Advertisement

On the Boundary Value Problem for Discontinuous Sturm–Liouville Operator

  • Ozge Akcay
Article
  • 21 Downloads

Abstract

In this paper, we study the Sturm–Liouville problem which has not only the discontinuous coefficient, but also the discontinuity condition at an interior point of a finite interval. The new integral representation for the solution of the discontinuous Sturm–Liouville equation is constructed and the properties of the its kernel function are given. The asymptotic formulas of the eigenvalues and eigenfunctions of this problem are examined. The completeness theorem and expansion theorem of eigenfunctions are proved.

Keywords

Sturm–Liouville problem discontinuous coefficient and discontinuity condition integral representation asymptotic formulas of eigenvalues and eigenfunctions completeness theorem 

Mathematics Subject Classification

34B24 34K10 47E05 

Notes

References

  1. 1.
    Akhmedova, E.N., Huseynov, H.M.: On eigenvalues and eigenfunctions of one class of Sturm–Liouville operators with discontinuous coefficient. Trans. NAS Azerb. 23, 7–18 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Amirov, R.K.: On Sturm–Liouville operators with discontinuity conditions inside an interval. J. Math. Anal. Appl. 317, 163–176 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anderssen, R.S.: The effect of discontinuous in density and shear velocity on the asymptotic overtone structure of torsional eigenfrequencies of the earth. Geophys. J. R. Astron. Soc. 50, 303–309 (1997)CrossRefGoogle Scholar
  4. 4.
    Freiling, G., Yurko, V.A.: Inverse Sturm–Liouville Problems and Their Applications. Nova Science Publishers Inc, Huntington (2001)zbMATHGoogle Scholar
  5. 5.
    Gomilko, A., Pivovarchik, V.: On basis properties of a part of eigenfunctions of the problem of vibrations of a smooth inhomogeneous string damped at the midpoint. Math. Nachr. 245, 72–93 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hald, O.H.: Discontinuous inverse eigenvalue problems. Commun. Pure Appl. Math. 37, 539–577 (1984)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kruger, R.J.: Inverse problems for nonabsorbing media with discontinuous material properties. J. Math. Phys. 23, 396–404 (1982)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lapwood, F.R., Usami, T.: Free Oscillations of the Earth. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar
  9. 9.
    Levitan, B.M., Sargsjan, I.S.: Sturm–Liouville and Dirac Operators. Kluwer Academic Publishers, Dordrecht (1991)CrossRefGoogle Scholar
  10. 10.
    Lykov, A.V., Mikhailov, Y.A.: The Theory of Heat and Mass Transfer. Qosenergaizdat, Moscow (1963). (Russian)Google Scholar
  11. 11.
    Mamedov, K.R., Cetinkaya, F.A.: Inverse problem for a class of Sturm–Liouville operator with spectral parameter in boundary condition. Bound. Value Probl. 183, 16 (2013).  https://doi.org/10.1186/1687-2770-2013-183 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Marchenko, V.A.: Sturm–Liouville Operators and Their Applications. AMS Chelsea Publishing, Providence (2011)CrossRefGoogle Scholar
  13. 13.
    Mochizuki, K., Trooshin, I.: Inverse problem for interior spectral data of Sturm–Liouville operator. J. Inverse Ill Posed Probl. 9, 425–433 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mukhtarov, O.S., Aydemir, K.: Eigenfunction expansion for Sturm–Liouville problems with transmission conditions at one interior point. Acta Math. Sci. 35, 639–649 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nabiev, A.A., Amirov, R.K.: On the boundary value problem for the Sturm–Liouville equation with the discontinuous coefficient. Math. Methods Appl. Sci. 36, 1685–1700 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Olgar H., Muhtarov F.S.: The basis property of the system of weak eigenfunctions of a discontinuous Sturm–Liouville problem. Mediterr. J. Math. (2017).  https://doi.org/10.1007/s00009-017-0915-9
  17. 17.
    Pöschel, J., Trubowitz, E.: Inverse Spectral Theory. Academic Press, New York (1987)zbMATHGoogle Scholar
  18. 18.
    Shepelsky, D.G.: The inverse problem of reconstruction of medium’s conductivity in a class of discontinuous and increasing functions. Adv. Soviet Math. 19, 209–231 (1994)MathSciNetGoogle Scholar
  19. 19.
    Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics. Dover Books on Physics and Chemistry, Dover (1990)zbMATHGoogle Scholar
  20. 20.
    Willis, C.: Inverse problems for torsional modes. Geophys. J. R. Astron. Soc. 78, 847–853 (1984)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsMersin UniversityMersinTurkey

Personalised recommendations