About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 324989, 20 pages
http://dx.doi.org/10.1155/2012/324989
Research Article

New 4(3) Pairs Diagonally Implicit Runge-Kutta-Nyström Method for Periodic IVPs

Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 14 February 2012; Revised 24 June 2012; Accepted 8 July 2012

Academic Editor: Taher S. Hassan

Copyright © 2012 Norazak Senu 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. P. J. van der Houwen and B. P. Sommeijer, “Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions,” SIAM Journal on Numerical Analysis, vol. 24, no. 3, pp. 595–617, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. P. J. van der Houwen and B. P. Sommeijer, “Diagonally implicit Runge-Kutta-Nyström methods for oscillatory problems,” SIAM Journal on Numerical Analysis, vol. 26, no. 2, pp. 414–429, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. A. B. Sideridis and T. E. Simos, “A low-order embedded Runge-Kutta method for periodic initial value problems,” Journal of Computational and Applied Mathematics, vol. 44, no. 2, pp. 235–244, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. García, P. Martín, and A. B. González, “New methods for oscillatory problems based on classical codes,” Applied Numerical Mathematics, vol. 42, no. 1–3, pp. 141–157, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. N. Senu, M. Suleiman, and F. Ismail, “An embedded explicit Runge–Kutta–Nyström method for solving oscillatory problems,” Physica Scripta, vol. 80, no. 1, 2009. View at Publisher · View at Google Scholar
  6. H. Van de Vyver, “A symplectic Runge-Kutta-Nyström method with minimal phase-lag,” Physics Letters A, vol. 367, no. 1-2, pp. 16–24, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. N. Senu, M. Suleiman, F. Ismail, and M. Othman, “A zero-dissipative Runge-Kutta-Nyström method with minimal phase-lag,” Mathematical Problems in Engineering, vol. 2010, Article ID 591341, 15 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. J. M. Franco, “A 5(3) pair of explicit ARKN methods for the numerical integration of perturbed oscillators,” Journal of Computational and Applied Mathematics, vol. 161, no. 2, pp. 283–293, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. P. W. Sharp, J. M. Fine, and K. Burrage, “Two-stage and three-stage diagonally implicit Runge-Kutta Nyström methods of orders three and four,” IMA Journal of Numerical Analysis, vol. 10, no. 4, pp. 489–504, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. S. O. Imoni, F. O. Otunta, and T. R. Ramamohan, “Embedded implicit Runge-Kutta Nyström method for solving second-order differential equations,” International Journal of Computer Mathematics, vol. 83, no. 11, pp. 777–784, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. R. A. Al-Khasawneh, F. Ismail, and M. Suleiman, “Embedded diagonally implicit Runge-Kutta-Nystrom 4(3) pair for solving special second-order IVPs,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1803–1814, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Z. A. Anastassi and T. E. Simos, “An optimized Runge-Kutta method for the solution of orbital problems,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 1–9, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. T. E. Simos, “Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1616–1621, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S. Stavroyiannis and T. E. Simos, “Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2467–2474, 2009. View at Publisher · View at Google Scholar
  15. T. E. Simos, “Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1331–1352, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. T. E. Simos, “New stable closed Newton-Cotes trigonometrically fitted formulae for long-time integration,” Abstract and Applied Analysis, vol. 2012, Article ID 182536, 15 pages, 2012. View at Publisher · View at Google Scholar
  17. T. E. Simos, “Optimizing a hybrid two-step method for the numerical solution of the Schrödinger equation and related problems with respect to phase-lag,” Journal of Applied Mathematics, vol. 2012, Article ID 420387, 17 pages, 2012. View at Publisher · View at Google Scholar
  18. T. E. Simos, “Embedded Runge-Kutta methods for periodic initial value problems,” Mathematics and Computers in Simulation, vol. 35, no. 5, pp. 387–395, 1993. View at Publisher · View at Google Scholar
  19. G. Avdelas and T. E. Simos, “Block Runge-Kutta methods for periodic initial-value problems,” Computers & Mathematics with Applications, vol. 31, no. 1, pp. 69–83, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. G. Avdelas and T. E. Simos, “Embedded methods for the numerical solution of the Schrödinger equation,” Computers & Mathematics with Applications, vol. 31, no. 2, pp. 85–102, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. D. F. Papadopoulos, Z. A. Anastassi, and T. E. Simos, “A phase-fitted Runge-Kutta-Nyström method for the numerical solution of initial value problems with oscillating solutions,” Computer Physics Communications, vol. 180, no. 10, pp. 1839–1846, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. J. M. Franco and M. Palacios, “High-order P-stable multistep methods,” Journal of Computational and Applied Mathematics, vol. 30, no. 1, pp. 1–10, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. J. M. Franco, “A class of explicit two-step hybrid methods for second-order IVPs,” Journal of Computational and Applied Mathematics, vol. 187, no. 1, pp. 41–57, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. J. R. Dormand, M. E. A. El-Mikkawy, and P. J. Prince, “Families of Runge-Kutta-Nyström formulae,” IMA Journal of Numerical Analysis, vol. 7, no. 2, pp. 235–250, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. L. Bursa and L. Nigro, “A one-step method for direct integration of structural dynamic equations,” International Journal for Numerical Methods in Engineering, vol. 15, pp. 685–699, 1980.
  26. I. Gladwell and R. M. Thomas, “Damping and phase analysis for some methods for solving second-order ordinary differential equations,” International Journal for Numerical Methods in Engineering, vol. 19, no. 4, pp. 495–503, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. E. Hairer and G. Wanner, “A theory for Nyström methods,” Numerische Mathematik, vol. 25, no. 4, pp. 383–400, 1975/76. View at Publisher · View at Google Scholar
  28. J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Chichester, UK, 2nd edition, 2008. View at Publisher · View at Google Scholar
  29. J. R. Dormand, Numerical Methods for Differential Equations, CRC Press, Boca Raton, Fla, USA, 1996.
  30. J. C. Butcher and D. J. L. Chen, “A new type of singly-implicit Runge-Kutta method,” Applied Numerical Mathematics, vol. 34, no. 2-3, pp. 179–188, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. R. C. Allen, Jr. and G. M. Wing, “An invariant imbedding algorithm for the solution of inhomogeneous linear two-point boundary value problems,” Journal of Computational Physics, vol. 14, pp. 40–58, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. H. Van de Vyver, “A Runge-Kutta-Nyström pair for the numerical integration of perturbed oscillators,” Computer Physics Communications, vol. 167, no. 2, pp. 129–142, 2005.
  33. J. D. Lambert and I. A. Watson, “Symmetric multistep methods for periodic initial value problems,” Journal of the Institute of Mathematics and its Applications, vol. 18, no. 2, pp. 189–202, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. E. Stiefel and D. G. Bettis, “Stabilization of Cowell's method,” Numerische Mathematik, vol. 13, no. 2, pp. 154–175, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. N. H. Cong, “A-stable diagonally implicit Runge-Kutta-Nyström methods for parallel computers,” Numerical Algorithms, vol. 4, no. 3, pp. 263–281, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH