Computation of the eigenvalues of Sturm-Liouville problems with parameter dependent boundary conditions using the regularized sampling method
Author:
Bilal Chanane
Journal:
Math. Comp. 74 (2005), 1793-1801
MSC (2000):
Primary 34B24, 34L15, 34B07
DOI:
https://doi.org/10.1090/S0025-5718-05-01717-5
Published electronically:
March 18, 2005
MathSciNet review:
2164097
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/S0893-9659%2899%2900111-1
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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
- Johann Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary condition, Math. Z. 133 (1973), 301–312. MR 335935, DOI https://doi.org/10.1007/BF01177870
- Ahmed I. Zayed, Advances in Shannon’s sampling theory, CRC Press, Boca Raton, FL, 1993. MR 1270907
Retrieve articles in Mathematics of Computation with MSC (2000): 34B24, 34L15, 34B07
Retrieve articles in all journals with MSC (2000): 34B24, 34L15, 34B07
Additional Information
Bilal Chanane
Affiliation:
Department of Mathematical Sciences, KFUPM, Dhahran 31261, Saudi Arabia
Email:
chanane@kfupm.edu.sa
Keywords:
Second order Sturm-Liouville problems,
eigenvalue problems,
Whittaker-Shannon-Kotel′nikov theorem,
parameter dependent boundary conditions,
regularized sampling method
Received by editor(s):
June 23, 2003
Received by editor(s) in revised form:
March 18, 2004
Published electronically:
March 18, 2005
Article copyright:
© Copyright 2005
American Mathematical Society