Computing High-Index Eigenvalues of Singular Sturm–Liouville Problems

  • Anis Haytham Saleh TaherEmail author
Original Paper


This paper deals with the computation of the high-index eigenvalues of singular Sturm–Liouville problems using the Chebyshev spectral collocation method. The singular Sturm–Liouville problem is transformed into generalized eigenvalue problem by using the spectral differentiation matrices to compute derivatives of Chebyshev polynomials at Chebyshev Gauss–Lobatto nodes. A few different examples shall be solved numerically to demonstrate reliability and efficiency of the proposed technique.


Singular Sturm–Liouville problems Chebyshev differentiation matrix Spectral collocation method High-index eigenvalues 



  1. 1.
    Pryce, J.D.: Numerical Solution of Sturm–Liouville Problems. Oxford University Press, New York (1993)zbMATHGoogle Scholar
  2. 2.
    Zettl, A.: Sturm–Liouville Theory. American Mathematical Society, Providence (2005)zbMATHGoogle Scholar
  3. 3.
    Zettl, A.: Sturm–Liouville problems. In: Hinton, D., Schaefer, P.W. (eds.) Spectral Theory and Computational Methods of Sturm–Liouville Problems, Lecture Notes in Pure and Appl. Math., vol. 191, pp. 1–104. Marcel Dekker, New York (1997)Google Scholar
  4. 4.
    Bailey, P.B., Everitt, W.N., Zettl, A.: Computing eigenvalues of singular Sturm–Liouville problems. Results Math. 20, 391–423 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Lutgen, J.P.: Eigenvalue accumulation for singular Sturm–Liouville problems nonlinear in the spectral paramete. J. Differ. Equ. 159, 515–542 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Homer, M.S.: Boundary value problems for the Laplace tidal wave equation. Proc. R. Soc. Lond. A 428, 157–180 (1990)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Patra, A., Srivastava, P.D.: Relative perturbation bounds for matrix eigenvalues and singular values. Int. J. Appl. Comput. Math. 4, 138 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Banks, D., Kurowski, G.: Computation of eigenvalues of singular Sturm–Liouville systems. Math. Comput. 22, 304–310 (1968)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bas, E., Metin, F.: Spectral analysis for fractional hydrogen atom equation. Adv. Pure Math. 5, 767–773 (2015)CrossRefGoogle Scholar
  10. 10.
    Ledoux, V., Daele, M.V.: Solution of Sturm–Liouville problems using modified Neumann schemes. SIAM J. Sci. Comput. 32, 564–584 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Aydemir, K., Mukhtarov, O.: Asymptotic distribution of eigenvalues and eigenfunctions for a multi-point discontinuous Sturm–Liouville problem. Electron. J. Differ. Equ. 2016, 1–14 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Paine, J., de Hoog, F.: Uniform estimation of the eigenvalues of Sturm–Liouville problems. J. Aust. Math. Soc. Ser. B 21, 365–383 (1980)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Paine, J.: Correction of Sturm–Liouville eigenvalue estimates. Math. Comput. 39, 415–420 (1982)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Andrew, A., Paine, J.: Correction of finite element estimates for Sturm–Liouville eigenvalues. Numer. Math. 50, 205–215 (1982)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Taher, A.H.S., Malek, A.: A new algorithm for solving sixth-order Sturm–Liouville problems. Int. J. Appl. Math. 24, 631–639 (2011)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Taher, A.H.S., Malek, A.: An efficient algorithm for solving high-order Sturm–Liouville problems using variational iteration method. Fixed Point Theory 14, 193–210 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Taher, A.H.S., Malek, A., Thabet, A.S.A.: Semi-analytical approximation for solving high-order Sturm–Liouville problems. Br. J. Math. Comput. Sci. 23, 3345–3357 (2014)CrossRefGoogle Scholar
  18. 18.
    Chanane, B.: Computing the eigenvalues of singular Sturm–Liouville problems using the regularized sampling method. Appl. Math. Comput. 184, 972–978 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Singh, R., Kumar, J.: Computation of eigenvalues of singular Sturm–Liouville problems using modified Adomian decomposition method. Int. J. Nonlinear Sci. 15, 247–258 (2013)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Makarov, V.L., Dragunov, D.V., Klimenko, Y.V.: The FD-method for solving Sturm–Liouville problems with special singular differential operator. Math. Comput. 82, 953–973 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ledoux, V., Daele, M.V., Berghe, G.V.: Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics. Comput. Phys. Commun. 180, 241–250 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Dehghan, M.: An efficient method to approximate eigenfunctions and high-index eigenvalues of regular Sturm–Liouville problems. Appl. Math. Comput. 279, 249–257 (2016)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Bailey, P.B., Gordon, M., Shampine, L.: Automatic solution for Sturm–Liouville problems. ACM Trans. Math. Softw. 4, 193–208 (1978)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Fulton, C., Pruess, S.: Mathematical software for Sturm–Liouville problems. ACM Trans. Math. Softw. 19, 360–376 (1993)CrossRefGoogle Scholar
  25. 25.
    Marletta, M., Pryce, J.: A new multipurpose software package for Schrödinger and Sturm–Liouville computations. Comput. Phys. Commun. 62, 42–52 (1991)CrossRefGoogle Scholar
  26. 26.
    Ledoux, V., Daele, M.V., Berghe, G.V.: MATSLISE: a software package for the numerical solution of Sturm–Liouville and Schrödinger problems. ACM Trans. Math. Softw. 31, 532–554 (2005)CrossRefGoogle Scholar
  27. 27.
    Bildik, N., Deniz, S.: Applications of Taylor collocation method and Lambert W function to the systems of delay differential equations. Turk. J. Math. Comput. Sci. 1, 1–13 (2013)Google Scholar
  28. 28.
    Deniz, S., Bildik, N., Sezer, M.: A note on stability analysis of Taylor collocation method. CBU J. Sci. 13, 149–153 (2017)Google Scholar
  29. 29.
    Bildik, N., Deniz, S.: New analytic approximate solutions to the generalized regularized long wave equations. Bull. Korean Math. Soc. 55, 749–762 (2018)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Taher, A.H.S., Malek, A., Momeni-Masuleh, S.H.: Chebyshev differentiation matrices for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems. Appl. Math. Model. 37, 4634–4642 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Fluid Dynamics. Springer, New York (1988)CrossRefGoogle Scholar
  32. 32.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer, Berlin (2007)zbMATHGoogle Scholar
  33. 33.
    Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  34. 34.
    Weideman, J.A.C., Reddy, S.C.: A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26, 465–519 (2000)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Groves, G.V.: Notes on obtaining the eigenvalues of Laplace’s tidal equation. Planet. Space Sci. 29, 1339–1344 (1981)CrossRefGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AdenKhormaksar, AdenYemen

Personalised recommendations