Skip to main content
SpringerLink
Log in

Construction of Eigenfunctions of the Tricomi--Neumann Problem for Equations of Mixed Type with Characteristic Degeneration and Their Application

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

For the equation of mixed type

$$\user1{L}\alpha \user1{u} \equiv \user1{u}_{\user1{xx}} + \user1{yu}_{\user1{yy}} + \alpha \user1{u}_\user1{y} + \lambda \user1{u}\user2{ = 0,}$$

where 0 < α < 1 and λ is a complex parameter, we obtain eigenvalues in a special domain by the method of separation of variables and construct the system of corresponding eigenfunctions of the Tricomi--Neumann spectral problem. We construct the solution of the Tricomi--Neumann problem as a sum of 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. M. M. Smirnov, Equations of Mixed Type [in Russian], Moscow, 1985.

  2. S. M. Ponomarev, The Spectral Theory of the Main Boundary-Value Problem for Equations of Mixed Lavrent' ev–Bitsadze Type [in Russian] Extended Abstract of Doctoral (Phys.–Math.) Dissertation, Moscow State University, Moscow, 1981.

    Google Scholar 

  3. E. I. Moiseev, Equations of Mixed Type with a Spectral Parameter [in Russian], Moscow State University, Moscow, 1998.

    Google Scholar 

  4. E. I. Moiseev, “The solution of the Tricomi problem in special domains,” Differentsial' nye Uravneniya [Differential Equations], 26 (1990), no. 1, 93–103.

    Google Scholar 

  5. K. B. Sabitov and V. V. Tikhomirov, “On the construction of eigenvalues and eigenfunctions for the Frankl problem in gas dynamics,” Mat. Modelirovanie, 2 (1990), no. 10, 100–109.

    Google Scholar 

  6. Ya. N. Mamedov, “On some eigenvalue problems for equations of mixed type,” Differentsial' nye Uravneniya [Differential Equations], 26 (1990), no. 1, 163–168.

    Google Scholar 

  7. E. I. Moiseev, “On the representation of the solution of the Tricomi problem in the form of a biorthogonal series,” Differentsial' nye Uravneniya [Differential Equations], 27 (1991), no. 7, 1229–1237.

    Google Scholar 

  8. K. B. Sabitov and V. Z. Vagapov, “On the construction of particular solutions of degenerate equations of mixed type,” in: Complex Analysis, Differential Equations, and Related Questions [in Russian], Third Internat. Scientific Conference, Ufa, 1996, pp. 99–106.

  9. S. L. Bibakova, “The eigenvalue problem for equations of mixed type,” in: “Physics of the Condensed State” [in Russian], vol. 1, Sterlitamak, 1997, pp. 14–18.

    Google Scholar 

  10. H. Bateman and A. Erdélyi, Higher Transcendental Functions, McGraw–Hill, New York–Toronto– London, 1953.

    Google Scholar 

  11. G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1945.

    Google Scholar 

  12. E. I. Moiseev, “On the basis property of a system of sines,” Differentsial' nye Uravneniya [Differential Equations], 23 (1987), no. 1, 177–179.

    Google Scholar 

  13. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Their Applications [in Russian] Nauka and Tekhnika, Minsk, 1987.

    Google Scholar 

  14. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series [in Russian], Nauka, Moscow, 1981.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sterlitamak Division, Academy of Sciences of Bashkortostan, Russia

    K. B. Sabitov

  2. Sterlitamak State Pedagogical Institute, Russian

    S. L. Bibakova

Authors
  1. K. B. Sabitov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. L. Bibakova
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sabitov, K.B., Bibakova, S.L. Construction of Eigenfunctions of the Tricomi--Neumann Problem for Equations of Mixed Type with Characteristic Degeneration and Their Application. Mathematical Notes 74, 70–80 (2003). https://doi.org/10.1023/A:1025019216707

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1025019216707

Search

Navigation