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Computing eigenvalues of Sturm–Liouville problems by Hermite interpolations

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Abstract

We establish a new method to compute the eigenvalues of Sturm–Liouville problems by the use of Hermite interpolations at equidistant nodes. We rigorously give estimates for the error by considering both truncation and amplitude errors. We compare the results of the new technique with those involving the classical sinc method as well as a SLEIGN2-based method. We also introduce curves that illustrate the enclosure intervals.

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References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Annaby, M.H., Asharabi, R.M.: Approximating eigenvalues of discontinuous problems by sampling theorems J. Numer. Math. 16, 163–183 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Annaby, M.H., Asharabi, R.M.: Error analysis associated with uniform Hermite interpolations of bandlimited functions. J. Korean Math. Soc. 47, 1299–1316 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Annaby, M.H., Tharwat, M.M.: On computing eigenvalues of second-order linear pencils. IMA J. Numer. Anal. 27, 366–380 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bailey, P.B., Everitt, W.N., Zettl, A.: The SLEIGN2 Sturm–Liouville code. ACM Trans. Math. Softw. 21, 143–192 (2001)

    Article  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boumenir, A.: Higher approximation of eigenvalues by sampling. BIT 40, 215–225 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Boumenir, A., Chanane, B.: Eigenvalues of Sturm–Liouville systems using sampling theory. Appl. Anal. 62, 323–334 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chadan, K., Sabatier, P.C.: Inverse Problems in Quantum Scattering Theory. Springer-Verlag (1989)

  10. Chanane, B.: Computing eigenvalues of regular Sturm–Liouville problems. Appl. Math. Lett. 12, 119–125 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chanane, B.: High order approximations of the eigenvalues of regular Sturm–Liouville problems. J. Math. Anal. Appl. 226, 121–129 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eastham, M.S.P.: Theory of Ordinary Differential Operators. Van Nostrand Reinhold, London (1970)

    Google Scholar 

  13. Eggert, N., Jarratt, M., Lund, J.: Sinc function computation of the eigenvalues of Sturm–Liouville problems. J. Comput. Phys. 69, 209–229 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grozev, G.R., Rahman, Q.I.: Reconstruction of entire functions from irregularly spaced sample points. Can. J. Math. 48, 777–793 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Higgins, J.R., Schmeisser, G., Voss, J.J.: The sampling theorem and several equivalent results in analysis. J. Comput. Anal. Appl. 2, 333–371 (2000)

    MathSciNet  MATH  Google Scholar 

  16. Hinsen, G.: Irregular sampling of bandlimited L p-functions. J. Approx. Theory 72, 346–364 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jagerman, D., Fogel, L.: Some general aspects of the sampling theorem. IRE Trans. Inf. Theory 2, 139–146 (1956)

    Article  Google Scholar 

  18. Levitan, B.M., Sargsjan, I.S.: Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, In: Translation of Mathematical Monographs, vol. 39. American Mathematical Society, Providence (1975)

    Google Scholar 

  19. Lund, J., Bowers, K.: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  20. Lund, J., Riley, B.: A sinc-collocation method for the computation of the eigenvalues of the radial Schrödinger equation. IMA J. Numer. Anal. 4(1), 83–98 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  21. Stenger, F.: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Rev. 23, 156–224 (1981)

    MathSciNet  Google Scholar 

  22. Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer-Verlag, New York (1993)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Physics, Qatar University, P.O. Box 2713, Doha, Qatar

    Mahmoud H. Annaby

  2. Department of Mathematics Faculty of Science, Cairo University, Giza, Egypt

    Mahmoud H. Annaby

  3. Department of Mathematics, Faculty of Science, Sana’a University, Sana, Yemen

    Rashad M. Asharabi

Authors
  1. Mahmoud H. Annaby
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  2. Rashad M. Asharabi
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Correspondence to Mahmoud H. Annaby.

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Annaby, M.H., Asharabi, R.M. Computing eigenvalues of Sturm–Liouville problems by Hermite interpolations. Numer Algor 60, 355–367 (2012). https://doi.org/10.1007/s11075-011-9518-x

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  • DOI: https://doi.org/10.1007/s11075-011-9518-x

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