Numerical treatment of Periodic and Oscillatory Problems Using Leapfrog Method

S. Sekar; M. Vijayarakavan

doi:https://doi.org/10.14445/22315373/IJMTT-V26P506

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 26 | Number 1 | Year 2015 | Article Id. IJMTT-V26P506 | DOI : https://doi.org/10.14445/22315373/IJMTT-V26P506

Numerical treatment of Periodic and Oscillatory Problems Using Leapfrog Method

S. Sekar, M. Vijayarakavan

Citation :

S. Sekar, M. Vijayarakavan, "Numerical treatment of Periodic and Oscillatory Problems Using Leapfrog Method," International Journal of Mathematics Trends and Technology (IJMTT), vol. 26, no. 1, pp. 24-28, 2015. Crossref, https://doi.org/10.14445/22315373/IJMTT-V26P506

Abstract

In this paper, the Leapfrog method is used to study the periodic and oscillatory problems. Results obtained using Leapfrog and Single-term Haar wavelet series (STHW) [10] methods are compared with the exact solutions of the periodic and oscillatory problems. The results obtained using Leapfrog is found to be very closer to the exact solutions of these problems. Error graphs for the obtained results and exact solutions are presented in a graphical form to highlight the efficiency of this method. This Leapfrog can be easily implemented in a digital computer and the solution can be obtained for any length of time.

Keywords

Periodic problems, Oscillatory problems, Ordinary differential equations, Leapfrog Method, Singleterm Haar wavelet series.

References

[1] J. C. Butcher, “The Numerical Methods for Ordinary Differential Equations”, 2003, John Wiley & Sons, U.K.
[2] Y. Fang, Y. Song and X. Wu, “A robust trigonometrically fitted embedded pair for perturbed oscillators”, J. Comput. Appl. Math., 225 (2009) 347-355.
[3] J. M. Franco, “Runge-Kutta-Nystrom methods adapted to the numerical integration of perturbed oscillators”, Comput. Phys. Comm., 147 (2002) 770-787.
[4] E. Hairer and G. Wanner, “Solving Ordinary Differential Equations II”, Springer, New York, 1996.
[5] E. Hairer,” A One-step Method of Order 10 for y00 = f(x; y)”, IMA J. Numer. Anal. 2, (1982) 83-94.
[6] S. N. Jator, S. Swindle, and R. French, ” Trigonometrically fitted block Numerov type method for y00 = f(x; y; y0)”, Numerical Algorithms, (2012), DOI 10.1007/s11075-012-9562-1.
[7] J. D. Lambert, ”Numerical methods for ordinary differential systems”, John Wiley, New York, 1991.
[8] J. D. Lambert, ”Computational methods in ordinary differential equations”, John Wiley, New York, 1973.
[9] H. S. Nguyen, R. B. Sidje and N. H. Cong, “Analysis of trigonometric implicit Runge-Kutta methods”, J. Comput. Appl. Math. 198 (2007) 187-207.
[10] S. Sekar and E. Paramanathan, “A study on periodic and oscillatory problems using single-term Haar wavelet series”, International Journal of Current Research, vol. 2, no 1, 2011, pp. 097-105.
[11] S. Sekar and K. Prabhavathi, “Numerical solution of first order linear fuzzy differential equations using Leapfrog method”, IOSR Journal of Mathematics, vol. 10, no. 5 Ver. I, (Sep-Oct. 2014), pp. 07-12.
[12] S. Sekar and K. Prabhavathi, “Numerical Solution of Second Order Fuzzy Differential Equations by Leapfrog Method”, International Journal of Mathematics Trends and Technology, vol. 16, no. 2, 2014, pp. 74-78.
[13] S. Sekar and K. Prabhavathi, “Numerical Strategies for the nth –order fuzzy differential equations by Leapfrog Method”, International Journal of Mathematical Archive, vol. 6, no. 1, 2014, pp. 162-168.
[14] S. Sekar and M. Vijayarakavan, “Numerical Investigation of first order linear Singular Systems using Leapfrog Method”, International Journal of Mathematics Trends and Technology, vol. 12, no. 2, 2014, pp. 89-93.
[15] S. Sekar and M. Vijayarakavan, “Numerical Solution of Stiff Delay and Singular Delay Systems using Leapfrog Method”, International Journal of Scientific & Engineering Research, vol. 5, no. 12, December-2014, pp. 1250-1253.
[16] T. E. Simos,” An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions”, Comput. Phys. Commun. 115 (1998) 1-8.
[17] B. P. Sommeijer, ” Explicit, high-order Runge-Kutta-Nystrom methods for parallel computers”, Appl. Numer. Math. 13 (1993) 221-240.
[18] G. Vanden, L. Gr. Ixaru, and M. van Daele, “Optimal implicit exponentially-fitted Runge-Kutta”, Comput. Phys. Commun. 140 (2001) 346-357.