Sains Malaysiana 42(6)(2013): 845–853

One-Step Exponential-rational Methods for the Numerical Solution of First Order Initial Value Problems

(Kaedah Eksponen-Bisbah Satu Langkah Bagi Penyelesaian Masalah Nilai Awal

Peringkat Pertama Secara Berangka)

Teh Yuan Ying*

School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia

06010 Sintok, Kedah Darul Aman, Malaysia

Nazeeruddin Yaacob

Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia

81310 Skudai, Johor Darul Ta’zim, Malaysia

Received: 27 June 2012/Accepted: 2 October 2012

ABSTRACT

In this study, a new class of exponential-rational methods (ERMs) for the numerical solution of first order initial value problems has been developed. Developments of third order and fourth order ERMs, as well as their corresponding local truncation error have been presented. Each ERM was found to be consistent with the differential equation and L-stable. Numerical experiments showed that the third order and fourth order ERMs generates more accurate numerical results compared with the existing rational methods in solving first order initial value problems.

Keywords: Exponential function; exponential-rational method; problem whose solution possesses singularity; rational function; rational method

ABSTRAK

Dalam kajian ini, satu kelas kaedah eksponen-nisbah untuk penyelesaian masalah nilai awal peringkat pertama secara berangka telah dibangunkan. Pembangunan kaedah eksponen-nisbah yang berperingkat ketiga dan keempat telah dibentangkan bersama dengan ralat pangkasan setempat yang sepadan. Setiap kaedah eksponen-nisbah didapati adalah tekal dengan persamaan pembezaan dan mempunyai kestabilan jenis L. Eksperimen secara berangka telah menunjukkan bahawa kaedah eksponen-nisbah berperingkat ketiga dan keempat menjana keputusan berangka yang lebih tepat berbanding dengan kaedah-kaedah nisbah yang sedia ada dalam menyelesaikan masalah nilai awal peringkat pertama.

Kata kunci: Fungsi eksponen; fungsi nisbah; kaedah eksponen-nisbah; kaedah nisbah; masalah dengan penyelesaian yang mempunyai singular

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*Corresponding author; email: yuanying@uum.edu.my