Russian Mathematics

, Volume 63, Issue 10, pp 66–76 | Cite as

The Second Initial-Boundary Value Problem for a B-hyperbolic Equation

  • K. B. SabitovEmail author
  • N. V. ZaitsevaEmail author


For a hyperbolic equation with a Bessel operator in a rectangular domain, we study the initial-boundary value problem in dependence of the numeric parameter that enters in the operator. We represent the solution as the Fourier-Bessel series. Using the method of integral identities, we prove the uniqueness of the problem solution. For proving the existence of the solution, we use estimates of coefficients of the series and the system of eigenfunctions; we establish them on the base of asymptotic formulas for the Bessel function and its zeros. We state sufficient conditions with respect to the initial conditions that guarantee the convergence of the constructed series in the class of regular solutions. We prove the theorem on the stability of the solution to the stated problem.

Key words

hyperbolic equation Bessel operator initial-boundary value problem uniqueness existence Fourier Bessel series uniform convergence stability 


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This work was supported by the Regional Scientific and Educational Mathematical Center of Kazan (Volga Region) Federal University, project no. 0212/02.12.10179.001.


  1. 1.
    Keldysh, M.V. “On Certain Classes of Elliptic Equations with Singularity on the Boundary of the Domain”, Dokl. Akad. Nauk SSSR77(2), 181–183 (1951).Google Scholar
  2. 2.
    Weinstein, A. “Discontinuous Integrals and Generalized Theory of Potential”, Trans. Amer. Math. Soc.63(2), 342–354 (1948).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Weinstein, A. “Generalized Axially Symmetric Potential Theory”, Bull. Amer. Math. Soc.59, 20–38 (1953).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bers, L. “On a Class of Differential Equations in Mechanics of Continua”, Quart. Appl. Math.5(1), 168–188 (1943).MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bers, L. “A Remark on an Applications of Pseudo-Analytic Functions”, Amer. J. Math.78(3), 486–496 (1956).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bers, L., Gelbart, A. “On a Class of Functions Defined by Partial Differential Equations”, Trans. Amer. Math. Soc.56, 67–93 (1944).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gilbert, R.P. Function Theoretic Method in Partial Differential Equations (Academic Press, New York-London, 1969).zbMATHGoogle Scholar
  8. 8.
    Gurevich, M.I. Theory of Jets in Ideal Fluids (Nauka, Moscow, 1979) [in Russian].Google Scholar
  9. 9.
    Bitsadze, A.V., Pashkovskii, V.I. “Theory of the Maxwell-Einstein Equations”, Dokl. Akad. Nauk SSSR216(2), 9–10 (1974).MathSciNetGoogle Scholar
  10. 10.
    Bitsadze, A.V., Pashkovskii, V.I. “Certain Classes of the Solutions of the Maxwell-Einstein Equation”, Tr. MIAN SSSR134, 26–30 (1975).MathSciNetGoogle Scholar
  11. 11.
    Dzhayani, G.V. Solution of Some Problems for a Degenerate Singular Elliptic Equation and Application to Prismatic Shells (Izd-vo Tbilis. un-ta, Tbilisi, 1982) [in Russian].Google Scholar
  12. 12.
    Kipriyanov, I.A. Singular Elliptic Boundary Value Problems (Nauka, Fizmatlit, Moscow, 1997) [in Russian].zbMATHGoogle Scholar
  13. 13.
    Carroll, R.W., Showalter, R.E. Singular and Degenerate Cauchy Problems (Academic Press, New York, 1976).Google Scholar
  14. 14.
    Katrakhov, V.V., Sitnik, S.M. “The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations”, Sovremennaya matem. Fundament, napravleniya64(2), 211–426 (2018).MathSciNetGoogle Scholar
  15. 15.
    Koshlyakov, N.S., Gliner, E.B., Smirnov, M.M. Partial Differential Equations of Mathematical Physics (Vyssh. Shkola, Moscow, 1970) [in Russian].zbMATHGoogle Scholar
  16. 16.
    Pul’kin, S.P. “Certain Boundary-Value Problems for the Equations \({u_{xx}} \pm {u_{yy}} + {p \over x}{u_x} = 0\)”, Uchen. zap. Kuibyshebsk. Gos. Pedagogicheskogo In-ta.21, 3–55 (1958).Google Scholar
  17. 17.
    Sabitov, K.B., Il’yasov, R.R. “On the Ill-Posedness of Boundary Value Problems for a Class of Hyperbolic Equations”, Russian Mathematics45(5), 56–60 (2001).MathSciNetzbMATHGoogle Scholar
  18. 18.
    Sabitov, K.B., Il’yasov, R.R. “Solution of the Tricomi Problem for an Equation of Mixed Type with a Singular Coefficient by the Spectral Method”, Russian Mathematics48(2), 61–68 (2004).MathSciNetCrossRefGoogle Scholar
  19. 19.
    Safina, R.M. “Keldysh problem for a mixed-type equation of the second kind with the Bessel operator”, Differential Equations51(10), 1347–1359 (2015).MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sabitov, K.B., Safina, R.M. “The First Boundary-Value Problem for an Equation of Mixed Type with a Singular Coefficient”, Izv. Ross. Akad. Nauk. Ser. Matem.82(2), 79–112 (2018).zbMATHGoogle Scholar
  21. 21.
    Zaitseva, N.V. “Keldysh-Type Problem for B-Hyperbolic Equation with Integral Boundary Value Condition of the First Kind”, Lobachevskii J. Math.38(1), 162–169 (2017).MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sabitov, K.B., Zaitseva, N.V. “Initial Value Problem for B-Hyperbolic Equation with Integral Condition of the Second Kind”, Differential Equations54(1), 121–133 (2018).MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pul’kin, S.P. “Uniqueness of the Solution of a Singular Problem of Gellerstedt-Tricomi”, Izv. Vuz. Matem.6, 214–225 (1960).MathSciNetzbMATHGoogle Scholar
  24. 24.
    Sabitov, K.B. On the Theory of Equations of the Mixed Type (Fizmatlit, Moscow, 2014) [in Russian].Google Scholar
  25. 25.
    Watson, G.N. A Treatise on the Theory of Bessel Functions. Part. 1 (IL, Moscow, 1949) [in Russian].Google Scholar
  26. 26.
    Olver, F.W.J. The Introduction to Asymptotic Methods and Special Functions (Mir, Moscow, 1986) [in Russian].Google Scholar
  27. 27.
    Vladimirov, V.S. Equations of Mathematical Physics. Ed. 4 (Nauka, Fizmatlit, Moscow, 1981) [in Russian].Google Scholar
  28. 28.
    Sabitov, K.B., Vagapova, E.V. “Dirichlet problem for an equation of mixed type with two degeneration lines in a rectangular domain”, Differential Equations49(1), 68–78 (2013).MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sabitov K.B., Zaitseva N.V. “Initial-Boundary Value Problem for Hyperbolic Equation with Singular Coefficient and Integral Condition of Second Kind”, Lobachevskii J. Math.39(9), 1419–1427 (2018).MathSciNetCrossRefGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Sterlitamak branch of Bashkir State UniversitySterlitamakRussia
  2. 2.Sterlitamak branch of the Institute for Strategic Studies of the Republic of BashkortostanSterlitamakRussia
  3. 3.Kazan Federal UniversityKazanRussia

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