Differential Equations

, Volume 54, Issue 1, pp 121–133 | Cite as

Initial Value Problem for B-Hyperbolic Equation with Integral Condition of the Second Kind

  • K. B. Sabitov
  • N. V. Zaitseva
Partial Differential Equations


For the hyperbolic equation with Bessel operator, we study the initial boundaryvalue problem with integral nonlocal condition of the second kind in a rectangular domain. The integral identity method is used to prove the uniqueness of the solution to the posed problem. The solution is constructed as a Fourier–Bessel series. To justify the existence of the solution to the nonlocal problem, we obtain sufficient conditions to be imposed on the initial conditions to ensure the convergence of the constructed series in the class of regular solutions.


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  1. 1.
    Kipriyanov, I.A., Singulyarnye ellipticheskie kraevye zadachi (Singular Elliptic Boundary Value Problems), Moscow: Fizmatlit, 1997.zbMATHGoogle Scholar
  2. 2.
    Koshlyakov, N.S., Gliner, E.B., and Smirnov, M.M., Uravneniya v chastnykh proizvodnykh matematicheskoi fiziki (Partial Differential Equations of Mathematical Physics), Moscow: Vysshaya Shkola, 1970.Google Scholar
  3. 3.
    Pul’kin, S.P., On the uniqueness of solution of the singular Gellerstedt problem, Izv. Vyssh. Uchebn. Zaved. Mat., 1960, no. 6 (19), pp. 214–225.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Sabitov, K.B. and Il’yasov, R.R., On ill-posedness of boundary-value problems for a class of hyperbolic equations, Russ. Math. (Izv. Vyssh. Uchebn. Zaved. Mat.), 2001, vol. 45, no. 5, pp. 56–60.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Sabitov, K.B. and Il’yasov, R.R., Solution of the Tricomi problem for an equation of mixed type with a singular coefficient by the spectral method, Russ. Math. (Izv. Vyssh. Uchebn. Zaved. Mat.), 2004, vol. 48, no. 2, pp. 61–68.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Pul’kina, L.S., A nonlocal problem with integral conditions for a hyperbolic equation, Differ. Equations, 2004, vol. 40, no. 7, pp. 947–953.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Pul’kina, L.S., Boundary-value problems for a hyperbolic equation with nonlocal conditions of I and II kind, Russ. Math. (Izv. Vyssh. Uchebn. Zaved. Mat.), 2012, vol. 56, no. 4, pp. 62–69.zbMATHGoogle Scholar
  8. 8.
    Pul’kina, L.S., Zadachi s neklassicheskimi usloviyami dlya giperbolicheskikh uravnenii (Problems with Nonclassical Conditions for Hyperbolic Equations), Samara: Samara Univ., 2012.Google Scholar
  9. 9.
    Benuar, N.E. and Yurchuk, N.I., Mixed problem with integral conditions for hyperbolic equations with Bessel operator, Differ. Equations, 1991, vol. 27, no. 12, pp. 1482–1487.zbMATHGoogle Scholar
  10. 10.
    Sabitova, Yu.K., Nonlocal initial boundary-value problems for a degenerate hyperbolic equation, Russ. Math. (Izv. Vyssh. Uchebn. Zaved. Mat.), 2009, vol. 53, no. 12, pp. 41-49.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Sabitov, K.B., Boundary-value problem for a parabolic-hyperbolic equation with a nonlocal integral condition, Differ. Equations, 2010, vol. 46, no. 10, pp. 1472–1481.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sabitov, K.B., Nonlocal problem for a parabolic-hyperbolic equation in a rectangular domain, Math. Notes, 2011, vol. 89, nos. 3–4, pp. 562–567.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sabitov, K.B., Boundary-value problem with nonlocal integral condition for mixed-type equations with degeneracy on the transition line, Math. Notes, 2015, vol. 98, nos. 3–4, pp. 454–465.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Keldysh, M.V., On some cases of degeneracy of elliptic-type equations on the boundary of a domain, Dokl. Akad. Nauk SSSR, 1951, vol. 77, no. 2, pp. 181–183.Google Scholar
  15. 15.
    Sabitov, K.B., K teorii uravnenii smeshannogo tipa (To the Theory of Mixed-Type Equations), Moscow: Fizmalit, 2014.Google Scholar
  16. 16.
    Bateman, H. and Erdelyi, A., Higher Transcendental Functions. Tables of Integral Transforms, McGraw-Hill, 1954.Google Scholar
  17. 17.
    Watson, G.N., A Treatise on the Theory of Bessel Functions, 2nd.ed., Cambridge: Cambridge University, 1966, Pt. 1.zbMATHGoogle Scholar
  18. 18.
    Olver, F.W.J., Asymptotics and Special Functions, New York: Academic, 1974.zbMATHGoogle Scholar
  19. 19.
    Sabitov, K.B. and Vagapova, E.V., Dirichlet problem for an equation of mixed type with two degeneration lines in a rectangular domain, Differ. Equations, 2013, vol. 49, no. 1, pp. 68–78.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Safina, R.M., Keldysh problem for a mixed-type equation of the second kind with the Bessel operator, Differ. Equations, 2015, vol. 51, no. 10, pp. 1347–1359.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Natanson, I.P., Teoriya funktsii veshchestvennoi peremennoi (Theory of Functions of a Real Variable), Moscow: Nauka, 1974.Google Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Bashkortostan Institute for Strategic ResearchSterlitamak, BashkortostanRussia
  2. 2.Sterlitamak Branch of Bashkir State UniversitySterlitamak, BashkortostanRussia
  3. 3.Kazan (Volga) Federal UniversityKazan, TatarstanRussia

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