Sains Malaysiana 44(3)(2015): 473–482

Zero-Dissipative Trigonometrically Fitted Hybrid Method for Numerical Solution of Oscillatory Problems

(Kaedah Hibrid Penyuaian Trigonometri Lesapan-Sifar untuk Penyelesaian Berangka Masalah Berayun)

YUSUF DAUDA JIKANTORO, FUDZIAH ISMAIL* & NORAZAK SENU

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

Serdang 43400, Selangor Darul Ehsan, Malaysia

Received: 7 January 2014/Accepted: 3 October 2014

ABSTRACT

In this paper, an improved trigonometrically fitted zero-dissipative explicit two-step hybrid method with fifth algebraic order is derived. The method is applied to several problems which solutions are oscillatory in nature. Numerical results obtained are compared with existing methods in the scientific literature. The comparison shows that the new method is more effective and efficient than the existing methods of the same order.

Keywords: Dispersion; hybrid method; oscillatory problems; oscillatory solution; trigonometrically fitted

ABSTRAK

Dalam kertas ini, suatu penyuaian trigonometri lesapan sifar kaedah hibrid dua langkah penambahbaikan peringkat kelima diterbitkan. Kaedah ini digunakan untuk beberapa masalah yang penyelesaiannya berayun. Keputusan berangka yang diperoleh dibandingkan dengan kaedah sedia ada dalam maklumat saintifik. Perbandingan tersebut menunjukkan kaedah yang yang baharu ini adalah lebih efektif dan cekap berbanding kaedah sedia ada dengan peringkat yang sama.

Kata kunci: Kaedah hibrid; masalah berayun; penyelesaian berayun; penyuaian trigonmetri; serakan

REFERENCES

Ahmad, S.Z., Ismail, F., Senu, N. & Suleiman, M. 2013. Zero-dissipative phase-fitted hybrid method for solving oscillatory second order ordinary differential equations. Applied Mathematics and Computation 219(19): 10096-10104.

Al-Khasawneh, R.A., Ismail, F. & Suleiman, M. 2007. Embedded diagonally implicit Runge–Kutta–Nyström 4(3) pair for solving special second-order IVPs. Applied Mathematics and Computation 190(2): 1803-1814.

Butcher, J.C. 2008. Numerical Methods for Ordinary Differential Equations. New York: John Wiley & Sons.

Coleman, J.P. 2003. Order conditions for a class of two-step Methods for y′′ = f (x, y). IMA Journal of Numerical Analysis 23(2): 197-220.

Dizicheh, A.K., Ismail, F., Md. Arifin, N. & Abu Hassan, M. 2012. A class of two-step hybrid trigonometrically fitted explicit Numerov-type method for second-order IVPs. Acta Comptare 1: 182-192.

Dormand, J.R., El-Mikkawy, M.E.A. & Prince, P.J. 1987. High-order embedded Runge-Kutta- Nystrom formulae. IMA Journal of Numerical Analysis 7(4): 423-430.

Fang, Y. & Wu, X. 2007. A trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems. Applied Mathematics and Computation 189(1): 178-185.

Franco, J.M. 2006. A class of explicit two-step hybrid methods for second-order IVPs. Journal of Computational and Applied Mathematics 187(1): 41-57.

Ramos, H. & Vigo-Aguiar, J. 2010. On the frequency choice in trigonometrically fitted methods. Applied Mathematics Letters 23: 1378-1381.

Ming, Q., Yang, Y. & Fang, Y. 2012. An Optimized Runge-Kutta Method for the Numerical Solution of the Radial Schrödinger Equation. Mathematical Problems in Engineering 2012: Article ID 867948.

Mohamad, M., Senu, N., Suleiman, M. & Ismail, F. 2012. Fifth order explicit Runge-Kutta-Nyström methods for periodic initial value problems. World of Applied Sciences Journal 17: 16-20.

Senu, N., Suleiman, M., Ismail, F. & Othman, M. 2010. Kaedah pasangan 4(3) Runge-Kutta-Nyström untuk masalah nilai awal berkala. Sains Malaysiana 39(4): 639-646.

Senu, N., Suleiman, M. & Ismail, F. 2009. An embedded explicit Runge–Kutta–Nyström method for solving oscillatory problems. Physica Scripta 80(1): 015005.

Simos, T.E. 2012. 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 2012: Article ID 420387.

Simos, T.E. 2002. Exponentially-fitted Runge-Kutta-Nyström method for the numerical solution of initial value problems with oscillating solutions. Applied Mathematics Letters 15(2): 217-225.

Simos, T.E. 1999. Explicit eighth order methods for the numerical integration of initial-value problems with periodic or oscillating solutions. Computer Physics Communications 119(1): 32-44.

Tsitouras, Ch. 2002. Explicit two-step methods for second-order linear IVPs. Computers & Mathematics with Applications 43(8): 943-949.

Van der Houwen, P.J. & Sommeijer, B.P. 1987. Explicit Runge- Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions. SIAM Journal on Numerical Analysis 24(3): 595-617.

Van de Vyver, H. 2005. A Runge-Kutta Nyström pair for the numerical integration of perturbed oscillators. Computer Physics Communications 167(2): 129-142.

Vanden Berghe, G., Ixaru L.G. & Van Daele, M. 2001. Optimal implicit exponentially fitted Runge-Kutta methods. Computational Physics Comm. 140: 346-357.

Yusuf Dauda Jikantoro. 2014. Numerical solution of special second initial value problems by hybrid type methods. MSc Thesis, Universiti Putra Malaysia (Unpublished).

*Corresponding author; email: fudziah_i@yahoo.com.my

 content