SAINS MALAYSIANA

Sains Malaysiana 49(11)(2020): 2859-2870

http://dx.doi.org/10.17576/jsm-2020-4911-24

Explicit Schemes based on Rational Approximant for Solving First Order Initial Value Problems

(Skim tak Tersirat berdasarkan Pendekatan Nisbah bagi Menyelesaikan Masalah Nilai Awal Peringkat Pertama)

A’IN NAZIFA FAIRUZ¹, ZANARIAH ABDUL MAJID^1,2* & ZARINA BIBI IBRAHIM²

¹Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

²Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

Received: 17 January 2020/Accepted: 20 May 2020

ABSTRACT

A class of rational methods of the second, third and fourth-order are proposed in this study. The formulas are developed based on a rational function with the denominator of degree one. Besides that, the concept of the closest points of approximation is also emphasized in formulating these methods. The derived methods are not self-starting; thus, an existing rational method is applied to calculate the starting values. The stability regions of the methods are also illustrated in this paper and suggest that only the second-order method is A-stable, while the third and fourth-order methods are not. The proposed formulas are examined on different problems, in which the solution possesses singularity, stiff and singularly perturbed problems. The numerical results show the capability of the proposed methods in solving problems with singularity. It also suggests that the developed schemes are more accurate than the existing rational multistep methods for problems with integer singular point. It is also shown that the derived schemes are suitable for solving stiff and singularly perturbed problems, although some of the formulas are not A-stable.

Keywords: Problem which solution possesses singularity; rational methods; singularly perturbed problem; stiff problem

ABSTRAK

Suatu kelas kaedah nisbah bagi peringkat kedua, ketiga dan keempat dicadangkan dalam kajian ini. Kaedah ini dirumus berdasarkan fungsi nisbah yang mempunyai penyebut berdarjah satu. Selain itu, konsep titik penghampiran terdekat juga ditekankan dalam merumus kaedah ini. Kaedah yang dirumus ini merupakan kaedah yang tidak bermula dengan sendirinya. Justeru, suatu kaedah nisbah sedia ada digunakan untuk menghitung nilai pemula. Rantau kestabilan bagi kaedah nisbah tersebut juga dijelaskan dan mencadangkan bahawa hanya kaedah peringkat kedua adalah A-stabil, manakala kaedah peringkat ketiga dan keempat pula bukan A-stabil. Kaedah yang dicadangkan telah diuji pada masalah yang berbeza, iaitu masalah dengan penyelesaian yang mempunyai kesingularan, kekakuan dan pengusikan singular. Hasil berangka menunjukkan kebolehan kaedah tersebut dalam menyelesaikan masalah dengan kesingularan. Hasil juga mencadangkan bahawa penghampiran yang diberikan oleh kaedah yang dirumus adalah lebih jitu berbanding kaedah multilangkah nisbah bagi masalah dengan titik singular integer. Keputusan juga menunjukkan bahawa kaedah yang dicadangkan sesuai untuk menyelesaikan masalah kekakuan dan masalah dengan pengusikan singular walaupun sebahagian daripada rumus tersebut bukanlah A-stabil.

Kata kunci: Kaedah nisbah; masalah dengan pengusikan singular; masalah dengan penyelesaian yang mempunyai kesingularan; masalah kekakuan

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*Corresponding author; email: am_zana@upm.edu.my

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