Skip to main content
SpringerLink
Log in

Method of solving the Cauchy problem for one-dimensional polywave equation with singular Bessel operator

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We study the Cauchy problem for an equation with singular Bessel operator. Unlike traditional methods to solve this problem, we apply Erde´ lyi–Kober fractional operator and find an explicit formula for the desired solution. We prove that the resulting formula is a unique classical solution to the problem.

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. Bitsadze, A. V. Selected Works (KBNTs RAN, Nal’hik 2012) [in Russian].

    Google Scholar 

  2. Zhegalov, V. I. “Bounadary-Value Problem for Equation of Mixed Type of Higher Order”, Dokl. Akad. Nauk SSSR 136, No. 2, 274–276 (1962) [in Russian].

    Google Scholar 

  3. Smirnov, M. M. Model Equation ofMixed Type of Fourth Order (Leningrad Univ. Press, Leningrad, 1972) [in Russian].

    Google Scholar 

  4. Meredov, M. M. “Uniqueness of Solution to Boundary-Value Problems for Equation of Mixed Type of the Fourth Order”, Izv. Akad. Nauk Turkmen SSR. Ser. fiz.-tekhn., khem. i geol. nauk, No. 4, 11–16 (1967) [in Russian].

    Google Scholar 

  5. Zhegalov, V. I. “CertainDirection in Theory of EquationsWith PartialDerivatives”, in Proceedings of Intern. Scientific conference ‘Boundary-Value Problems for Differential Equations and Analytic Functions‘, Kazan, September 29–October 1, 2014 (Kazan Math. Soc. 2014), pp. 13–15 [in Russian].

    Google Scholar 

  6. Sabitov, K. B. “Positivity of Solution to Inhomogeneous Equation of Mixed Type of Higher Order”, in Proceedings of Intern. Scientific conference ‘Boundary-Value Problems for Differential Equations and Analytic Functions‘, Kazan, September 29–October 1, 2014 (Kazan Math. Soc. 2014), pp. 64–67 [in Russian].

    Google Scholar 

  7. Gal’perin, S. A., Kondrashov, V. E. “The Cauchy Problem for DifferentialOperators Decomposing IntoWave Factors”, Trans.Mosc. Math. Soc. 16, 117–145 (1967).

    Google Scholar 

  8. Aldashev, S. A. “On the Cauchy Problem for Operators Splitting into Factors With Singularities”, Differ. Uravn. 17, 247–255 (1981) [in Russian].

    MATH  Google Scholar 

  9. Ivanov, L. A. “The Cauchy Problem for Some Operators With Singularities”, Differ. Uravn. 18, 1020–1028 (1982) [in Russian].

    MathSciNet  MATH  Google Scholar 

  10. Pul’kin, S. P. “Certain Boundary-Value Problems for Equations u xx ± u yy + (p/x)u x = 0”, Uchen. Zap. Kuibyshev Ped. Inst., No. 21, 3–55 (1958) [in Russian].

    Google Scholar 

  11. Il’in, V. A., Sadovnichii, V. A., Sendov, Bl. Ch. Mathematical Analysis. Continuation of the Course (Moscow Univ. Press, Moscow, 1987) [in Russian].

    Google Scholar 

  12. Samko, S. G., Kilbas, A. A., Marichev, O. I. Integrals and Derivatives of Fractional Order With Applications (Nauka i Tekhnika,Minsk, 1987) [in Russian].

    MATH  Google Scholar 

  13. Bateman, G., Erdélyi, A. Higher Transcendental Functions (Nauka, Moscow, 1973), Vol. 1 [Russian translation].

    MATH  Google Scholar 

  14. Karimov, Sh. T. “New Properties of Generalized Erdélyi–Kober Operator With Application”, Dokl. Akad. Nauk Uzbek Republic, No. 5, 11–13 (2014) [in Russian].

    Google Scholar 

  15. Karimov, Sh. T. “Multidimensional Generalized Erdélyi–Kober Operator and its Application to Solving Cauchy Problems for Differential Equations With Singular Coefficients”, Fract. Calc. Appl. Anal. 18, No. 4, 845–861 (2015).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fergana State University, ul. Murabbiylar 19, Fergana, 150100, Uzbekistan

    Sh. T. Karimov

Authors
  1. Sh. T. Karimov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sh. T. Karimov.

Additional information

Original Russian Text © Sh.T. Karimov, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 8, pp. 27–41.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karimov, S.T. Method of solving the Cauchy problem for one-dimensional polywave equation with singular Bessel operator. Russ Math. 61, 22–35 (2017). https://doi.org/10.3103/S1066369X17080035

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X17080035

Keywords

Search

Navigation