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Improved
Runge-Kutta Methods for Solving Ordinary Differential Equations
(Penambahbaikan Kaedah Runge-Kutta untuk Menyelesaikan Persamaan Pembezaan Biasa)
FARANAK RABIEI1, FUDZIAH
ISMAIL2* & MOHAMED
SULEIMAN3 1Department of Mathematics, Faculty of Science, Universiti Putra
Malaysia 43400 UPM Serdang, Selangor, Malaysia 2Department
of Mathematics, Faculty of Science and Institute for Mathematical
Research, Universiti
Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
3Institute
for Mathematical Research, Universiti Putra Malaysia, 43400 UPM
Serdang, Selangor Malaysia
Diserahkan:
24 Februari 2012/Diterima: 29 Mei 2013
ABSTRACT
In this article we proposed three explicit Improved Runge-Kutta (IRK)
methods for solving first-order ordinary differential equations. These methods
are two-step in nature and require lower number of stages compared to the
classical Runge-Kutta method. Therefore the new scheme is computationally more
efficient at achieving the same order of local accuracy. The order conditions
of the new methods are obtained up to order five using Taylor series expansion
and the third and fourth order methods with different stages are derived based
on the order conditions. The free parameters are obtained through minimization
of the error norm. Convergence of the method is proven and the stability
regions are presented. To illustrate the efficiency of the method a number of
problems are solved and numerical results showed that the method is more
efficient compared to the existing
Runge-Kutta method.
Keywords: Convergence and stability region; improved Runge-Kutta
methods; order conditions; ordinary differential equations; two-step methods
ABSTRAK
Dalam artikel ini kami mencadangkan tiga kaedah Runge-Kutta tak
tersirat penambahbaikan untuk menyelesaikan persamaan pembezaan peringkat
pertama. Kaedah ini adalah dalam bentuk dua langkah dan memerlukan bilangan
tahap yang kurang berbanding kaedah Runge-Kutta klasik. Maka kaedah yang baru
ini adalah lebih cekap bagi mencapai peringkat kejituan setempat yang sama.
Syarat peringkat untuk kaedah ini hingga peringkat kelima diterbitkan
menggunakan kembangan siri Taylor dan kaedah peringkat ketiga dan keempat
dengan tahap yang berbeza diterbitkan berdasarkan syarat peringkat tersebut.
Parameter bebasnya diperoleh melalui norma ralat yang diminimumkan. Penumpuan
kaedah ini dibuktikan dan kestabilannya dipersembahkan. Untuk menunjukkan
kecekapan kaedah ini, beberapa masalah diselesaikan dan keputusan berangka
menunjukkan kaedah ini lebih cekap berbanding kaedah Runge-Kutta sedia ada.
Kata kunci: Kaedah dua
langkah; penambahbaikan kaedah Runge-Kutta; penumpuan dan rantau kestabilan;
persamaan pembezaan biasa; syarat peringkat
RUJUKAN
Butcher,
J.C. 2008. The Numerical Methods for
Ordinary Differential Equations. John Wiley and Sons.
Goeken,
D. & Johnson, O. 2000. Runge-Kutta with higher order derivative
approximations. Applied Numerical Mathematics. 34:207- 218.
Hull,
T.E., Enright, W.H., Fellen, B.M. & Sedgwick, A.E. 1982. Comparing
numerical. Journal of Numerical Analysis 9(4): 603-637.
Phohomsiri,
P. & Udwadia, F.E. 2004. Acceleration of Runge- Kutta integeration schemes. Discrete Dynamics in Nature and Society 2: 307-314.
Rabiei,
F. & Ismail, F. 2011. Third-order Improved Runge- Kutta method for solving
ordinary differential equation. International Journal of Applied Physics and
Mathematics 1(3): 191-194.
Udwadia,
F.E. & Farahani, A. 2008. Accelerated Runge- Kutta methods. Discrete
Dynamics in Nature and Society doi:10.1155/2008/790619.
Xinyuan,
W. 2003. A class of Runge-Kutta formulae of order three and four with reduced
evaluations of function. Applied Mathematics and Computation 146:
417-432.
*Pengarang untuk surat-menyurat; email: fudziah@science.upm.edu.my
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