Sains Malaysiana 45(11)(2016): 1747–1754

 

A Sixth-Order RKFD Method with Four-Stage for Directly Solving Special Fourth-Order ODEs

(Kaedah RKFD Peringkat Keenam dengan Tahap Empat untuk Menyelesaikan Secara Terus PPB Khas Peringkat Keempat)

 

FUDZIAH ISMAIL1,2*, KASIM HUSSAIN3 & NORAZAK SENU1,2

 

1Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor Darul Ehsan, Malaysia

 

2Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor

Darul Ehsan, Malaysia

 

3Department of Mathematics, College of Science, Al-Mustansiriyah University, Baghdad

Iraq

 

Received: 28 October 2015/Accepted: 23 March 2016

 

ABSTRACT

In this article, the general form of Runge-Kutta method for directly solving a special fourth- order ordinary differential equations denoted as RKFD method is given. The order conditions up to order seven are derived, based on the order conditions, we construct a new explicit four-stage sixth-order RKFD method denoted as RKFD6 method. Zero-stability of the method is proven. Comparisons are made using the existing Runge–Kutta methods after the problems are reduced to a system of first order ordinary differential equations. Numerical results are presented to illustrate the efficiency and competency of the new method.

 

Keywords: Ordinary differential equations; special fourth order; RKFD method; Runge-Kutta method

 

ABSTRAK

Dalam kertas ini, bentuk umum kaedah Runge-Kutta untuk menyelesaikan secara terus persamaan pembezaan biasa khas peringkat keempat yang ditandakan sebagai kaedah RKFD diberikan. Syarat tertib hingga ke peringkat ketujuh diterbitkan, berasaskan syarat ini, kami bina kaedah baharu RKFD tahap empat peringkat keenam yang ditandakan sebagai RKFD6. Kestabilan sifar kaedah ini dibuktikan. Perbandingan dijalankan menggunakan kaedah Runge-Kutta sedia ada setelah masalah tersebut diturunkan kepada sistem persamaan pembezaan peringkat pertama. Keputusan berangka dipersembahkan untuk menunjukkan kecekapan dan kompetensi kaedah yang baharu tersebut.

 

Kata kunci: Kaedah RKFD; kaedah Runge-Kutta; peringkat keempat khas; persamaan pembezaan biasa

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

 

 

 

 

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