SpringerLink
Log in

On Solvability of a Poincare–Tricomi Type Problem for an Elliptic–Hyperbolic Equation of the Second Kind

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

Abstract

In this paper we study a boundary value problem with the Poincare–Tricomi condition for a degenerate partial differential equation of elliptic-hyperbolic type of the second kind. In the hyperbolic part of a degenerate mixed differential equation of the second kind the line of degeneracy is a characteristic. For this type of differential equations a class of generalized solutions is introduced in the characteristic triangle. Using the properties of generalized solutions, the modified Cauchy and Dirichlet problems are studied. The solutions of these problems are found in the convenient form for further investigations. A new method has been developed for a differential equation of mixed type of the second kind, based on energy integrals. Using this method, the uniqueness of the considering problem is proved. The existence of a solution of the considering problem reduces to investigation of a singular integral equation and the unique solvability of this problem is proved by the Carleman–Vekua regularization method.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. F. I. Frankl, Selected Works on Gas Dynamics (Nauka, Moscow, 1973) [in Russian].

    Google Scholar 

  2. I. N. Vekua, Generalized Analytic Functions (Fizmatgiz, Moscow, 1959) [in Russian].

    MATH  Google Scholar 

  3. E. I. Moiseev and M. Mogimi, ‘‘On the completeness of eigenfunctions of the Neumann–Tricomi problem for a degenerate equation of mixed type,’’ Differ. Equat. 41, 1789–1791 (2005).

    Article  MathSciNet  Google Scholar 

  4. E. I. Moiseev and M. Mogimi, ‘‘On the completeness of eigenfunctions of the Tricomi problem for a degenerate equation of mixed type,’’ Differ. Equat. 41, 1462–1466 (2005).

    Article  MathSciNet  Google Scholar 

  5. E. I. Moiseev, T. E. Moiseev, and A. A. Kholomeeva, ‘‘On the non-uniqueness of the solution of the inner Neumann-Hellerstedt problem for the Lavrent’ev–Bitsadze equation,’’ Sib. Zh. Chist. Prikl. Mat. 17 (3), 52–57 (2017).

    MATH  Google Scholar 

  6. A. B. Okboeb, ‘‘Tricomy problem for second kind parabolic-hyperbolic type equation,’’ Lobachevskii J. Math. 41, 58–70 (2020).

    Article  MathSciNet  Google Scholar 

  7. O. A. Repin and S. K. Kumykova, ‘‘A nonlocal problem for degenerate hyperbolic equation,’’ Russ. Math. (Iz. VUZ) 61 (7), 43–48 (2017).

  8. K. B. Sabitov, ‘‘On the theory of the Frankl problem for equations of mixed type,’’ Russ. Math. (Iz. VUZ) 81 (1), 99–136 (2017).

  9. K. B. Sabitov and V. A. Novikova, ‘‘Nonlocal A. A. Dezin’s problem for Lavrent’ev–Bitsadze equation,’’ Russ. Math. (Iz. VUZ) 60 (6), 52–62 (2016).

  10. K. B. Sabitov and I. A. Khadzhi, ‘‘The boundary-value problem for the Lavrent’ev-Bitsadze equation with unknown right-hand side,’’ Russ. Math. (Iz. VUZ) 55 (5), 35–42 (2011).

  11. M. S. Salakhitdinov and B. Islomov, ‘‘Boundary value problems for an equation of mixed type with two inner lines of degeneracy,’’ Dokl. Math. 43, 235–238 (1991).

    MathSciNet  MATH  Google Scholar 

  12. M. S. Salakhitdinov and M. Mirsaburov, ‘‘The Bitsadze-Samarskii problem for a class of degenerate hyperbolic equations,’’ Differ. Equat. 38), 288–293 (2002).

  13. M. S. Salakhitdinov and A. K. Urinov, ‘‘Eigenvalue problems for a mixed-type equation with two singular coefficients,’’ Sib. Math. J. 48, 707–717 (2007).

    Article  Google Scholar 

  14. M. S. Salakhitdinov and M. Mirsaburov, ‘‘A problem with a nonlocal boundary condition on the characteristic for a class of equations of mixed type,’’ Math. Notes 86, 704–715 (2009).

    Article  MathSciNet  Google Scholar 

  15. M. S. Salakhitdinov and B. I. Islomov, ‘‘A nonlocal boundary-value problem with conormal derivative for a mixed-type equation with two inner degeneration lines and various orders of degeneracy,’’ Russ. Math. (Iz. VUZ) 55 (1), 42–49 (2011).

  16. M. S. Salakhitdinov and N. B. Islamov, ‘‘Nonlocal boundary-value problem with Bitsadze–Samarskii condition for equation of parabolic-hyperbolic type of the second kind,’’ Russ. Math. (Iz. VUZ) 59 (6), 34–42 (2015).

  17. F. G. Tricomi, Lezioni sulle equazioni a derivate parziali (Gheroni, Torino, 1954).

    MATH  Google Scholar 

  18. A. V. Bitsadze, Some Classes of Partial Differential Equations (Nauka, Moscow, 1981) [in Russian].

    MATH  Google Scholar 

  19. M. M. Smirnov, Equations of a Mixed Type (Nauka, Moscow, 1970) [in Russian].

    Google Scholar 

  20. I. L. Karol, ‘‘On a boundary value problem for an equation of mixed elliptic-hyperbolic type,’’ Dokl. Akad. Nauk SSSR 88, 197–200 (1953).

