Sains Malaysiana 35(2): 63-68 (2006)

Direct Integration Implicit Variable Steps Method for Solving Higher

Order Systems of Ordinary Differential Equations Directly

(Kaedah Kamiran Terus Langkah Berubah Tersirat Bagi Menyelesaikan Sistem

Persamaan Terbitan Peringkat Tinggi Secara Langsung)

Zanariah Abdul Majid

Institut Perguruan Teknik

Jalan Yaacob Latif, Bandar Tun Razak

56000 Kuala Lumpur

Mohamed Suleiman

Mathematics Department

Faculty of Science

Universiti Putra Malaysia

43400 Serdang, Selangor D.E.

### Abstract

In this paper, a direct integration implicit variable step size method in the form of Adams Moulton Method is developed for solving directly the second order system of ordinary differential equations (ODEs) using variable step size. The existing multistep method involves the computations of the divided differences and integration coefficients in the code when using the variable step size or variable step size and order. The idea of  developing this method is to store all the coefficients involved in the code. Thus, this strategy can avoid the lengthy computation of the coefficients during the implementation of the code as well as to improve the execution time. Numerical results are given to compare the efficiency of the developed method with the 1-point method of variable step size and order code (1PDVSO) in Omar (1999).

Keywords: implicit method; variable steps method; ordinary differential equations

Abstrak

Dalam makalah ini, suatu kaedah kamiran terus dengan saiz langkah berubah tersirat dalam bentuk Kaedah Adams Moulton dibangunkan bagi menyelesaikan secara terus sistem persamaan peringkat dua menerusi saiz langkah berubah. Kaedah multilangkah yang sedia ada melibatkan pengiraan beza pembahagi dan pekali kamiran dalam kod apabila menggunakan saiz langkah berubah atau saiz  langkah berubah dan peringkat. Ide di sebalik kaedah ini ialah untuk menyimpan kesemua pekali yang terlibat dalam kod. Maka strategi ini boleh menghindarkan pengiraan berpanjangan pekali berkaitan semasa implementasi kod tersebut disamping memperbaiki masa pelaksanaan. Hasil berangka diberikan untuk membandingkan keberkesanan kaedah yang telah dibangunkan itu dengan kaedah 1-titik bagi saiz langkah berubah dan  peringkat (1PDVSO) dalam Omar (1999).

Kata kunci: kaedah tersirat; kaedah langkah berubah; persamaan terbitan biasa

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