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A nonstandard finite difference technique for singular Lane-Emden type equations

Michael Chapwanya (Department of Mathematics, University of Pretoria, Pretoria, South Africa)
Robert Dozva (University of Zimbabwe, Harare, Zimbabwe)
Gift Muchatibaya (University of Zimbabwe, Harare, Zimbabwe)

Engineering Computations

ISSN: 0264-4401

Article publication date: 25 July 2019

Issue publication date: 15 August 2019

94

Abstract

Purpose

This paper aims to design new finite difference schemes for the Lane–Emden type equations. In particular, the authors show that the schemes are stable with respect to the properties of the equation. The authors prove the uniqueness of the schemes and provide numerical simulations to support the findings.

Design/methodology/approach

The Lane–Emden equation is a well-known highly nonlinear ordinary differential equation in mathematical physics. Exact solutions are known for a few parameter ranges and it is important that any approximation captures the properties of the equation it represent. For this reason, designing schemes requires a careful consideration of these properties. The authors apply the well-known nonstandard finite difference methods.

Findings

Several interesting results are provided in this work. The authors list these as follows. Two new schemes are designed. Mathematical proofs are provided to show the existence and uniqueness of the solution of the discrete schemes. The authors show that the proposed method can be extended to singularly perturbed equations.

Originality/value

The value of this work can be measured as follows. It is the first time such schemes have been designed for the kind of equations.

Keywords

Acknowledgements

The work was initiated when M.C. was on sabbatical leave at the University of Zimbabwe (UZ) in 2018. M.C. is grateful to the Department of Mathematics at UZ for making his visit the most pleasant and productive and to the support of South African DST/NRF SARChI Chair on Mathematical Models and Methods in Bioengineering and Biosciences (M3B2) of the University of Pretoria.

Thanks are also addressed to the anonymous reviewers whose suggestions have contributed to the improvement of the paper.

Citation

Chapwanya, M., Dozva, R. and Muchatibaya, G. (2019), "A nonstandard finite difference technique for singular Lane-Emden type equations", Engineering Computations, Vol. 36 No. 5, pp. 1566-1578. https://doi.org/10.1108/EC-08-2018-0344

Publisher

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Emerald Publishing Limited

Copyright © 2019, Emerald Publishing Limited

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