Approximation of Sturm-Liouville Problems by Exponentially Weighted Legendre-Gauss Tau Method

Authors: Mohamed K. El Daou

Abstract:

We construct an exponentially weighted Legendre- Gauss Tau method for solving differential equations with oscillatory solutions. The proposed method is applied to Sturm-Liouville problems. Numerical examples illustrating the efficiency and the high accuracy of our results are presented.

Keywords: Oscillatory functions, Sturm-Liouville problems, legendre polynomial, gauss points.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1055799

References:

[1] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Series in Appl. Math., SIAM, Philadelphia, Pa., 1977
[2] L. Gr. Ixaru, Numerical Methods for Differential Equations and Applications, D.Reidel Publishing Co, Dordrecht, 1984.
[3] L. Gr. Ixaru, CP methods for Schr¨odinger equation, J. Comput. Appl. Math. 125(2000) 347-357.
[4] L. Gr. Ixaru and G. Vanden Berghe, Exponential Fitting, Kluwer Academic Publishers, Dordrecht, 2004
[5] J. D. Lambert, Computational Methods in Ordinary Differential Equations, Jhon Wiley and Sons, London, 1973.
[6] C. Lanczos, Applied Analysis, Prentice-Hall, Englewood Cliffs, New Jersy, 1956.
[7] V. Ledoux, M. Rizea, L. Ixaru, G. Vanden Berghe, M. Van Daele, Solution of the Schr¨odinger equation by a high order perturbation method based on linear reference potential, Comput. Phys. Commun. 175:424-439 2006
[8] E .L. Ortiz, The Tau Method, , SIAM J. Numer. Anal. 6(1969) 480-492.
[9] S. Pruess, Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equations, SIAM J. Numer. Anal. 10:55-68 (1973) .
[10] J. D. Pryce, Numerical Solution of Sturm-Liouville Problems, Clarendon Press, Oxford Science Publications, 1993.
[11] T.E. Simos, An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions, Comput. Phys. Comm., 115:1-8 (1998)