Sains Malaysiana 39(4)(2010): 639–646

Kaedah Pasangan 4(3) Runge-Kutta-Nystršm untuk Masalah Nilai Awal Berkala

(A 4(3) Pair Runge-Kutta-Nystršm Method for Periodic Initial Value Problems)

Norazak Senu*, Mohamed Suleiman & Fudziah Ismail

Jabatan Matematik, Fakulti Sains, Universiti Putra Malaysia

43400 UPM Serdang, Selangor, Malaysia

Mohamed Othman

Jabatan Teknologi Komunikasi dan Rangkaian

Fakulti Sains Komputer dan Teknologi Maklumat

Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

Received: 14 August 2009 / Accepted: 4 November 2009

ABSTRAK

Kaedah baru pasangan benaman 4(3) tahap-empat berperingkat empat tak tersirat Runge-Kutta-Nyström (RKN) diterbitkan untuk mengamir persamaan pembezaan peringkat dua berbentuk yʺ = f (x, y) dengan penyelesaian bentuk berkala. Dipersembahkan kaedah yang bercirikan serakan berperingkat tinggi serta pekali ralat pangkasan utama yang ‘kecil’. Analisis kestabilan bagi kaedah yang diterbitkan juga diberikan. Perbandingan keputusan berangka antara kaedah yang dihasilkan dengan kaedah RK4(3) dan RKN4(3)D menunjukkan kaedah yang baru ini berkecekapan lebih baik daripada segi penilaian fungsi dan masa pelaksanaan.

Kata kunci: Kaedah Runge-Kutta-Nyström; penyelesaian berkala; serakan

ABSTRACT

A new embedded 4(3) pair explicit four-stage fourth-order Runge-Kutta-Nyström (RKN) method was developed to integrate second-order differential equations of the form yʺ = f (x, y) where the solution was oscillatory. Presented is a method which has high order dispersion with a ‘small’ principal local truncation error coefficient. The stability analysis of the method derived is also given. Numerical comparisons of this new method with RK4(3) and RKN4(3)D methods show its clear advantage in terms of function evaluations and computation time.

Keywords: Oscillatory solutions; Phase-lag; Runge-Kutta-Nyström methods

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*Corresponding author; email: razak@math.upm.edu.my