    MathSciNet  Google Scholar 

  21. N. K. Mamadaliev, ‘‘On representation of solution of the modified Cauchy problem,’’ Sib. Math. J. 41, 889–899 (2000).

    Article  MathSciNet  Google Scholar 

  22. S. A. Tersenov, ‘‘On the theory of hyperbolic equations with data on lines of degeneration type,’’ Sib. Mat. Zh. 2, 931–935 (1961).

    MathSciNet  Google Scholar 

  23. S. S. Isamuhamedov and Zh. Oromov, ‘‘On boundary value problems for a mixed type equation of the second kind with a nonsmooth line of degeneration,’’ Differ. Uravn. 18, 324–334 (1982).

    Google Scholar 

  24. N. K. Mamadaliev, ‘‘Tricomi problem for strongly degenerate equations of parabolic-hyperbolic type,’’ Math Notes 66, 310–315 (1999).

    Article  MathSciNet  Google Scholar 

  25. R. S. Khairullin, The Tricomi Problem for an Equation of the Second Kind with Strong Degeneracy (Kazan. Gos. Univ., Kazan, 2015) [in Russian].

    Google Scholar 

  26. M. S. Salakhitdinov and N. B. Islamov, ‘‘A nonlocal boundary-value problem with the Bitsadze–Samarskii conditon for a parabolic-hyperbolic equation of the second kind,’’ Russ. Math. (Iz. VUZ) 59 (6), 34–42 (2015).

  27. K. B. Sabitov and I. P. Egorova, ‘‘On the correctness of boundary value problems with periodicity conditions for an equation of mixed type of the second kind,’’ Vestn. Samar. Tekh. Univ., Ser. Fiz.-Mat. Nauki 23, 430–451 (2019).

    Google Scholar 

  28. A. M. Nakhushev, ‘‘Loaded equations and their applications,’’ Differ. Uravn. 19, 86–94 (1983).

    MathSciNet  MATH  Google Scholar 

  29. A. A. Samarsky, ‘‘On some problems of the theory of differential equations,’’ Differ. Uravn. 16, 1925–1935 (1980).

    MathSciNet  Google Scholar 

  30. M. S. Salakhitdinov and T. G. Ergashev, ‘‘Integral representation of a generalized solution of the Cauchy problem in the class for one equation of hyperbolic type of the second kind,’’ Uzb. Mat. Zh., No. 1, 67–75 (1995).

  31. K. B. Sabitov and A. K. Suleimanova, ‘‘The Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain,’’ Russ. Math. (Iz. VUZ) 51 (4), 42–50 (2007).

  32. K. B. Sabitov and A. K. Suleimanova, ‘‘The Dirichlet problem for a mixed-type equation with characteristic degeneration in a rectangular domain,’’ Russ. Math. (Iz. VUZ) 53 (11), 37–45 (2009).

  33. T. K. Yuldashev, ‘‘Nonlocal inverse problem for a pseudohyperbolic-pseudoelliptic type integro-differential equations,’’ Axioms 9 (2), 45-1–21 (2020).

  34. T. K. Yuldashev and B. J. Kadirkulov, ‘‘Boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator,’’ Axioms 9 (2), 68-1–19 (2020).

  35. B. I. Islomov and A. A. Abdullayev, ‘‘On a problem for an elliptic type equation of the second kind with a conormal and integral condition,’’ J. Nanosyst.: Phys. Chem. Math. 9, 307–318 (2018).

    Google Scholar 

  36. M. S. Salakhitdinov and A. A. Abdullaev, ‘‘Nonlocal boundary value problem for a mixed type equation of the second kind,’’ Dokl. Akad. Nauk Uzb., No. 1, 3–5 (2013).

  37. M. M. Smirnov, Equations of a Mixed Type (Vysshaya Shkola, Moscow, 1985) [in Russian].

    Google Scholar 

  38. N. I. Muskhelishvili, Singular Integral Equations (Nauka, Moscow, 1968).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Uzbek–Israel Joint Faculty of High Technology and Engineering Mathematics, National University of Uzbekistan, 100174, Tashkent, Uzbekistan

    T. K. Yuldashev

  2. Department of Differential Equations and Mathematical Physics, National University of Uzbekistan, 100174, Tashkent, Uzbekistan

    B. I. Islomov

  3. Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, 100000, Tashkent, Uzbekistan

    A. A. Abdullaev

Authors
  1. T. K. Yuldashev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. I. Islomov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. A. Abdullaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to T. K. Yuldashev, B. I. Islomov or A. A. Abdullaev.

Additional information

(Submitted by A. M. Elizarov)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yuldashev, T.K., Islomov, B.I. & Abdullaev, A.A. On Solvability of a Poincare–Tricomi Type Problem for an Elliptic–Hyperbolic Equation of the Second Kind. Lobachevskii J Math 42, 663–675 (2021). https://doi.org/10.1134/S1995080221030239

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080221030239

Keywords:

Search

Navigation