## Abstract and Applied Analysis

### Computing Eigenvalues of Discontinuous Sturm-Liouville Problems with Eigenparameter in All Boundary Conditions Using Hermite Approximation

#### Abstract

The eigenvalues of discontinuous Sturm-Liouville problems which contain an eigenparameter appearing linearly in two boundary conditions and an internal point of discontinuity are computed using the derivative sampling theorem and Hermite interpolations methods. We use recently derived estimates for the truncation and amplitude errors to investigate the error analysis of the proposed methods for computing the eigenvalues of discontinuous Sturm-Liouville problems. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented. Moreover, it is shown that the proposed methods are significantly more accurate than those based on the classical sinc method.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 498457, 14 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512016

Digital Object Identifier
doi:10.1155/2013/498457

Mathematical Reviews number (MathSciNet)
MR3095348

Zentralblatt MATH identifier
1291.65236

#### Citation

Tharwat, M. M.; Bhrawy, A. H.; Alofi, A. S. Computing Eigenvalues of Discontinuous Sturm-Liouville Problems with Eigenparameter in All Boundary Conditions Using Hermite Approximation. Abstr. Appl. Anal. 2013 (2013), Article ID 498457, 14 pages. doi:10.1155/2013/498457. https://projecteuclid.org/euclid.aaa/1393512016

