SOLVABILITY OF A NONLOCAL PROBLEM WITH INTEGRAL CONDITIONS OF THE II KIND FOR ONE-DIMENSIONAL HYPERBOLIC EQUATION



Cite item

Full Text

Abstract

In this paper we consider a nonlocal problem with integral conditions of the II kind for one-dimensional hyperbolic equation. Nonlocal conditions of the second kind differ in type of non-integral terms, that may contain traces of required solution and traces of derivatives. This difference turns out to be significant for choosing a method for investigating the solvability of the problem. In this work we consider the case when
nonintegral terms are traces of required solution on boundary of the domain. To investigate the solvability of the problem we use method of reduction to the boundary problem for loaded equation. This method allowed us to define a generalized solution, to obtaim apriori estimates and to prove existence of unique generalized solution of the given problem.

About the authors

V. A. Kirichek

Samara National Research University, 34, Moskovskoye shosse, Samara, 443086, Russian Federation.

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

postgraduate student of the Department of Differential Equations and Control Theory

Russian Federation

References

  1. Cannon J.R. The solution of the heat equation subject to the specification of energy. Quart. Appl. Math., 1963, vol. 21, no. 2, pp. 155–160. doi: 10.1090/qam/160437. .
  2. Kamynin L.I. Ob odnoi kraevoi zadache teorii teploprovodnosti s neklassicheskimi granichnymi usloviyami . Zh. vychisl. matem. i matem. fiz. , 1964, 4:6, pp. 33–59. DOI: https://doi.org/10.1016/0041-5553(64)90080-1 .
  3. Pulkina L.S. Ob odnoi neklassicheskoi zadache dlya vyrozhdayushchegosya giperbolicheskogo uravneniya . Izvestiya vuzov. Matematika , 1991, vol. 35, no. 11, pp. 49–51. Available at: http://www.mathnet.ru/links/117798e729d69dadb1c531679395a347/ivm5192.pdf .
  4. Pul’kina L.S. Ob odnoi nelokal’noi zadache dlya vyrozhdayushchegosya giperbolicheskogo uravneniya . Matematicheskie zametki , 1992, 51:3, pp. 286–290. DOI: https://doi.org/10.1007/BF01206393 .
  5. Il’in V.A., Moiseev E.I. O edinstvennosti resheniya smeshannoi zadachi dlya volnovogo uravneniya s nelokal’nymi granichnymi usloviyami . Differents. uravneniya , 2000, 36:5, pp. 728–733. DOI: https://doi.org/10.1007/BF02754231 .
  6. Pulkina L.S. Smeshannaya zadacha s integral’nym usloviem dlya giperbolicheskogo uravneniya . Matematicheskie zametki , 2003, vol. 74, no. 3, pp. 411–421. DOI: https://doi.org/10.4213/mzm277 .
  7. Pulkina L.S. Nachal’no-kraevaya zadacha s nelokal’nym granichnym usloviem dlya mnogomernogo giperbolicheskogo uravneniya . Differents. uravneniya , 2008, vol. 44, no. 8, pp. 1119–1125. doi: 10.1134/S0012266108080090 .
  8. Lazetic N.L. O klassicheskoi razreshimosti smeshannoi zadachi dlya odnomernogo giperbolicheskogo uravneniya vtorogo poryadka . Differents. uravneniya , 2006, 42:8, pp. 1134–1139. DOI: https://doi.org/10.1134/S0012266106080088 .
  9. Pulkina L.S. Zadachi s neklassicheskimi usloviyami dlya giperbolicheskikh uravnenii . Samara: Izdatel’stvo "Samarskii universitet 2012, 194 p. .
  10. Kozhanov A.I., Pulkina L.S. O razreshimosti kraevykh zadach s nelokal’nym granichnym usloviem integral’nogo vida dlya mnogomernykh giperbolicheskikh uravnenii . Differents. uravneniya , 2006, 42:9, pp. 1233–1246. DOI: https://doi.org/10.1134/S0012266106090023 .
  11. Pulkina L.S. Nonlocal problems for hyperbolic equations with degenerate integral condition. Electronic Journal of Differential Equations, 2016, vol. 2016, p. 193. Available at: https://pdfs.semanticscholar.org/5550/c097496f428d827925bfb987497a291bee78.pdf?_ga=2.141811954.1367780915.1591514604-1525477732.1586505106. .
  12. Pulkina L.S., Kirichek V.A. Razreshimost’ nelokal’noi zadachi dlya giperbolicheskogo uravneniya s vyrozhdayushchimisya integral’nymi usloviyami . Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki , 2019, vol. 23, no. 2, pp. 229–245. doi: 10.14498/vsgtu1707 .
  13. Ladyzhenskaya O.A. Kraevye zadachi matematicheskoi fiziki . Moscow: Nauka, 1973, 407 p. Available at: http://bookre.org/reader?file=442669 .

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2019 Kirichek V.A.

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies