Sains Malaysiana 48(8)(2019): 1761–1769

http://dx.doi.org/10.17576/jsm-2019-4808-23

Variable Order Block Method for Solving Second Order Ordinary Differential Equations

(Kaedah Blok Peringkat Berubah untuk Penyelesaian Persamaan Pembezaan Biasa Peringkat Kedua)

ZARINA BIBI IBRAHIM^1,2, NOORAINI ZAINUDDIN³*, KHAIRIL ISKANDAR OTHMAN⁴,

MOHAMED SULEIMAN² & ISKANDAR SHAH MOHD ZAWAWI⁵

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

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

³Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS (UTP), 32610 Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia

⁴Department of Mathematics, Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor Darul Ehsan, Malaysia

⁵Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Seremban Campus, 70300 Seremban, Negeri Sembilan Darul Khusus, Malaysia

Received: 1 February 2019/Accepted: 7 May 2019

ABSTRACT

This paper proposed 2-point block backward differentiation formulas (BBDF) of order 3, 4, and 5 for direct solution of second order ordinary differential equations. These methods were derived via backward difference interpolation polynomial with two solutions are produced simultaneously at each step. All the three different orders of 2-point BBDF is implemented in variable order scheme. The scheme utilizes the local truncation error, which is generated by the single order of 2-point BBDF method. Numerical results are presented to illustrate the validity of the proposed scheme.

Keywords: Block method; initial value problem; second order ODEs; variable order

ABSTRAK

Kertas ini membangunkan formula 2-titik blok pembezaan kebelakang (FBPK) peringkat 3, 4, dan 5 untuk menyelesaikan persamaan pembezaan biasa peringkat kedua. Kaedah ini diterbitkan melalui polinomial interpolasi beza kebelakang dengan dua penyelesaian diberikan secara serentak untuk setiap langkah. Ketiga-tiga peringkat 2-titik FBPK dijalankan dengan skema peringkat berubah. Skema ini menggunakan ralat pangkasan setempat, yang dijanakan oleh setiap peringkat kaedah 2-titik FBPK. Keputusan berangka ditunjukkan untuk menggambarkan kesahihan skema yang dicadangkan.

Kata kunci: Kaedah blok; masalah nilai awal; PBB peringkat kedua; peringkat berubah

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*Corresponding author; email: aini_zainuddin@utp.edu.my