Rational methods for solving first-order initial value problems

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 approximat...

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Published inInternational journal of computer mathematics Vol. 98; no. 2; pp. 252 - 270
Main Authors Fairuz, A. N., Majid, Z. A.
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 01.02.2021
Taylor & Francis Ltd
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ISSN0020-7160
1029-0265
DOI10.1080/00207160.2020.1737862

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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.
AbstractList 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.
Author Majid, Z. A.
Fairuz, A. N.
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StartPage 252
SubjectTerms Approximation
Boundary value problems
Chemical reactions
Differential equations
Problem solving
Problem whose solutions possesses singularity
rational method
singularly perturbed problem
stiff problem
system of differential equations
Van der Pol's equation
Title Rational methods for solving first-order initial value problems
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