## Computation of the eigenvalues of Sturm-Liouville problems with parameter dependent boundary conditions using the regularized sampling method

HTML articles powered by AMS MathViewer

- by Bilal Chanane PDF
- Math. Comp.
**74**(2005), 1793-1801 Request permission

## Abstract:

The purpose in this paper is to compute the eigenvalues of Sturm-Liouville problems with quite general separated boundary conditions nonlinear in the eigenvalue parameter using the*regularized sampling method*, an improvement on the method based on Shannon sampling theory, which does not involve any multiple integration and provides higher order estimates of the eigenvalues at a very low cost. A few examples shall be presented to illustrate the power of the method and a comparison made with the the exact eigenvalues obtained as squares of the zeros of the exact characteristic functions.

## References

- P. A. Binding, P. J. Browne, and K. Seddighi,
*Sturm-Liouville problems with eigenparameter dependent boundary conditions*, Proc. Edinburgh Math. Soc. (2)**37**(1994), no. 1, 57–72. MR**1258031**, DOI 10.1017/S0013091500018691 - P. A. Binding and Patrick J. Browne,
*Oscillation theory for indefinite Sturm-Liouville problems with eigenparameter-dependent boundary conditions*, Proc. Roy. Soc. Edinburgh Sect. A**127**(1997), no. 6, 1123–1136. MR**1489428**, DOI 10.1017/S0308210500026974 - Amin Boumenir,
*The sampling method for Sturm-Liouville problems with the eigenvalue parameter in the boundary condition*, Proceedings of the International Conference on Fourier Analysis and Applications (Kuwait, 1998), 2000, pp. 67–75. MR**1759988**, DOI 10.1080/01630560008816940 - A. Boumenir, B. Chanane,
*Eigenvalues of Sturm-Liouville systems using sampling theory*, Appl. Anal., 62, (1996), 323–334. - A. Boumenir and B. Chanane,
*Computing eigenvalues of Sturm-Liouville systems of Bessel type*, Proc. Edinburgh Math. Soc. (2)**42**(1999), no. 2, 257–265. MR**1697397**, DOI 10.1017/S001309150002023X - A. Boumenir and B. Chanane,
*The computation of negative eigenvalues of singular Sturm-Liouville problems*, IMA J. Numer. Anal.**21**(2001), no. 2, 489–501. MR**1825833**, DOI 10.1093/imanum/21.2.489 - B. Chanane,
*Computing eigenvalues of regular Sturm-Liouville problems*, Appl. Math. Lett.**12**(1999), no. 7, 119–125. MR**1750070**, DOI 10.1016/S0893-9659(99)00111-1 - B. Chanane,
*High order approximations of the eigenvalues of regular Sturm-Liouville problems*, J. Math. Anal. Appl.**226**(1998), no. 1, 121–129. MR**1646473**, DOI 10.1006/jmaa.1998.6049 - B. Chanane,
*The Paley-Wiener-Levinson theorem and the computation of Sturm-Liouville eigenvalues: irregular sampling*, Appl. Anal.**75**(2000), no. 3-4, 261–266. MR**1801687**, DOI 10.1080/00036810008840847 - Bilal Chanane,
*High order approximations of the eigenvalues of Sturm-Liouville problems with coupled self-adjoint boundary conditions*, Appl. Anal.**80**(2001), no. 3-4, 317–330. MR**1914684**, DOI 10.1080/00036810108840995 - B. Chanane,
*On a class of random Sturm-Liouville problems*, Int. J. Appl. Math.**8**(2002), no. 2, 171–182. MR**1869540** - B. Chanane,
*Approximation of the eigenvalues of regular fourth order Sturm-Liouville problems using interpolation theory*, Approximation theory, X (St. Louis, MO, 2001) Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2002, pp. 155–166. MR**1924880** - Charles T. Fulton,
*Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions*, Proc. Roy. Soc. Edinburgh Sect. A**77**(1977), no. 3-4, 293–308. MR**593172**, DOI 10.1017/S030821050002521X - Don Hinton and Philip W. Schaefer (eds.),
*Spectral theory and computational methods of Sturm-Liouville problems*, Lecture Notes in Pure and Applied Mathematics, vol. 191, Marcel Dekker, Inc., New York, 1997. MR**1460546** - M. A. Naĭmark,
*Linear differential operators. Part II: Linear differential operators in Hilbert space*, Frederick Ungar Publishing Co., New York, 1968. With additional material by the author, and a supplement by V. È. Ljance; Translated from the Russian by E. R. Dawson; English translation edited by W. N. Everitt. MR**0262880** - John D. Pryce,
*Numerical solution of Sturm-Liouville problems*, Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR**1283388** - Frank Stenger,
*Numerical methods based on sinc and analytic functions*, Springer Series in Computational Mathematics, vol. 20, Springer-Verlag, New York, 1993. MR**1226236**, DOI 10.1007/978-1-4612-2706-9 - Johann Walter,
*Regular eigenvalue problems with eigenvalue parameter in the boundary condition*, Math. Z.**133**(1973), 301–312. MR**335935**, DOI 10.1007/BF01177870 - Ahmed I. Zayed,
*Advances in Shannon’s sampling theory*, CRC Press, Boca Raton, FL, 1993. MR**1270907**

## Additional Information

**Bilal Chanane**- Affiliation: Department of Mathematical Sciences, KFUPM, Dhahran 31261, Saudi Arabia
- Email: chanane@kfupm.edu.sa
- Received by editor(s): June 23, 2003
- Received by editor(s) in revised form: March 18, 2004
- Published electronically: March 18, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Math. Comp.
**74**(2005), 1793-1801 - MSC (2000): Primary 34B24, 34L15, 34B07
- DOI: https://doi.org/10.1090/S0025-5718-05-01717-5
- MathSciNet review: 2164097