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International Journal of Mathematics and Mathematical Sciences
Volume 2016, Article ID 9823147, 8 pages
http://dx.doi.org/10.1155/2016/9823147
Research Article

A 4-Point Block Method for Solving Higher Order Ordinary Differential Equations Directly

1Faculty of Electronic & Computer Engineering, Universiti Teknikal Malaysia Melaka (UTeM), 76100 Melaka, Malaysia
2Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
3Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 18 April 2016; Revised 23 June 2016; Accepted 30 June 2016

Academic Editor: Harvinder S. Sidhu

Copyright © 2016 Nazreen Waeleh and Zanariah Abdul Majid. 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. M. T. Darvishi, S. Kheybari, and F. Khani, “A numerical solution of the Korteweg-de Vries equation by pseudospectral method using Darvishi's preconditionings,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 98–105, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. L. Jin, “Application of variational iteration method to the fifth-order KdV equation,” International Journal of Contemporary Mathematical Sciences, vol. 3, no. 5, pp. 213–221, 2008. View at Google Scholar · View at MathSciNet
  3. L. Kaur, “Generalized (G′/G)—expansion method for generalized fifth order KdV equation with time-dependent coefficients,” Mathematical Sciences Letters, vol. 3, no. 3, pp. 255–261, 2014. View at Publisher · View at Google Scholar
  4. M. Suleiman, Z. B. Ibrahim, and A. F. N. Bin Rasedee, “Solution of higher-order ODEs using backward difference method,” Mathematical Problems in Engineering, vol. 2011, Article ID 810324, 18 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  5. U. Goktas and W. Hereman, “Symbolic computation of conserved densities for systems of nonlinear evolution equations,” Journal of Symbolic Computation, vol. 24, no. 5, pp. 591–621, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. N. Khanal, R. Sharma, J. Wu, and J.-M. Yuan, “A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations,” Discrete and Continuous Dynamical Systems, pp. 442–450, 2009. View at Google Scholar
  7. J. Li and Z. Qiao, “Explicit soliton solutions of the Kaup-Kupershmidt equation through the dynamical system approach,” Journal of Applied Analysis and Computation, vol. 1, no. 2, pp. 243–250, 2011. View at Google Scholar · View at MathSciNet
  8. S. J. Kayode and D. O. Awoyemi, “A multiderivative Collocation method for 5th order ordinary differential equations,” Journal of Mathematics and Statistics, vol. 6, no. 1, pp. 60–63, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. S. J. Kayode, “An order seven continuous explicit method for direct solution of general fifth order ordinary differential equations,” International Journal of Differential Equations and Applications, vol. 13, no. 2, pp. 71–80, 2014. View at Google Scholar
  10. J. B. Rosser, “A Runge-Kutta for all seasons,” SIAM Review, vol. 9, no. 3, pp. 417–452, 1967. View at Publisher · View at Google Scholar · View at MathSciNet
  11. M. B. Suleiman, “Solving nonstiff higher order ODEs directly by the direct integration method,” Applied Mathematics and Computation, vol. 33, no. 3, pp. 197–219, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. D. O. Awoyemi, “A new sixth-order algorithm for general second order ordinary differential equations,” International Journal of Computer Mathematics, vol. 77, no. 1, pp. 117–124, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. S. J. Kayode, “An efficient zero-stable numerical method for fourth-order differential equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2008, Article ID 364021, 10 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. B. T. Olabode and Y. Yusuph, “A new block method for special third order ordinary differential equations,” Journal of Mathematics and Statistics, vol. 5, no. 3, pp. 167–170, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. S. N. Jator and J. Li, “A self-starting linear multistep method for a direct solution of the general second-order initial value problem,” International Journal of Computer Mathematics, vol. 86, no. 5, pp. 827–836, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. N. Waeleh, Z. A. Majid, F. Ismail, and M. Suleiman, “Numerical solution of higher order ordinary differential equations by direct block code,” Journal of Mathematics and Statistics, vol. 8, no. 1, pp. 77–81, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. L. F. Shampine and H. A. Watts, “Block implicit one-step methods,” Mathematics of Computation, vol. 23, pp. 731–740, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. J. R. Cash, “Block Runge-Kutta methods for the numerical integration of initial value problems in ordinary differential equations. Part I. The nonstiff case,” Mathematics of Computation, vol. 40, no. 161, pp. 175–191, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  19. S. N. Jator, “Solving second order initial value problems by a hybrid multistep method without predictors,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 4036–4046, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. N. Waeleh, Z. A. Majid, and F. Ismail, “A new algorithm for solving higher order IVPs of ODEs,” Applied Mathematical Sciences, vol. 5, no. 53–56, pp. 2795–2805, 2011. View at Google Scholar · View at MathSciNet · View at Scopus
  21. J. Vigo-Aguiar and H. Ramos, “Variable stepsize implementation of multistep methods for y′′=f(x, y, y′),” Journal of Computational and Applied Mathematics, vol. 192, no. 1, pp. 114–131, 2006. View at Publisher · View at Google Scholar
  22. Z. A. Majid, N. A. Azmi, M. Suleiman, and Z. B. Ibrahaim, “Solving directly general third order ordinary differential equations using two-point four step block method,” Sains Malaysiana, vol. 41, no. 5, pp. 623–632, 2012. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. J. D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley & Sons, London, UK, 1973. View at MathSciNet
  24. S. Ola Fatunla, “Block methods for second order ODEs,” International Journal of Computer Mathematics, vol. 41, no. 1-2, pp. 55–63, 1991. View at Publisher · View at Google Scholar
  25. Z. A. Majid and M. B. Suleiman, “Implementation of four-point fully implicit block method for solving ordinary differential equations,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 514–522, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus