About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 498457, 14 pages
http://dx.doi.org/10.1155/2013/498457
Research Article

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

1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt

Received 23 March 2013; Revised 24 July 2013; Accepted 28 July 2013

Academic Editor: Jose L. Gracia

Copyright © 2013 M. M. Tharwat et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1820–1832, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  4. A. H. Bhrawy, A. S. Alofi, and S. I. El-Soubhy, “An extension of the Legendre-Galerkin method for solving sixth-order differential equations with variable polynomial coefficients,” Mathematical Problems in Engineering, vol. 2012, Article ID 896575, 13 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  5. 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.
  6. 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 Analysis, vol. 2012, Article ID 925134, 13 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  7. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. Boumenir and B. Chanane, “Eigenvalues of Sturm-Liouville systems using sampling theory,” Applied Analysis, vol. 62, pp. 323–334, 1996.
  9. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. B. Chanane, “Eigenvalues of Sturm-Liouville problems with discontinuity conditions inside a finite interval,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1725–1732, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. B. Chanane, “Sturm-Liouville problems with impulse effects,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 610–626, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. M. Tharwat, “Discontinuous Sturm-Liouville problems and associated sampling theories,” Abstract and Applied Analysis, vol. 2011, Article ID 610232, 30 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  16. V. Kotelnikov, “On the carrying capacity of the “ether” and wire in telecommunications,,” in Proceedings of the 1st all union conference on questions of communications, Izd. Red. Upr. Svyazi RKKA, Moscow, Russia, 1933.
  17. C. E. Shannon, “Communication in the presence of noise,” Proceedings of the IEEE, vol. 37, pp. 10–21, 1949. View at MathSciNet
  18. 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.
  19. G. R. Grozev and Q. I. Rahman, “Reconstruction of entire functions from irregularly spaced sample points,” Canadian Journal of Mathematics, vol. 48, no. 4, pp. 777–793, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J. R. Higgins, G. Schmeisser, and J. J. Voss, “The sampling theorem and several equivalent results in analysis,” Journal of Computational Analysis and Applications, vol. 2, no. 4, pp. 333–371, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. G. Hinsen, “Irregular sampling of bandlimited Lp-functions,” Journal of Approximation Theory, vol. 72, no. 3, pp. 346–364, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  22. D. Jagerman and L. Fogel, “Some general aspects of the sampling theorem,” IRE Transactions on Information Theory, vol. 2, pp. 139–146, 1956.
  23. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford University Press, Oxford, UK, 1996.
  25. 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. View at Zentralblatt MATH · View at MathSciNet
  26. 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. View at MathSciNet
  27. A. Boumenir, “Higher approximation of eigenvalues by the sampling method,” BIT, vol. 40, no. 2, pp. 215–225, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J. Lund and K. L. Bowers, Sinc Methods for Quadrature and Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  30. F. Stenger, “Numerical methods based on Whittaker cardinal, or sinc functions,” SIAM Review, vol. 23, no. 2, pp. 165–224, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, vol. 20, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  32. 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. View at Zentralblatt MATH · View at MathSciNet
  33. D. Jagerman, “Bounds for truncation error of the sampling expansion,” SIAM Journal on Applied Mathematics, vol. 14, pp. 714–723, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  35. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  36. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. 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. View at Zentralblatt MATH · View at MathSciNet
  38. 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. View at Publisher · View at Google Scholar · View at MathSciNet