Advertisement

Journal of Mathematical Sciences

, Volume 213, Issue 6, pp 910–916 | Cite as

The Gellerstedt Equation with Integral Perturbation in the Cauchy Data

  • N. E. Tokmagambetov
Article
  • 23 Downloads

We establish the well-posedness of the problem for a degenerate hyperbolic equation of the first kind in the characteristic triangle with integral perturbation in the Cauchy data along the degeneracy line. Bibliography: 16 titles.

Keywords

Hyperbolic Equation Cauchy Data Nonlocal Boundary Condition Transparent Boundary Condition Weakly Singular Kernel 
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.
    A. M. Nakhushev, Problems with Shifts for Partial Differential Equations [in Russian], Nauka, Moscow (2006).Google Scholar
  2. 2.
    J. R. Cannon, “The solution of the heat equation subject to the specification of energy” Q. Appl. Math. 22, 155–160 (1964).MathSciNetMATHGoogle Scholar
  3. 3.
    L. I. Kamynin, “A boundary value problem in the theory of heat conduction with a nonclassical boundary condition” [in Russian], Zh. Vychisl. Mat. Mat. Fiz. 4, 1006–1024 (1964); English transl.: U.S.S.R. Comput. Math. Math. Phys. 4, No. 6, 33–59 (1967).Google Scholar
  4. 4.
    N. I. Ionkin, “Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition” [in Russian], Differ. Uravn. 13, No. 2, 294–304 (1977); English transl.: Differ. Equations 13, No. 2, 204–211 (1977).Google Scholar
  5. 5.
    L. A. Muravei and A. V. Filinovskii, “On the non-local boundary-value problem for a parabolic equation” [in Russian], Mat. Zametki 54, No. 4, 98–116 (1993); English transl.: Math. Notes 54, No. 4, 1045–1057 (1993).Google Scholar
  6. 6.
    A. I. Kozhanov, “On the solvability of certain spatially nonlocal boundary-value problems for linear hyperbolic equations of second order” [in Russian], Mat. Zametki 90, No. 2, 254–268 (2011); English transl.: Math. Notes 90, No. 2, 238–249 (2011).Google Scholar
  7. 7.
    A. I. Kozhanov, “On the solvability of a boundary-value problem with a nonlocal boundary condition for linear parabolic equations” [in Russian], Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki 30, 63–69 (2004).Google Scholar
  8. 8.
    A. I. Kozhanov and L. S. Pul’kina, “On the solvability of boundary value problems with a nonlocal boundary condition of integral form for multidimensional hyperbolic equations” [in Russian], Differ. Uravn. 42, No. 9, 1166–1179 (2006); English transl.: Differ. Equ. 42, No. 9, 1233–1246 (2006).Google Scholar
  9. 9.
    L. S. Pul’kina, “A mixed problem with an integral condition for a hyperbolic equation” [in Russian], Mat. Zametki 74, No. 3, 435–445 (2003); English transl.: Math. Notes 74, No. 3, 411–421 (2003).Google Scholar
  10. 10.
    L. S. Pul’kina, “Mixed problem with a nonlocal condition for a hyperbolic equation” [in Russian], In: Nonclassical Equations of Mathematical Physics, pp. 176–184, Novosibirsk (2002).Google Scholar
  11. 11.
    A. V. Bitsadze, Some Classes of Partial Differential Equations [in Russian], Nauka, Moscow (1981); English transl.: Gordon & Breach, New York (1988).Google Scholar
  12. 12.
    M. M. Smirnov, Degenerate Hyperbolic Equations [in Russian], Vysshaya Shkola, Minsk (1977).Google Scholar
  13. 13.
    A. Erd´elyi (Ed.), Higher Transcendental Functions. I, McGraw Hill, New York etc. (1953).Google Scholar
  14. 14.
    T. Sh. Kal’menov and N. E. Tokmagambetov, “On a nonlocal boundary value problem for the multidimensional heat equation in a noncylindrical domain” [in Russian], Sib. Mat. Zh. 54, No. 6, 1287–1293 (2013); English transl.: Sib. Mat. J. 54, No. 6, 1023–1028 (2013).Google Scholar
  15. 15.
    B. E. Kanguzhin, D. B. Nurakhmetov, and N. E. Tokmagambetov, “Laplace operator with δ-like potentials” [in Russian], Izv. Vyssh. Uchebn. Zaved., Mat. No. 2, 9–16 (2014); English transl.: Rus. Math. 58, No. 2, 6–12 (2014).Google Scholar
  16. 16.
    D. Suragan and N. Tokmagambetov, “On transparent boundary conditions for the highorder heat equation” Sib. Èlektron. Mat. Izv. 10, 141–149 (2013).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Mathematics and Mathematical Modeling MES RKAlmatyKazakhstan

Personalised recommendations