Sains Malaysiana 46(5)(2017): 817–824

http://dx.doi.org/10.17576/jsm-2017-4605-16

Numerical Algorithm of Block Method for General Second Order ODEs using Variable

Step Size

(Algoritma Berangka Kaedah Blok bagi ODE Umum Peringkat Kedua Menggunakan Pemboleh Ubah Saiz Langkah)

NAZREEN WAELEH¹* & ZANARIAH ABDUL MAJID²

¹Faculty of Electronic & Computer Engineering, Universiti Teknikal Malaysia Melaka (UTeM), Hang Tuah Jaya, 76100 Durian Tunggal, Melaka Bandaraya Bersejarah, Malaysia

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

 Diserahkan: 26 Januari 2016/Diterima: 1 November 2016

 ABSTRACT

This paper outlines an alternative algorithm for solving general second order ordinary differential equations (ODEs). Normally, the numerical method was designed for solving higher order ODEs by converting it into an n-dimensional first order equations with implementation of constant step length. Nevertheless, this involved a lot of computational complexity which led to consumption a lot of time. Consequently, a direct block multistep method with utilization of variable step size strategy is proposed. This method was developed for computing the solution at four points simultaneously and the derivation based on numerical integration as well as using interpolation approach. The convergence of the proposed method is justified under suitable conditions of stability and consistency. Five numerical examples are considered and some comparisons are made with the existing methods for demonstrating the validity and reliability of the proposed algorithm.

Keywords: Block method; general second order ordinary differential equations; variable step size

ABSTRAK

Kertas ini menggariskan satu algoritma alternatif untuk menyelesaikan persamaan pembezaan biasa (ODE) umum peringkat kedua. Kebiasaannya, kaedah berangka untuk menyelesaikan ODE peringkat tinggi direka dengan menukarkan ia ke dalam n-dimensi persamaan peringkat pertama dengan perlaksanaan panjang langkah kekal. Walau bagaimanapun, ini melibatkan kerumitan pengiraan yang membawa kepada penggunaan masa yang banyak. Oleh yang demikian, satu kaedah langsung pelbagai langkah blok dengan penggunaan strategi saiz langkah berubah dicadangkan. Kaedah ini dibangunkan bagi menghitung penyelesaian pada empat titik secara serentak dan terbitannya berdasarkan integrasi berangka serta menggunakan pendekatan interpolasi. Penumpuan kaedah yang dicadangkan dijustifikasi mengikut syarat kestabilan dan tekal yang sesuai. Terdapat lima contoh berangka dipertimbangkan dan beberapa perbandingan telah dibuat dengan kaedah yang sedia ada untuk menunjukkan kesahan dan kebolehpercayaan algoritma yang dicadangkan.

Kata kunci: Kaedah blok; persamaan pembezaan biasa umum peringkat kedua; saiz langkah berubah

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*Pengarang untuk surat-menyurat; email: nazreen@utem.edu.my