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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 136961, 10 pages
Semi Implicit Hybrid Methods with Higher Order Dispersion for Solving Oscillatory Problems
1Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
2Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
Received 11 January 2013; Accepted 4 March 2013
Academic Editor: Juan Carlos Cortés López
Copyright © 2013 S. Z. Ahmad 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.
- J. M. Franco, “An explicit hybrid method of Numerov type for second-order periodic initial-value problems,” Journal of Computational and Applied Mathematics, vol. 59, no. 1, pp. 79–90, 1995.
- L. K. Yap, F. Ismail, M. Suleiman, and S. Md. Amin, “Block methods based on Newton interpolations for solving special second order ordinary differential equations directly,” Journal of Mathematics and Statistics, vol. 4, no. 3, pp. 174–180, 2008.
- 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.
- J. P. Coleman, “Order conditions for a class of two-step methods for ,” IMA Journal of Numerical Analysis, vol. 23, no. 2, pp. 197–220, 2003.
- 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.
- 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.
- N. Senu, M. Suleiman, F. Ismail, and M. Othman, “A singly diagonally implicit Runge-Kutta-Nyström method for solving oscillatory problems,” IAENG International Journal of Applied Mathematics, vol. 41, no. 2, pp. 155–161, 2011.
- L. Brusa and L. Nigro, “A one-step method for direct integration of structural dynamic equations,” International Journal for Numerical Methods in Engineering, vol. 15, no. 5, pp. 685–699, 1980.
- 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.
- N. Senu, M. Suleiman, F. Ismail, and M. Othman, “A fourth-order diagonally implicit Runge-Kutta- Nyström method with dispersion of high order,” in Proceedings of the 4th International Conference on Applied Mathematics, Simulation, Modelling (ASM '10), pp. 78–82, July 2010.
- A. A. Kosti, Z. A. Anastassi, and T. E. Simos, “Construction of an optimized explicit Runge-Kutta-Nyström method for the numerical solution of oscillatory initial value problems,” Computers & Mathematics with Applications, vol. 61, no. 11, pp. 3381–3390, 2011.
- 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.
- B. P. Sommeijer, “A note on a diagonally implicit Runge-Kutta-Nyström method,” Journal of Computational and Applied Mathematics, vol. 19, no. 3, pp. 395–399, 1987.
- J. R. Dormand, Numerical Methods for Differential Equations, Library of Engineering Mathematics, CRC Press, Boca Raton, Fla, USA, 1996.
- E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations 1, Springer, Berlin, Germany, 2010.
- J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Chichester, UK, 2nd edition, 2008.
- M. M. Chawla and P. S. Rao, “High-accuracy -stable methods for ,” IMA Journal of Numerical Analysis, vol. 5, no. 2, pp. 215–220, 1985.
- 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.
- E. Stiefel and D. G. Bettis, “Stabilization of Cowell's method,” Numerische Mathematik, vol. 13, pp. 154–175, 1969.
- 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.
- 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.
- 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.