Skip to main content
SpringerLink
Log in

The boundary-value problem for Lavrent’Ev–Bitsadze equation with two internal lines of change of a type

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We study the problem with boundary conditions of the first and second kind on the boundary of a rectangular domain for an equation with two internal perpendicular lines of change of a type. With the use of the spectral method we prove the unique solvability of the mentioned problem. The eigenvalue problem for an ordinary differential equation obtained by separation of variables is not self-adjoint, and the system of root functions is not orthogonal. We construct the corresponding biorthogonal system of functions and prove its completeness. This allows us to establish a criterion for the uniqueness of the solution to the problem under consideration. We construct the solution as the sum of the biorthogonal series.

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

    Google Scholar 

  2. Shabat, B.V. “Examples of Solutions to the DirichletProblem forMixed-TypeEquations”, Dokl. Akad.Nauk SSSR 112, No. 3, 386–389 (1957).

    MathSciNet  MATH  Google Scholar 

  3. Bitsadze, A. V. “Ill-Posedness of the Dirichlet Problem for Equations of Mixed Type”, Dokl. Akad. Nauk SSSR 122, No. 2, 167–170 (1958).

    MathSciNet  MATH  Google Scholar 

  4. Cannon, J. R. “Dirichlet Problem for an Equation of Mixed Type with a Discontinuous Coefficient”, Ann. Mat. Pura Appl. 62, 371–377 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  5. Nakhushev, A. M. “Uniqueness Criterion for the Dirichlet Problem for an Equation of Mixed Type in a Cylindrical Domain”, Diff. Uravn. 6, No. 1, 190–191 (1970).

    MATH  Google Scholar 

  6. Soldatov, A. P. “Problems of Dirichlet Type for the Lavrent’ev–Bitsadze Equation. I: Uniqueness Theorems”, Russ. Acad. Sci., Dokl., Math. 48, No. 2, 410–414 (1994).

    MathSciNet  Google Scholar 

  7. Soldatov, A. P. “Problems of Dirichlet Type for the Lavrent’ev–Bitsadze Equation. I: Existence Theorems”, Russ. Acad. Sci., Dokl., Math. 48, No. 3, 433–437 (1994).

    MathSciNet  Google Scholar 

  8. Zhegalov, V. I. “On Dirichlet-Type Problems with Shifts for the Lavrent’ev–Bitsadze Equation”, Trudy Semin. po Kraevym Zadacham 23, 81–88 (Kazan Univ. Press, Kazan, 1987).[in Russian].

    MathSciNet  Google Scholar 

  9. Khachev, M. M. The First Boundary-Value Problem for LinearMixed-Type Equations (El’brus, Nal’chik, 1998).[in Russian].

    Google Scholar 

  10. Sabitov, K. B. “Dirichlet Problem for Mixed-Type Equations in a Rectangular Domain”, Dokl.Math. 75, No. 2, 193–196 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  11. Sabitov, K. B. and Suleimanova, A. Kh. “The Dirichlet Problem for a Mixed-Type Equation of the Second Kind in a Rectangular Domain”, RussianMathematics (Iz. VUZ) 51, No. 4, 42–50 (2007).

    MathSciNet  MATH  Google Scholar 

  12. Sabitov, K. B. and Suleimanova, A. Kh. “The Dirichlet Problem for aMixed-Type Equation with Characteristic Degeneration in a Rectangular Domain”, RussianMathematics (Iz. VUZ) 53, No. 11, 37–45 (2009).

    MathSciNet  MATH  Google Scholar 

  13. Sabitov, K. B. and Vagapova, E. V. “Dirichlet Problem for an Equation ofMixed Type With TwoDegeneration Lines in a Rectangular Domain”, Differ. Equ. 49, No. 1, 68–78 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  14. Khairullin, R. S. “On the Dirichlet Problem for a Mixed-Type Equation of the Second Kind With Strong Degeneration”, Differ. Equ. 49, No. 4, 510–516 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  15. Sabitov, K. B. and Melisheva, E. P. “The Dirichlet Problem for a Loaded Mixed-Type Equation in a Rectangular Domain”, RussianMathematics (Iz. VUZ) 57, No. 7, 53–65 (2013).

    MathSciNet  MATH  Google Scholar 

  16. Il’in, V. A. “Proof of Uniqueness and Membership in W1 2 of the Classical Solution of a Mixed Problem for a Self-Adjoint Hyperbolic Equation”, Math.Notes 17, 53–58 (1975).

    Article  MATH  Google Scholar 

  17. Lomov, I. S. “Small Denominators in Analytic Theory of Degenerate Differential Equations”, Differ. Equations 29, No. 12, 1811–1820 (1993).

    MathSciNet  Google Scholar 

  18. Lomov, I. S. “Nonsmooth Eigenfunctions in Problems of Mathematical Physics”, Differ. Equ. 47, No. 3, 355–362 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  19. Keldysh, M. V. “On the Completeness of the Eigenfunctions of Some Classes of Non-Selfadjoint Linear Operators”, Usp.Mat. Nauk 26, No. 4, 15–41 (1971).[in Russian].

    MATH  Google Scholar 

  20. Bari, N. K. “Biorthogonal Systems and Bases in Hilbert Space”, Uchen. Zap. Moskovsk.Univ. 4, 69–107 (1971).[in Russian].

    Google Scholar 

  21. Il’in, V. A. “On the Unconditional Basis Property for Systems of Eigenfunctions and Associated Functions of a Second-Order Differential Operator on a Closed Interval”, Sov.Math., Dokl. 28, 743–747 (1983).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sterlitamak Branch of the Bashkir State University, pr. Lenina 37, Sterlitamak, 453103, Russia

    A. A. Gimaltdinova & K. V. Kurman

Authors
  1. A. A. Gimaltdinova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. V. Kurman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Gimaltdinova.

Additional information

Original Russian Text © A.A. Gimaltdinova, K.V. Kurman, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 3, pp. 23–37.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gimaltdinova, A.A., Kurman, K.V. The boundary-value problem for Lavrent’Ev–Bitsadze equation with two internal lines of change of a type. Russ Math. 60, 18–31 (2016). https://doi.org/10.3103/S1066369X16030038

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X16030038

Keywords

Search

Navigation