Sains Malaysiana 47(9)(2018): 2223–2230

http://dx.doi.org/10.17576/jsm-2018-4709-33

 

Block Hybrid Method with Trigonometric-Fitting for Solving Oscillatory Problems

(Kaedah Blok Hibrid dengan Penyuaian-Trigonometri untuk Menyelesaikan Masalah Berayun)

 

FUDZIAH ISMAIL^2*, SUFIA ZULFA AHMAD¹, YUSUF DAUDA JIKANTORO^1,3 & NORAZAK SENU^1,2

 

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 Mathematics/Computer Science, Faculty of Science, Ibrahim Badamasi Babangida University, P.M.B. 11, Lapai, Nigeria

 

Received: 13 February 2018/Accepted: 22 May 2018

 

ABSTRACT

In this paper, we develop algebraic order conditions for two-point block hybrid method up to order five using the approach of B-series. Based on the order conditions, we derive fifth order two-point block explicit hybrid method for solving special second order ordinary differential equations (ODEs), where the existing explicit hybrid method of order five is used to be the method at the first point. The method is then trigonometrically fitted so that it can be suitable for solving highly oscillatory problems arising from special second order ODEs. The new trigonometrically-fitted block method is tested using a set of oscillatory problems over a very large interval. Numerical results clearly showed the superiority of the method in terms of accuracy and execution time compared to other existing methods in the scientific literature.

 

Keywords: B-Series; explicit block hybrid method; oscillatory problems

 

ABSTRAK

Dalam kertas ini, kami membangunkan syarat peringkat aljabar kaedah blok hibrid dua titik sehingga peringkat kelima menggunakan pendekatan siri-B. Berdasarkan syarat peringkat tersebut, kami menerbitkan kaedah blok hibrid tak tersirat dua titik peringkat kelima untuk menyelesaikan persamaan pembezaan biasa (PPB) khas peringkat kedua, dengan kaedah hibrid tak tersirat sedia ada peringkat kelima digunakan sebagai kaedah pada titik pertama. Kaedah ini kemudiannya difasa-suaikan secara trigonometri supaya sesuai untuk menyelesaikan masalah berayun yang timbul daripada persamaan pembezaan khas peringkat kedua. Kaedah baru blok trigonometri fasa-suai ini diuji menggunakan satu set masalah berayun bagi selang yang sangat besar. Keputusan berangka dengan jelas menunjukkan keunggulan kaedah tersebut daripada segi ketepatan dan masa pengiraan berbanding kaedah sedia ada yang lain dalam kepustakaan saintifik.

 

Kata kunci: Kaedah blok hibrid tak tersirat; masalah berayun; Siri-B

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