Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 896575, 13 pages
http://dx.doi.org/10.1155/2012/896575
Research Article

An Extension of the Legendre-Galerkin Method for Solving Sixth-Order Differential Equations with Variable Polynomial Coefficients

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
3Department of Mathematics, Faculty of Science, Taibah University, Madinah 20012, Saudi Arabia

Received 26 April 2011; Accepted 12 December 2011

Academic Editor: Alexei Mailybaev

Copyright © 2012 A. H. Bhrawy 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. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Mechanics, Springer, New York, NY, USA, 2006.
  2. C. Bernardi and Y. Maday, Approximations Spectrales de Problèmes aux Limites Elliptiques, vol. 10, Springer, Paris, France, 1992.
  3. J. Shen, “Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials,” SIAM Journal on Scientific Computing, vol. 15, no. 6, pp. 1489–1505, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. Boutayeb and E. Twizell, “Numerical methods for the solution of special sixth-order boundary value problems,” International Journal of Computer Mathematics, vol. 45, pp. 207–233, 1992. View at Google Scholar
  5. E. H. Twizell and A. Boutayeb, “Numerical methods for the solution of special and general sixth-order boundary value problems, with applications to Bénard layer eigenvalue problems,” Proceedings of the Royal Society London Series A, vol. 431, no. 1883, pp. 433–450, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. P. Baldwin, “Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global phase-integral methods,” Philosophical Transactions of the Royal Society of London Series A, vol. 322, no. 1566, pp. 281–305, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. S. S. Siddiqi and E. H. Twizell, “Spline solutions of linear sixth-order boundary-value problems,” International Journal of Computer Mathematics, vol. 60, no. 3-4, pp. 295–304, 1996. View at Google Scholar
  8. M. El-Gamel, J. R. Cannon, and A. I. Zayed, “Sinc-Galerkin method for solving linear sixth-order boundary-value problems,” Mathematics of Computation, vol. 73, no. 247, pp. 1325–1343, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific Publishing, Singapore, 1986. View at Zentralblatt MATH
  10. E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials,” Numerical Algorithms, vol. 42, no. 2, pp. 137–164, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials,” Applied Numerical Mathematics, vol. 58, no. 8, pp. 1224–1244, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. E. H. Doha and A. H. Bhrawy, “A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations,” Numerical Methods for Partial Differential Equations, vol. 25, no. 3, pp. 712–739, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. E. H. Doha, A. H. Bhrawy, and W. M. Abd-Elhameed, “Jacobi spectral Galerkin method for elliptic Neumann problems,” Numerical Algorithms, vol. 50, no. 1, pp. 67–91, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. 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
  15. 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
  16. A. H. Bhrawy and W. M. Abd-Elhameed, “New algorithm for the numerical solutions of nonlinear third-order differential equations using Jacobi-Gauss collocation method,” Mathematical Problems in Engineering, vol. 2011, Article ID 837218, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. E. H. Doha, A. H. Bhrawy, and M. A. Saker, “Integrals of Bernstein polynomials: an application for the solution of high even-order differential equations,” Applied Mathematics Letters, vol. 24, no. 4, pp. 559–565, 2011. View at Publisher · View at Google Scholar
  18. E. H. Doha, A. H. Bhrawy, and M. A. Saker, “On the derivatives of Bernstein polynomials: an application for the solution of high even-order differential equations,” Boundary Value Problems, vol. 2011, Article ID 829543, 16 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. G. B. Loghmani and M. Ahmadinia, “Numerical solution of sixth order boundary value problems with sixth degree B-spline functions,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 992–999, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. S. S. Siddiqi and G. Akram, “Septic spline solutions of sixth-order boundary value problems,” Journal of Computational and Applied Mathematics, vol. 215, no. 1, pp. 288–301, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. A. Wazwaz, “The numerical solution of sixth-order boundary value problems by the modified decomposition method,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 311–325, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. A. H. Bhrawy, “Legendre-Galerkin method for sixth-order boundary value problems,” Journal of the Egyptian Mathematical Society, vol. 17, no. 2, pp. 173–188, 2009. View at Google Scholar · View at Zentralblatt MATH
  23. E. H. Doha, “On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials,” Journal of Physics A, vol. 37, no. 3, pp. 657–675, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. A. Graham, Kronecker Products and Matrix Calculus: With Applications, Ellis Horwood Ltd., England, UK, 1981.
  25. D. Funaro, Polynomial Approximation of Differential Equations, vol. 8 of Lecture Notes in Physics, Springer, Berlin, Germany, 1992.