Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2015, Article ID 217578, 11 pages
http://dx.doi.org/10.1155/2015/217578
Research Article

Optimized Hybrid Methods for Solving Oscillatory Second Order Initial Value Problems

1Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
2Department of Mathematics, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Received 25 October 2014; Accepted 6 January 2015

Academic Editor: Xiaohua Ding

Copyright © 2015 N. 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. J. P. Coleman, “Order conditions for a class of two-step methods for y=fx,y,” IMA Journal of Numerical Analysis, vol. 23, no. 2, pp. 197–220, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. 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 MathSciNet · View at Scopus
  3. L. Brusa and L. Nigro, “A one-step method for direct in tegration of structural dynamic equations,” International Journal for Numerical Methods in Engineering, vol. 15, no. 5, pp. 685–699, 1980. View at Publisher · View at Google Scholar · View at Scopus
  4. 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 MathSciNet · View at Scopus
  5. 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 MathSciNet
  6. R. M. Thomas, “Phase Properties of higher order, almost P-stable formulae,” BIT Numerical Mathematics, vol. 24, no. 2, pp. 225–238, 1984. View at Publisher · View at Google Scholar
  7. 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 MathSciNet · View at Scopus
  8. R. D'Ambrosio, M. Ferro, and B. Paternoster, “Trigonometrically fitted two-step hybrid methods for special second order ordinary differential equations,” Mathematics and Computers in Simulation, vol. 81, no. 5, pp. 1068–1084, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. F. Samat, F. Ismail, and M. Suleiman, “High order explicit hybrid methods for solving second-order ordinary differential equations,” Sains Malaysiana, vol. 41, no. 2, pp. 253–260, 2012. View at Google Scholar · View at Scopus
  10. D. F. Papadopoulos and T. E. Simos, “A new methodology for the construction of optimized Runge-Kutta-Nyström methods,” International Journal of Modern Physics C. Computational Physics and Physical Computation, vol. 22, no. 6, pp. 623–634, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. A. A. Kosti, Z. A. Anastassi, and T. E. Simos, “An optimized explicit Runge-Kutta-Nyström method for the numerical solution of orbital and related periodical initial value problems,” Computer Physics Communications, vol. 183, no. 3, pp. 470–479, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. D. F. Papadopoulos, Z. A. Anastassi, and T. E. Simos, “A modified phase-fitted and amplification-fitted Runge-Kutta-Nyström method for the numerical solution of the radial Schrödinger equation,” Journal of Molecular Modeling, vol. 16, no. 8, pp. 1339–1346, 2010. View at Publisher · View at Google Scholar · View at Scopus
  13. E. Hairer, S. P. Nrsett, and G. Wanner, Solving Ordinary Differential Equations, vol. 1, Springer, Berlin, Germany, 2010.
  14. J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, England, UK, 2008.
  15. M. M. Chawla and P. S. Rao, “High-accuracy P-stable methods for y=fx,y,” IMA Journal of Numerical Analysis, vol. 5, no. 2, pp. 215–220, 1985. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. B. S. Attili, K. Furati, and M. I. Syam, “An efficient implicit Runge-Kutta method for second order systems,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 229–238, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. J. 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 MathSciNet · View at Scopus
  18. 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 MathSciNet · View at Scopus
  19. 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 MathSciNet · View at Scopus
  20. E. Stiefel and D. G. Bettis, “Stabilization of Cowell's method,” Numerische Mathematik, vol. 13, pp. 154–175, 1969. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. S. Z. Ahmad, F. Ismail, N. Senu, and M. Suleiman, “Zero-dissipative phase-fitted hybrid methods for solving oscillatory second order ordinary differential equations,” Applied Mathematics and Computation, vol. 219, no. 19, pp. 10096–10104, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus