Rational methods for solving first-order initial value problems

ABSTRACT

In this paper, a class of rational methods of second to fourth order of accuracy are presented. The methods are developed by considering the concept of the closest points of approximation in its formulas. These methods require the application of a suitable method to calculate the starting approximation values as they are not self-starting, as well as higher derivatives of the problems. The illustrated region of absolute stability presented in the article shows that the rational schemes are all A-stable. The methods are tested on several initial value problems including systems of differential equations. From the numerical results, it appears that the proposed methods are comparable to some of the existing methods in terms of accuracy and efficiency. The capability of the schemes in solving problems whose solution possesses singularity and singularly perturbed problem can be obviously seen as they are compared to the existing rational multistep methods. The methods also show its reliability in solving stiff problems. Besides that, the schemes are tested on the Van der Pol's equation, which usually arises in physics and chemical reactions. The presented numerical solutions show that the proposed methods are comparably accurate to the existing methods.

KEYWORDS:

• Problem whose solutions possesses singularity
• rational method
• singularly perturbed problem
• stiff problem
• system of differential equations
• Van der Pol's equation

2010 AMS SUBJECT CLASSIFICATIONS:

• 65L05
• 65L04

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors would like to thank Universiti Putra Malaysia for supporting this research under the Putra Grant (GP-IPS/2018/9625300).