Skip to main content
SpringerLink
Log in

A Family of Singular Differential Equations

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

A family of singular differential equations with variable coefficients and parameter \(k\in\mathbb{R}\) is introduced into the consideration. The properties inherent in all differential equations of this family are investigated and theorems on the solvability of a number of initial problems for the considered family are formulated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (CRC, Boca Raton, FL, 1993).

    MATH  Google Scholar 

  2. V. V. Katrakhov and S. M. Sitnik, ‘‘The transmutation method and boundary value problems for singular differential equations,’’ Sovrem. Mat. Fundam. Napravl. 64, 211–428 (2018).

    MathSciNet  Google Scholar 

  3. S. M. Sitnik and E. L. Shishkina, Transformation Operator Method for Differential Equations with Bessel Operators (Fizmatlit, Moscow, 2019) [in Russian].

    Google Scholar 

  4. A. V. Glushak, V. I. Kononenko, and S. D. Shmulevich, ‘‘A singular abstract Cauchy problem,’’ Sov. Math. 30, 678–681 (1986).

    MATH  Google Scholar 

  5. S. A. Tersenov, Introduction to the Theory of Equations Degenerating on the Boundary (Novosibirsk. Gos. Univ., Novosibirsk, 1973) [in Russian].

    Google Scholar 

  6. L. A. Ivanov, ‘‘A Cauchy problem for some operators with singularities,’’ Differ. Equat. 18, 724–731 (1982).

    MathSciNet  MATH  Google Scholar 

  7. I. A. Kipriyanov and L. A. Ivanov, ‘‘The Cauchy problem for the Euler–Poisson–Darboux equation in a symmetric space,’’ Math. USSR Sb. 52, 41–51 (1985).

    Article  Google Scholar 

  8. A. V. Glushak and O. A. Pokruchin, ‘‘Criterion for the solvability of the Cauchy problem for an abstract Euler–Poisson–Darboux equation,’’ Differ. Equat. 52, 39–57 (2016).

    Article  MathSciNet  Google Scholar 

  9. S. M. Sitnik and E. L. Shishkina, ‘‘General form of the Euler–Poisson–Darboux equation and application of the transmutation method,’’ Electron. J. Differ. Equat. 177, 1–20 (2017).

    MathSciNet  MATH  Google Scholar 

  10. E. L. Shishkina, ‘‘Singular Cauchy problem for the general Euler–Poisson–Darboux equation,’’ Open Math. 16, 23–31 (2018).

    Article  MathSciNet  Google Scholar 

  11. E. L. Shishkina and M. Karabacak, ‘‘Singular Cauchy problem for generalized homogeneous Euler–Poisson–Darboux equation,’’ Mat. Zam. SVFU 25 (2), 85–96 (2018).

    MATH  Google Scholar 

  12. M. N. Olevskii, ‘‘On connections between solutions of the generalized wave equation and the generalized heat-conduction equation,’’ Dokl. Akad. Nauk SSSR 101, 21–24 (1955).

    MathSciNet  MATH  Google Scholar 

  13. I. A. Kipriyanov and L. A. Ivanov, ‘‘The Euler–Poisson–Darboux equation in a Riemannian space,’’ Sov. Math. Dokl. 24, 331–335 (1981).

    MATH  Google Scholar 

  14. V. I. Kononenko and L. A. Khinkis, ‘‘Transformation operators related to the Jacobi differential operator,’’ Available from VINITI, No. 1604-B89 (1989).

  15. A. V. Glushak, ‘‘The Legendre operator function,’’ Izv. Math. 65, 1073–1083 (2001).

    Article  MathSciNet  Google Scholar 

  16. V. Ya. Yaroslavtseva, ‘‘On one transformation operator class and its applications to differential equations,’’ Dokl. Akad. Nauk SSSR 227, 816–819 (1976).

    MathSciNet  Google Scholar 

  17. J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs (Oxford Univ. Press, Oxford, 1985).

  18. V. V. Vasil’ev, S. G. Krein, and S. I. Piskarev, ‘‘Operator semigroups, cosine operator functions and linear differential equations,’’ Itogi Nauki Tekh., Mat. Anal. 28, 87–202 (1990).

    Google Scholar 

  19. B. M. Levitan, ‘‘Expansion in Fourier series and integrals with Bessel functions,’’ Usp. Mat. Nauk 6 (2), 102–143 (1951).

    MathSciNet  Google Scholar 

  20. A. V. Glushak, ‘‘Abstract Cauchy problem for the Bessel-Struve equation,’’ Differ. Equat. 53, 864–878 (2017).

    Article  MathSciNet  Google Scholar 

  21. A. V. Glushak, ‘‘Uniquely solvable problems for abstract Legendre equation,’’ Russ. Math. 62 (7), 1–12 (2018).

    Article  MathSciNet  Google Scholar 

  22. A. V. Glushak, ‘‘Criterion for the solvability of the weighted Cauchy problem for an abstract Euler–Poisson–Darboux equation,’’ Differ. Equat. 54, 622–632 (2018).

    Article  MathSciNet  Google Scholar 

Download references

Funding

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

Author information

Authors and Affiliations

  1. Federal State Autonomous Belgorod State University, 308015, Belgorod, Russia

    A. V. Glushak

Authors
  1. A. V. Glushak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Glushak.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Glushak, A.V. A Family of Singular Differential Equations. Lobachevskii J Math 41, 763–771 (2020). https://doi.org/10.1134/S1995080220050030

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080220050030

Search

Navigation