Skip to main content
SpringerLink
Log in

On a Frankl-type problem for a mixed parabolic-hyperbolic equation

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We state a new nonlocal boundary value problem for a mixed parabolic-hyperbolic equation. The equation is of the first kind, i.e., the curve on which the equation changes type is not a characteristic. The nonlocal condition involves points in hyperbolic and parabolic parts of the domain. This problem is a generalization of the well-known Frankl-type problems. Unlike other close publications, the hyperbolic part of the domain agrees with a characteristic triangle. We prove unique solvability of this problem in the sense of classical and strong solutions.

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. Dzhuraev T. D., Boundary Value Problems for Mixed and Mixed-Composite Type Equations [Russian], Fan, Tashkent 1979).

    MATH  Google Scholar 

  2. Dzhuraev T. D., Sopuev A., and Mamazhanov A., Boundary Value Problems for Parabolic-Hyperbolic Type Equations [Russian], Fan, Tashkent 1986).

    MATH  Google Scholar 

  3. Berdyshev A. S., Boundary Value Problems and Their Spectral Properties for a Mixed Parabolic-Hyperbolic and Mixed- Composite Type Equation, Izdat. KazNPU im. Abaya, Almaty 2015).

    Google Scholar 

  4. Frankl F. I., “On Chaplygin’s problems for mixed subsonic and supersonic flows,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 9, no. 2, 121–142 (1945).

    MATH  Google Scholar 

  5. Frankl F. I., “Flow around airfoils by a stream of subsonic velocity with supersonic zones terminating in a straight-line condensation shock,” Prikl. Mat. Mekh., vol. 20, no. 2, 196–202 (1956).

    Google Scholar 

  6. Kapustin N. Yu., “On the generalized solvability of the Tricomi problem for a parabolic-hyperbolic equation,” Dokl. Akad. Nauk SSSR, vol. 274, no. 6, 1294–1298 (1984).

    MathSciNet  Google Scholar 

  7. Kapustin N. Yu., “Existence and uniqueness of the L 2-solution of the Tricomi problem for a parabolic-hyperbolic equation,” Dokl. Akad. Nauk SSSR, vol. 291, no. 2, 288–292 (1986).

    MathSciNet  Google Scholar 

  8. Sadybekov M. A. and Toĭzhanova G. D., “Spectral properties of a class of boundary value problems for a parabolic-hyperbolic equation,” Differ. Uravn., vol. 28, no. 1, 176–179 (1992).

    MathSciNet  MATH  Google Scholar 

  9. Berdyshev A. S., “The Volterra property of some problems with the Bitsadze–Samarskiĭ-type conditions for a mixed parabolic-hyperbolic equation,” Sib. Math. J., vol. 46, no. 3, 386–395 (2005).

    Article  MATH  Google Scholar 

  10. Rakhmanova L. Kh., “Solution of a nonlocal problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain by the spectral method,” Russian Math. (Iz. VUZ), vol. 51, no. 11, 35–39 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  11. Sabitov K. B. and Rakhmanova L. Kh., “Initial-boundary value problem for an equation of mixed parabolic-hyperbolic type in a rectangular domain,” Differ. Equ., vol. 44, no. 9, 1218–1224 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  12. Sabitov K. B., “Nonlocal problem for a parabolic-hyperbolic equation in a rectangular domain,” Math. Notes, vol. 89, no. 4, 562–567 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  13. Orazov M. B. and Sadybekov M. A., “On a class of problems of determining the temperature and density of heat sources given initial and final temperature,” Sib. Math. J., vol. 53, no. 1, 146–151 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  14. Kal’menov T. Sh. and Tokmagambetov N. E., “On a nonlocal boundary value problem for the multidimensional heat equation in a noncylindrical domain,” Sib. Math. J., vol. 54, no. 6, 1023–1028 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  15. Moiseev E. I., Nefedov P. V., and Kholomeeva A. A., “Analogs of the Tricomi and Frankl problems for the Lavrent’ev–Bitsadze equation in three-dimensional domains,” Differ. Equ., vol. 50, no. 12, 1677–1680 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  16. Nakhushev A. M., Problems with a Shift for Partial Differential Equations [Russian], Nauka, Moscow 2006).

    MATH  Google Scholar 

  17. Babich V. M., Kapilevich M. B., Mikhlin S. G., et al., Linear Equations of Mathematical Physics [Russian], Nauka, Moscow 1964).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    T. Sh. Kal’menov & M. A. Sadybekov

Authors
  1. T. Sh. Kal’menov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Sadybekov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. A. Sadybekov.

Additional information

The authors were supported by the Ministry of Education and Science of the Republic of Kazakhstan (Grants 0825/GF4 and 4075/GF4).

Almaty. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 58, No. 2, pp. 298–304, March–April, 2017; DOI: 10.17377/smzh.2017.58.205.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kal’menov, T.S., Sadybekov, M.A. On a Frankl-type problem for a mixed parabolic-hyperbolic equation. Sib Math J 58, 227–231 (2017). https://doi.org/10.1134/S0037446617020057

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446617020057

Keywords

Search

Navigation