Numerical Algorithms

, Volume 63, Issue 1, pp 27–48 | Cite as

Numerical computation of eigenvalues of discontinuous Sturm–Liouville problems with parameter dependent boundary conditions using sinc method

Original Paper


In this paper, we consider a Sturm–Liouville problem which contains an eigenparameter appearing linearly in two boundary conditions, in addition to an internal point of discontinuity. Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. We apply the sinc method, which is based on the sampling theory to compute approximations of the eigenvalues. An error analysis is exhibited involving rigorous error bounds. Using computable error bounds we obtain eigenvalue enclosures in a simple way. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.


Sturm–Liouville problems Discontinuity conditions Sinc methods Error analysis 

Mathematics Subject Classifications (2010)

34L16 94A20 65L15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Annaby, M.H., Asharabi, R.M.: Approximating eigenvalues of discontinuous problems by sampling theorems. J. Numer. Math. 16, 163–183 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Annaby, M.H., Asharabi, R.M.: On sinc-based method in computing eigenvalues of boundary-value problems. SIAM J. Numer. Anal. 46, 671–690 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Annaby, M.H., Tharwat, M.M.: On sampling theory and eigenvalue problems with an eigenparameter in the boundary conditions. SUT J. Math. 42, 157–176 (2006)MathSciNetMATHGoogle Scholar
  4. 4.
    Annaby, M.H., Tharwat, M.M.: On sampling and Dirac systems with eigenparameter in the boundary conditions. J. Appl. Math. Comput. 36(1–2), 291–317 (2011)MathSciNetGoogle Scholar
  5. 5.
    Annaby, M.H., Tharwat, M.M.: On computing eigenvalues of second-order linear pencils. IMA J. Numer. Anal. 27, 366–380 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Annaby, M.H., Tharwat, M.M.: Sinc-based computations of eigenvalues of Dirac systems. BIT 47, 699–713 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Annaby, M.H., Tharwat, M.M.: On the computation of the eigenvalues of Dirac systems. Calcolo (2011). doi:10.1007/s10092-011-0052-y
  8. 8.
    Binding, P.A., Browne, P.J., Watson, B.A.: Strum-Liouville problems with boundary conditions rationally dependent on the eigenparameter II. J. Comput. Appl. Math. 148, 147–169 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Boumenir, A.: The sampling method for Sturm–Liouville problems with the eigenvalue parameter in the boundary condition. Numer. Funct. Anal. Optim. 21 67–75 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Boumenir, A.: Higher approximation of eigenvalues by the sampling method. BIT 40, 215–225 (2000)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Butzer, P.L., Schmeisser, G., Stens, R.L.: An introduction to sampling analysis. In: Marvasti, F. (ed.) Non Uniform Sampling: Theory and Practice, pp. 17–121. Kluwer, New York (2001)Google Scholar
  12. 12.
    Butzer, P.L., Splettstösser, W., Stens, R.L.: The sampling theorem and linear prediction in signal analysis. Jahresber. Dtsch. Math.-Ver. 90, 1–70 (1988)MATHGoogle Scholar
  13. 13.
    Chadan, K., Sabatier, P.C.: Inverse Problems in Quantum Scattering Theory, 2nd edn. Springer (1989)Google Scholar
  14. 14.
    Chanane, B.: Computation of the eigenvalues of Sturm–Liouville problems with parameter dependent boundary conditions using the regularized sampling method. Math. Comput. 74, 1793–1801 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Chanane, B.: Eigenvalues of Sturm Liouville problems with discontinuity conditions inside a finite interval. Appl. Math. Comput. 188, 1725–1732 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Eastham, M.S.P.: Theory of Ordinary Differential Equations. Van Nostrand Reinhold, London (1970)MATHGoogle Scholar
  17. 17.
    Fulton, C.T.: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinb. 77A, 293–308 (1977)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hinton, D.B.: An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition. Q. J. Math. 30, 33–42 (1979)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jagerman, D.: Bounds for truncation error of the sampling expansion. SIAM. J. Appl. Math. 14, 714–723 (1966)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Likov, A.V., Mikhailov, Yu.A.: The Theory of Heat and Mass Transfer. Qosenergaizdat (Russian) (1963)Google Scholar
  21. 21.
    Linden, H.: Leighton’s bounds for Sturm–Liouville eigenvalues with eigenvalue parameter in the boundary conditions. J. Math. Anal. Appl. 156, 444–456 (1991)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lund, J., Bowers, K.: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia, PA (1992)Google Scholar
  23. 23.
    Mukhtarov, O.Sh., Kadakal, M., Altinisik, N.: Eigenvalues and eigenfunctions of discontinuous Sturm–Liouville problems with eigenparameter in the boundary conditions. Indian J. Pure Appl. Math. 34, 501–516 (2003)MathSciNetMATHGoogle Scholar
  24. 24.
    Kandemir, M., Mukhtarov, O.Sh.: Discontinuous Sturm–Liouville problems containing eigenparameter in the boundary conditions. Acta Math. Sinica 34, 1519–1528 (2006)Google Scholar
  25. 25.
    Pruess, S., Fulton, C.T.: Mathematical software for Sturm–Liouville problems. ACM Trans. Math. Softw. 19, 360–376 (1993)CrossRefMATHGoogle Scholar
  26. 26.
    Pruess, S., Fulton, C.T., Xie, Y.: An asymptotic numerical method for a class of singular Sturm–Liouville problems. SIAM J. Numer. Anal. 32, 1658–1676 (1995)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Pryce, J.D.: Numerical Solution of Sturm–Liouville Problems. OUP (1993)Google Scholar
  28. 28.
    Shkalikov, A.A.: Boundary value problems for ordinary diffierential equations with a parameter in boundary conditions. Trydy Sem. Imeny I. G. Petrowsgo 9, 190–229 (Russian) (1983)MathSciNetMATHGoogle Scholar
  29. 29.
    Stenger, F.: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Rev. 23, 165–224 (1981)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York (1993)CrossRefMATHGoogle Scholar
  31. 31.
    Tharwat, M.M.: Discontinuous Sturm–Liouville problems and associated sampling theories. Abstr. Appl. Anal. 2011, 30 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics. Pergamon, Oxford and New York (1963)MATHGoogle Scholar
  33. 33.
    Walter, J.: Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 133, 301–312 (1973)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • M. M. Tharwat
    • 1
    • 2
  • A. H. Bhrawy
    • 1
    • 2
  • Ahmet Yildirim
    • 3
  1. 1.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceBeni-Suef UniversityBeni-SuefEgypt
  3. 3.Urla-IzmirTurkey

Personalised recommendations