vol. 42, Issue 8, pp. 1134–1139 .
  7. Ilin V.A., Moiseev E.I. O edinstvennosti resheniia smeshannoi zadachi dlia volnovogo uravneniia s nelokal’nymi granichnymi usloviiami . Differents. uravneniia , 2000, Vol. 36, Issue 5, Pages 728–733 .
  8. Kozhanov A.I., Pulkina L.S. O razreshimosti nekotorykh granichnykh zadach so smeshcheniem dlia lineinykh giperbolicheskikh uravnenii . Matematicheskii zhurnal instituta matematiki MO i NRK, Almaty , 2009, Vol. 2(32), pp. 78–92 .
  9. Pulkina L.S., Dyuzheva A.V. Nelokal’naia zadacha s peremennymi po vremeni kraevymi usloviiami Steklova dlia giperbolicheskogo uravneniia . Vestnik SamGU , 2010, no. 4(78), pp. 56–64 .
  10. Cannon J.R. The solution of the heat equation subject to the specification of energy. Quart.Appl.Math., 1963, no. 21, pp. 155–160 .
  11. Kamynin L.I. Ob odnoi kraevoi zadache teorii teploprovodnosti s neklassicheskimi granichnymi usloviiami . Zhurnal vychisl. matem. i matem. fiz. , 1964, no. 4(6), pp. 1006–1024 .
  12. Gordeziani D.G., Avalishvili G.A. Resheniia nelokal’nykh zadach dlia odnomernykh kolebanii sredy . Matem. modelir. , 2000, Vol. 12, no.1, pp. 94–103 .
  13. Bouziani A. On the solvability of a nonlocal problems arising in dynamics of moisture transfer. Georgian Mathematical Journal, 2003, no. 4, pp. 607–622 .
  14. Avalishvili G., Avalishvili M., Gordeziani D. On integral nonlocal boundary problems for some partial differential equations. Bulletin of the Georgian National Academy of Sciences, 2011, no. 5(1), pp. 31–37 .
  15. Kozhanov A.I., Pulkina L.S. O razreshimosti kraevykh zadach s nelokal’nym granichnym usloviem integral’nogo vida dlia mnogomernykh giperbolicheskikh uravnenii . Differents. uravneniia , 2006, Vol. 42, no. 9, pp. 1233–1246 .
  16. Dmitriev V.B. Nelokal’naia zadacha s integral’nymi usloviiami dlia volnovogo uravneniia . Vestnik SamGU , 2006, no. 2(42), pp. 15–27 .
  17. Zdenek P. Bazant, Milan Jirasek. Nonlocal Integral Formulationof Plasticity And Damage: Survey of Progress. American Society of Civil Engineers. Journal of Engineering Mechanics, 2002, pp. 1119–1149 .
  18. Pulkina L.S. Kraevye zadachi dlia giperbolicheskogo uravneniia s nelokal’nymi usloviiami I i II roda . Izv. vuzov. Matematika , 2012, Vol. 56, no.4, pp. 62–69 .
  19. Tikhonov A.N., Samarskii A.A. Uravneniia matematicheskoi fiziki . M.: Nauka, 2004 .
  20. Korpusov O.M. Razrushenie v neklassicheskikh volnovykh uravneniiakh . M.: URSS, 2010 .
  21. Doronin G.G., Larkin N.A., Souza A.J. A hyperbolic problem with nonlinear second-order boundary damping. EJDE, 1998, no. 28, pp. 1–10 .
  22. Pulkina L.S. . EJDE, 2014, no. 116, pp. 1-9 .
  23. Pul’kina L.S. Zadacha s dinamicheskim nelokal’nym usloviem dlia psevdogiperbolicheskogo uravneniia . Izv. vuzov. Matematika , 2016, Vol. 60, Issue 9, pp. 38–45 .
  24. Ladyzhenskaya O.A. Kraevye zadachi matematicheskoi fiziki . M.: Nauka, 1973 .

Copyright (c) 2018 A. Б. Бейлин, Л. С. Пулькина

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