Differential Equations

, Volume 55, Issue 2, pp 243–259 | Cite as

Method of Integral Equations for Studying the Solvability of Boundary Value Problems for the System of Nonlinear Differential Equations of the Theory of Timoshenko Type Shallow Inhomogeneous Shells

  • S. N. TimergalievEmail author
Partial Differential Equations


The solvability of the boundary value problem for a system of second-order nonlinear partial differential equations with given boundary conditions which describes the equilibrium of elastic inhomogeneous shallow shells with free edges in the framework of the Timoshenko shear model is considered. The boundary value problem is reduced to a single nonlinear equation whose solvability is established by using the contraction mapping principle.


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  1. 1.
    Galimov, K.Z., Osnovy nelineinoi teorii tonkikh obolochek (Foundations of Nonlinear Theory of Thin Shells), Kazan: Kazan Gos. Univ., 1975.Google Scholar
  2. 2.
    Vorovich, I.I., Matematicheskie problemy nelineinoi teorii pologikh obolochek (Mathematical Problems of Nonlinear Theory of Shallow Shells), Moscow: Nauka, 1989.Google Scholar
  3. 3.
    Morozov, N.F., Izbrannyue dvumernye zadachi teorii uprugosti (Selected Two-Dimensional Problems of Elasticity), Leningrad: Leningrad Gos. Univ., 1978.Google Scholar
  4. 4.
    Karchevskii, M.M., Nonlinear problems of the theory of plates and shells and their difference approximation, Soviet Math. (Iz. VUZ), 1985, vol. 29, no. 10, pp. 21–38.MathSciNetGoogle Scholar
  5. 5.
    Karchevskii, M.M., On solvability of variational problems of nonlinear shallow shell theory, Differ. Equations, 1991, vol. 27, no. 7, pp. 841–847.MathSciNetGoogle Scholar
  6. 6.
    Karchevskii, M.M., Investigation of solvability of nonlinear equilibrium problem of a shallow unfixed shell, Uchen. Zap. Kazan Univ. Ser. Fiz.-Mat. Nauki, 2013, vol. 155, no. 3, pp. 105–110.Google Scholar
  7. 7.
    Timergaliev, S.N., Teoremy sushchestvovaniya v nelineinoi teorii tonkikh uprugikh obolochek (Existence Theorems in Nonlinear Theory of Thin Elastic Shells), Kazan: Kazan Gos. Univ., 2011.Google Scholar
  8. 8.
    Timergaliev, S.N., Solvability of geometrically nonlinear boundary-value problems for the Timoshenkotype anisotropic shells with rigidly clamped edges, Russian Math. (Iz. VUZ), 2011, vol. 55, no. 8, pp. 47–58.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Timergaliev, S.N., Proof of the solvability of a system of partial differential equations in the nonlinear theory of shallow shells of Timoshenko type, Differ. Equations, 2012, vol. 48, no. 3, pp. 458–463.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Timergaliev, S.N., On the existence of solutions to geometrically nonlinear problems for shallow Timoshenko-type shells with free edges, Russian Math. (Iz. VUZ), 2014, vol. 58, no. 3, pp. 31–46.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Timergaliev, S.N., On the existence of solutions of a nonlinear boundary value problem for the system of partial differential equations of the theory of Timoshenko type shallow shells with free edges, Differ. Equations, 2015, vol. 51, no. 3, pp. 377–390.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Timergaliev, S.N. and Kharasova, L.S., Study of the solvability of a boundary value problem for the system of nonlinear differential equations of the theory of shallow shells of the Timoshenko type, Differ. Equations, 2016, vol. 52, no. 5, pp. 630–643.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Timergaliev, S.N., A method of integral equations in nonlinear boundary-value problems for flat shells of the Timoshenko type with free edges, Russian Math. (Iz. VUZ), 2017, vol. 61, no. 4, pp. 49–64.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Vekua, I.N., Obobshchennye analiticheskie funktsii (Generalized Analytic Functions), Moscow: Nauka, 1988.Google Scholar
  15. 15.
    Vekua, I.N., Novye metody resheniya ellipticheskikh uravnenii (New Methods for Solving Elliptic Equations), Moscow: Gostekhizdat, 1948.Google Scholar
  16. 16.
    Muskhelishvili, N.I., Singulyarnye integral’nye uravneniya (Singular Integral Equations), Moscow: Nauka, 1968.Google Scholar
  17. 17.
    Gakhov, F.D., Kraevye zadachi (Boundary Value Problems), Moscow: Nauka, 1977.Google Scholar
  18. 18.
    Krasnosel’skii, M.A., Topologicheskie metody v teorii nelineinykh integral’nykh uravnenii (Mathematical Methods in the Theory of Nonlinear Integral Equations), Moscow: Gostekhizdat, 1956.Google Scholar
  19. 19.
    Duvant, G. and Lions, J.-L., Inequalities in Mechanics and Physics, Berlin: Springer, 1976. Translated under the title Neravenstva v mekhanike i fizike, Moscow: Nauka, 1980.Google Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Kazan State University of Architecture and EngineeringKazanRussia

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