Differential Equations

, Volume 55, Issue 8, pp 1069–1076 | Cite as

On the Solvability of Boundary Value Problems for an Abstract Bessel-Struve Equation

  • A. V. GlushakEmail author
Partial Differential Equations


We consider the Dirichlet and Neumann boundary value problems for the hyperbolic Bessel-Struve equation u″(t) + kt−1(u′(t) - u′(0)) = Au(t) on the half-line t > 0, where k > 0 is a parameter and A is a densely defined closed linear operator in a complex Banach space E. Generally speaking, these problems are ill posed. We establish sufficient conditions on the operator coefficient A and the boundary elements for these problems to be uniquely solvable.


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This work was supported by the Russian Foundation for Basic Research, project no. 19-01-00732.


  1. 1.
    Glushak, A.V., Abstract Cauchy problem for the Bessel-Struve equation Differ. Equations, 2017, vol. 53, no. 7, pp. 864–878.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Zheng, Q., Integrated cosine functions Int. J. Math. Math. Sci., 1996, vol. 19, no. 3, pp. 575–580.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Zhang, J. and Zheng, Q., On α-times integrated cosine functions Math. Jap., 1999, vol. 50, pp. 401–408.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Kostić, M. Generalized Semigroups and Cosine Functions, Beograd: Mat. Inst. SANU, 2011.zbMATHGoogle Scholar
  5. 5.
    Ivanov, V.K., Mel’nikova, I.V., and Filinkov, A.I. Differentsial’no-operatornye uravneniya i nekorrektnye zadachi (Operator-Differential Equations and Ill-Posed Problems), Moscow: Fizmatlit, 1995.zbMATHGoogle Scholar
  6. 6.
    Kabanikhin, S.I. and Krivorot’ko, O.I., A numerical method for solving the Dirichlet problem for the wave equation J. Appl. Ind. Math., 2012, vol. 15, no. 4, pp. 187–198.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Vasil’ev, V.I., Kardashevskii, A.M., and Popov, V.V., Solving the Dirichlet problem for vibrating string equation by the conjugate gradient method Vestn. Sev.-Vost. Fed. Univ., 2015, vol. 12, no. 2, pp. 43–50.Google Scholar
  8. 8.
    Lebedev, N.N. Spetsial’nye funktsii i ikh prilozheniya (Special Functions and Their Applications), Moscow-Leningrad: Gos. Izd. Fiz.-Mat. Lit., 1963.Google Scholar
  9. 9.
    Prilepko, A.I., Orlovsky, D.G., and Vasin, I.A. Methods for Solving Inverse Problems in Mathematical Physics, New York-Basel: Marcel Dekker, 2000.zbMATHGoogle Scholar
  10. 10.
    Cioranescu, I. and Lizama, C., Some applications of Fejér’s theorem to operator cosine functions in Banach spaces Proc. Amer. Math. Soc., 1997, vol. 125, no. 8, pp. 2353–2362.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Eidelman, Y., An inverse problem for a second-order differential equation in a Banach space Abstr. Appl. Anal., vol. 2004, no. 12, pp. 997–1005.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Glushak, A.V. and Pokruchin, O.A., Criterion for the solvability of the Cauchy problem for an abstract Euler-Poisson-Darboux equation Differ. Equations, 2016, vol. 52, no. 1, pp. 39–57.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I. Integraly i ryady. Dopolnitel’nye glavy (Integrals and Series. Additional Chapters), Moscow: Fizmatlit, 1986.zbMATHGoogle Scholar
  14. 14.
    Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I. Integraly i ryady. Spetsial’nye funktsii (Integrals and Series. Special Functions), Moscow: Nauka, 1983.zbMATHGoogle Scholar
  15. 15.
    Nebol’sina, M.N., Chebyshev orthogonal polynomials and the Neumann problem Differ. Equations, 2010, vol. 46, no. 3, pp. 455–457.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nebol’sina, M.N., Neumann problem for a second-order differential equation in a Banach space and orthogonal polynomials Mat. Model. Oper. Uravn., Voronezh, 2007, vol. 4, pp. 104–115.Google Scholar
  17. 17.
    Glushak, A.V., Abstract Euler-Poisson-Darboux equation with nonlocal condition Russ. Math., 2016, vol. 60, no. 6, pp. 21–28.MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Belgorod State UniversityBelgorodRussia

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