Advertisement

Siberian Mathematical Journal

, Volume 42, Issue 5, pp 961–974 | Cite as

Spectral Properties of a Solution to the Gellerstedt Problem for Mixed-Type Equations and Their Applications

  • K. B. Sabitov
  • A. N. Kuchkarova
Article
  • 15 Downloads

Abstract

We find eigenvalues and the corresponding eigenfunctions of the Gellerstedt problem for mixed-type equations with the Lavrent'ev–Bitsadze operator. We study the spectral properties of the system of eigenfunctions and present their applications to construction of a solution to the Gellerstedt problem in the form of the sum of a series.

Keywords

Spectral Property Gellerstedt Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gellerstedt S., Sur un Probleme aux Limites pour une Equation Lineaire aux Derivees Partielles du Second Ordre de Type Mixte, Thes. Doct., Uppsala (1935).Google Scholar
  2. 2.
    Gellerstedt S., “Quelgues problemes mixtes pour l'equation y mzxx +z yy = 0,” Ark. Mat. Astr. Fys., 26A, No. 3, 78-93 (1938).Google Scholar
  3. 3.
    Frankl 'F. I., Collected Works on Gas Dynamics [in Russian], Nauka, Moscow (1973).Google Scholar
  4. 4.
    Ovsyannikov L. V., Lectures on the Fundamentals of Gas Dynamics [in Russian], Nauka, Moscow (1981).Google Scholar
  5. 5.
    Bitsadze A. V., To the Problem of Equations of Mixed Type, Dis. Dokt. Fiz.-Mat. Nauk, Moscow (1951).Google Scholar
  6. 6.
    Volkodavov V. F. and Lerner M. E., “To the question on uniqueness of a solution to the Gellerstedt problem,” Differentsial'nye Uravneniya(Trudy Ped. Inst. RSFSR),Ryazan', 1975, No. 6, pp. 55-56.Google Scholar
  7. 7.
    Khe Kan Cher, “On the Gellerstedt problem,” Trudy Sem. S. L. Soboleva (Novosibirsk), No. 2, 139-145 (1976).Google Scholar
  8. 8.
    Khe Kan Cher, “On the Gellerstedt problem for a mixed-type equation,” Dinamika Sploshn. Sredy (Novosibirsk), 26, 134-141 (1976).Google Scholar
  9. 9.
    Khe Kan Cher, “On uniqueness of a solution to the Gellerstedt problem for a mixed-type equation,” Sibirsk. Mat. Zh., 18, No. 6, 1426-1429 (1977).Google Scholar
  10. 10.
    Smirnov M. M., Equations of Mixed Type [in Russian], Nauka, Moscow (1985).Google Scholar
  11. 11.
    Vragov V. N., “To the theory of boundary value problems for mixed-type equations on the plane and in space,” Dis. Dokt. Fiz.-Mat. Nauk, Novosibirsk (1978).Google Scholar
  12. 12.
    Moiseev E. I., “Application of the separation of variables for the solution of equations of mixed type,” Differentsial0nye Uravneniya, 26, No. 7, 1160-1172 (1990).Google Scholar
  13. 13.
    Sabitov K. B., “Construction in explicit form of the solutions of the Darboux problem for the telegraph equation and their application to the treatment of integral equations. I,” Differentsial0nye Uravneniya, 26, No. 6, 1023-1032 (1990).Google Scholar
  14. 14.
    Prudnikov A. P., Brychkov Yu. A., and Marichev O. I., Integrals and Series. Elementary Functions [in Russian], Nauka, Moscow (1983).Google Scholar
  15. 15.
    Watson G. N., A Treatise on the Property of Bessel Functions. Vol. 1 [Russian translation], Izdat. Inostr. Lit., Moscow (1949).Google Scholar
  16. 16.
    Sabitov K. B. and Tikhomirov V. V., “On construction of eigenvalues and functions of a Frankl'gas dynamical problem,” Mat. Modeling, 2, No. 10, 100-109 (1990).Google Scholar
  17. 17.
    Moiseev E. I., “On the basis property of a system of sines,” Differentsial0nye Uravneniya, 23, No. 1, 177-179 (1987).Google Scholar
  18. 18.
    Whittaker E. T. and Watson G. N., A Course of Modern Analysis. Vol. 2 [Russian translation], Fizmatgiz, Moscow (1963).Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • K. B. Sabitov
    • 1
  • A. N. Kuchkarova
    • 1
  1. 1.Sterlitamak

Personalised recommendations