#### References

• E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differ-ential equations,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1820–1832, 2011.
• E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “A Jacobi dual-Petrov-Galerkin method for solving some odd-order ordinary differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 947230, 21 pages, 2011.
• E. Tohidi, A. H. Bhrawy, and K. Erfani, “A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation,” Applied Mathematical Modelling, vol. 37, no. 6, pp. 4283–4294, 2013.
• A. H. Bhrawy, A. S. Alofi, and S. I. El-Soubhy, “An extension of the Legendre-Galerkin method for solving sixth-order differ-ential equations with variable polynomial coefficients,” Mathematical Problems in Engineering, vol. 2012, Article ID 896575, 13 pages, 2012.
• A. H. Bhrawy, “A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients,” Applied Mathematics and Computation, vol. 222, pp. 255–264, 2013.
• A. H. Bhrawy, M. M. Tharwat, and A. Al-Fhaid, “Numerical algorithms for computing eigenvalues of discontinuous Dirac system using sinc-Gaussian method,” Abstract and Applied Ana-lysis, vol. 2012, Article ID 925134, 13 pages, 2012.
• A. Imani, A. Aminataei, and A. Imani, “Collocation method via Jacobi polynomials for solving nonlinear ordinary differential equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 673085, 11 pages, 2011.
• A. Boumenir and B. Chanane, “Eigenvalues of Sturm-Liouville systems using sampling theory,” Applied Analysis, vol. 62, pp. 323–334, 1996.
• B. Chanane, “Computation of the eigenvalues of Sturm-Liouville problems with parameter dependent boundary conditions using the regularized sampling method,” Mathematics of Computation, vol. 74, no. 252, pp. 1793–1801, 2005.
• B. Chanane, “Computing the spectrum of non-self-adjoint Sturm-Liouville problems with parameter-dependent boundary conditions,” Journal of Computational and Applied Mathematics, vol. 206, no. 1, pp. 229–237, 2007.
• B. Chanane, “Computing the eigenvalues of singular Sturm-Liouville problems using the regularized sampling method,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 972–978, 2007.
• B. Chanane, “Eigenvalues of Sturm-Liouville problems with dis-continuity conditions inside a finite interval,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1725–1732, 2007.
• B. Chanane, “Sturm-Liouville problems with impulse effects,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 610–626, 2007.
• M. M. Tharwat, “Discontinuous Sturm-Liouville problems and associated sampling theories,” Abstract and Applied Analysis, vol. 2011, Article ID 610232, 30 pages, 2011.
• M. M. Tharwat, A. H. Bhrawy, and A. Yildirim, “Numerical computation of eigenvalues of discontinuous Sturm-Liouville problems with parameter dependent boundary conditions using sinc method,” Numerical Algorithms, vol. 63, no. 1, pp. 27–48, 2013.
• V. Kotelnikov, “On the carrying capacity of the “ether” and wire in telecommunications,” in Proceedings of the 1st all union con-ference on questions of communications, Izd. Red. Upr. Svyazi RKKA, Moscow, Russia, 1933.
• C. E. Shannon, “Communication in the presence of noise,” Pro-ceedings of the IEEE, vol. 37, pp. 10–21, 1949.
• E. Whittaker, “On the functions which are represented by the expansion of the interpolation theory,” Proceedings of the Royal Society of Edinburgh A, vol. 35, pp. 181–194, 1915.
• G. R. Grozev and Q. I. Rahman, “Reconstruction of entire func-tions from irregularly spaced sample points,” Canadian Journal of Mathematics, vol. 48, no. 4, pp. 777–793, 1996.
• J. R. Higgins, G. Schmeisser, and J. J. Voss, “The sampling theo-rem and several equivalent results in analysis,” Journal of Com-putational Analysis and Applications, vol. 2, no. 4, pp. 333–371, 2000.
• G. Hinsen, “Irregular sampling of bandlimited ${L}^{p}$-functions,” Journal of Approximation Theory, vol. 72, no. 3, pp. 346–364, 1993.
• D. Jagerman and L. Fogel, “Some general aspects of the sampling theorem,” IRE Transactions on Information Theory, vol. 2, pp. 139–146, 1956.
• M. H. Annaby and R. M. Asharabi, “Error analysis associated with uniform Hermite interpolations of bandlimited functions,” Journal of the Korean Mathematical Society, vol. 47, no. 6, pp. 1299–1316, 2010.
• J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford University Press, Oxford, UK, 1996.
• P. L. Butzer, J. R. Higgins, and R. L. Stens, “Sampling theory of signal analysis,” in Development of Mathematics 1950–2000, pp. 193–234, Birkhäuser, Basel, Switzerland, 2000.
• P. L. Butzer, G. Schmeisser, and R. L. Stens, “An introduction to sampling analysis,” in Nonuniform Sampling, F. Marvasti, Ed., pp. 17–121, Kluwer, New York, NY, USA, 2001.
• A. Boumenir, “Higher approximation of eigenvalues by the sampling method,” BIT, vol. 40, no. 2, pp. 215–225, 2000.
• M. M. Tharwat, A. H. Bhrawy, and A. Yildirim, “Numerical computation of the eigenvalues of a discontinuous Dirac system using the sinc method with error analysis,” International Journal of Computer Mathematics, vol. 89, no. 15, pp. 2061–2080, 2012.
• J. Lund and K. L. Bowers, Sinc Methods for Quadrature and Dif-ferential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1992.
• F. Stenger, “Numerical methods based on Whittaker cardinal, or sinc functions,” SIAM Review, vol. 23, no. 2, pp. 165–224, 1981.
• F. Stenger, Numerical Methods Based on Sinc and Analytic Func-tions, vol. 20, Springer, New York, NY, USA, 1993.
• P. L. Butzer, W. Splettstösser, and R. L. Stens, “The sampling theorem and linear prediction in signal analysis,” Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 90, no. 1, p. 70, 1988.
• D. Jagerman, “Bounds for truncation error of the sampling expansion,” SIAM Journal on Applied Mathematics, vol. 14, pp. 714–723, 1966.
• M. M. Tharwat and A. H. Bhrawy, “Computation of eigenvalues of discontinuous Dirac system using Hermite interpolation technique,” Advances in Difference Equations, vol. 2012, article 59, 2012.
• M. M. Tharwat, A. H. Bhrawy, and A. S. Alofi, “Approximation of eigenvalues of discontinuous Sturm-Liouville problems with eigenparameter in all boundary conditions,” Boundary Value Problems, vol. 2013, article 132, 2013.
• M. Kadakal and O. S. Mukhtarov, “Discontinuous Sturm-Liouville problems containing eigenparameter in the boundary conditions,” Acta Mathematica Sinica, vol. 22, no. 5, pp. 1519–1528, 2006.
• O. S. Mukhtarov, M. Kadakal, and N. Altinisik, “Eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problems with eigenparameter in the boundary conditions,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 3, pp. 501–516, 2003.
• M. M. Tharwat, A. Yildirim, and A. H. Bhrawy, “Sampling of discontinuous Dirac systems,” Numerical Functional Analysis and Optimization, vol. 34, no. 3, pp. 323–348, 2013.