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Goursat-type nonlocal problem for a fourth-order loaded equation

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Abstract

Initial and boundary value problems for partial differential equations have been sufficiently studied. However, it has been recently become apparent that various processes and phenomena of modern natural science lead to nonclassical problems for differential equations. The class of nonclassical problems involves nonlocal problems. We turn our attention on nonlocal problems, to be exact, on some problems with nonlocal integral conditions for a loaded equation. The majority of the works concerned with boundary value problems with integral conditions deals with second-order equations. In this article we consider a nonlocal problem with integral conditions for a loaded fourth-order equation. The factorization enables to transfer this problem to two problems for second order equations: the Goursat problem for an integrodifferential equation and an integral analogue of the Goursat problem for a simple hyperbolic equation. The unique solvability of the considered problem is proved.

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References

  1. Ashyralyev, A., Aggez, N.: Nonlocal boundary value hyperbolic problems involving integral conditions. Bound Value Probl. 2014, 205 (2014). https://doi.org/10.1186/s13661-014-0205-4

    Article  MathSciNet  MATH  Google Scholar 

  2. Assanova, A.T.: Nonlocal problem with integral conditions for the system of hyperbolic equations in the characteristic rectangle. Russ. Math. 61(5), 7–20 (2017). https://doi.org/10.3103/S1066369X17050024

    Article  MathSciNet  MATH  Google Scholar 

  3. Attaev, A.K.: The characteristic problem for the second-0rder hyperbolic equation loaded along one of its characteristics, Vestnik KRAUNC. Phys-Math. Nauki. 3(23), 14–18 (2018). https://doi.org/10.18454/2079-6641-2018-23-3-14-18. (in Russian)

    Article  MathSciNet  MATH  Google Scholar 

  4. Avalishvili, G., Avalishvili, M., Gordeziani, D.: On integral nonlocal boundary value problems for some partial differential equations. Bull. Georg. Natl. Acad. Sci. 5(1), 31–37 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Bogatov, A.V., Gilev, A.V., Pulkina, L.S.: A problem with nonlocal condition for fourth-order equation with multiple characteristics, Russian Universities Reports. Mathematics 27(139), 214–230 (2022). https://doi.org/10.20310/2686-9667-2022-27-139-214-230

  6. Fedotov, I.A., Polianin, A.D., Shatalov, MYu.: The theory based on Raley model of free and forced vibration of a solid. Doklady RAN 417(1), 56–61 (2007). (in Russian)

    Google Scholar 

  7. Ionkin, N.I.: A solution of certain boundary-value problem of heat conduction with nonclassical boundary condition. Differ. Equ. 13(2), 294–304 (1977). (in Russian)

    MathSciNet  MATH  Google Scholar 

  8. Iskenderov, A.D.: On a mixed problem for loaded quasilinear hyperbolic equations. Dokl. Acad. Nauk SSSR 199(6), 1237–1239 (1971). (in Russian)

    Google Scholar 

  9. Klimova, E.: Some non-local boundary-value problems and their relationship to problems for loaded equations. Z. Anal. Anwend. 30(1), 71–81 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Korzyuk, V.I., Kozlovskaya, I.S., Naumavets, S.N.: Classical solution of a problem with integral cognitions of the second kind for the one-dimensional wave equation. Differ. Equ. 55(3), 353–362 (2019). https://doi.org/10.1134/S0012266119030091

    Article  MathSciNet  MATH  Google Scholar 

  11. Kozhanov, A.I., Pulkina, L.S.: On the solvability of boundary value problems with a nonlocal boundary condition of integral form for multidimensional hyperbolic equations. Differ. Equ. 42(9), 1233–1246 (2006). https://doi.org/10.1134/S0012266106090023

    Article  MathSciNet  MATH  Google Scholar 

  12. Nakhushev, A.M.: Problems with Displacement for Partial Differential Equations. Nauka, Moscow (2006). (in Russian)

    MATH  Google Scholar 

  13. Nakhushev, A.M.: Loaded Equations and Their Applications. Nauka, Moscow (2012). (in Russian)

    MATH  Google Scholar 

  14. Pulkina, L.S.: On the solvability in \(L_2\) of a nonlocal problem with integral conditions for a hyperbolic equation. (Russian) Differ. Uravn. 36(2), 279–280, 288 (2000); translation in Differ. Equ. 36(2), 316–318 (2000). https://doi.org/10.1007/BF02754219

  15. Pulkina, L.S.: Boundary-value problems for a hyperbolic equation with nonlocal conditions of the I and II kind. Russ. Math. (Iz.VUZ) 56(4), 62–69 (2012). https://doi.org/10.3103/S1066369X12040081

  16. Pulkina, L.S.: Nonlocal problems for hyperbolic equations from the viewpoint of strongly regular boundary conditions. Electron. J. Differ. Equ. 2020(28), 1–20 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Pulkina, L.S.: Nonlocal Problems for Hyperbolic Equations, Mathematical analysis in Interdisciplinary Research, pp. 119–140. Springer, Berlin (2021). https://doi.org/10.1007/978-3-030-84721-0_28

  18. Pulkina, L.S., Beylin, A.B.: Nonlocal approach to problems on longitudinal vibration in a short bar. Electron. J. Differ. Equ. 2019(29), 1–9 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Samarskii, A.A.: On certain problems of the modern theory of differential equations. Differ. Equ. 16(11), 1925–1935 (1980). (in Russian)

    Google Scholar 

  20. Sobolev, S.L.: Equations of Mathematical Physics. Nauka, Moscow (1966)

    Google Scholar 

  21. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1999). https://doi.org/10.1002/9781118032954

  22. Zdeněk, P., Bažant, M.J.: Nonlocal integral formulation of plasticity and damage: survey of progress, American Society of Civil Engineers. J. Eng. Mech. 128(11), 1119–1149 (2002). https://doi.org/10.1061/(ASCE)0733-9399(2002)128:11(1119)

    Article  Google Scholar 

  23. Zhegalov, V.I., Mironov, A.N., Utkina, E.A.: Equations with a Dominant Partial Derivative. Kazan University Press, Kazan (2014). (in Russian)

    Google Scholar 

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Authors and Affiliations

  1. Department of Differential Equations and Control Theory, Samara University, 1, Ac. Pavlova st., Samara, 443011, Russia

    L. S. Pulkina

  2. Faculty of Informatics/Mathematics, Dresden University of Applied Sciences (HTW Dresden), Friedrich-List-Platz 1, 01069, Dresden, Saxony, Germany

    Elena Klimova

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  1. L. S. Pulkina
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  2. Elena Klimova
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Correspondence to L. S. Pulkina.

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Pulkina, L.S., Klimova, E. Goursat-type nonlocal problem for a fourth-order loaded equation. Bol. Soc. Mat. Mex. 29, 30 (2023). https://doi.org/10.1007/s40590-023-00500-8

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