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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 910624, 11 pages
http://dx.doi.org/10.1155/2013/910624
Research Article

The Use of Phase Lag and Amplification Error Derivatives for the Construction of a Modified Runge-Kutta-Nyström Method

1Department of Business Administration, Faculty of Management and Economy, Technological Educational Institute of Patras, 26 334 Patras, Greece
2Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
3Laboratory of Computational Sciences, Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, 22 100 Tripolis, Greece

Received 14 November 2012; Revised 16 March 2013; Accepted 3 April 2013

Academic Editor: Juan Carlos Cortés López

Copyright © 2013 D. F. Papadopoulos and T. E. Simos. 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 · View at MathSciNet
  2. P. J. van der Houwen and B. P. Sommeijer, “Predictor-corrector methods for periodic second-order initial-value problems,” IMA Journal of Numerical Analysis, vol. 7, no. 4, pp. 407–422, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. Van de Vyver, “An embedded phase-fitted modified Runge-Kutta method for the numerical integration of the radial Schrödinger equation,” Physics Letters A, vol. 352, no. 4, pp. 278–285, 2006.
  4. J. M. Franco, “Exponentially fitted explicit Runge-Kutta-Nyström methods,” Journal of Computational and Applied Mathematics, vol. 167, no. 1, pp. 1–19, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. T. E. Simos and J. V. Aguiar, “A modified Runge-Kutta method with phase-lag of order infinity for the numerical solution of the Schrödinger equation and related problems,” Computers and Chemistry, vol. 25, no. 3, pp. 275–281, 2001.
  6. Z. A. Anastassi, D. S. Vlachos, and T. E. Simos, “A family of Runge-Kutta methods with zero phase-lag and derivatives for the numerical solution of the Schrödinger equation and related problems,” Journal of Mathematical Chemistry, vol. 46, no. 4, pp. 1158–1171, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. 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.
  8. 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, vol. 22, no. 6, pp. 623–634, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Z. Kalogiratou, T. Monovasilis, and T. E. Simos, “Computation of the eigenvalues of the Schrödinger equation by exponentially-fitted Runge-Kutta-Nyström methods,” Computer Physics Communications, vol. 180, no. 2, pp. 167–176, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. I. Alolyan and T. E. Simos, “High algebraic order methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 48, no. 4, pp. 925–958, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. I. Alolyan and T. E. Simos, “A family of ten-step methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 49, no. 9, pp. 1843–1888, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. A. Konguetsof, “Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 48, no. 2, pp. 224–252, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. A. Konguetsof, “A hybrid method with phase-lag and derivatives equal to zero for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 49, no. 7, pp. 1330–1356, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  14. A. A. Kosti, Z. A. Anastassi, and T. E. Simos, “An optimized explicit Runge-Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation and related problems,” Journal of Mathematical Chemistry, vol. 47, no. 1, pp. 315–330, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. 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 · View at MathSciNet
  16. H. Kobeissi, K. Fakhreddine, and M. Kobeissi, “On a canonical functions approach to the elastic scattering phase-shift problem,” International Journal of Quantum Chemistry, vol. 40, no. 1, pp. 11–21, 1991.
  17. L. G. Ixaru and M. Rizea, “A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies,” Computer Physics Communications, vol. 19, no. 1, pp. 23–27, 1980.