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

A nonstandard finite difference technique for singular Lane-Emden type equations 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/approachThe 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.FindingsSeveral 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/valueThe value of this work can be measured as follows. It is the first time such schemes have been designed for the kind of equations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Engineering Computations Emerald Publishing

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

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Publisher
Emerald Publishing
Copyright
© Emerald Publishing Limited
ISSN
0264-4401
DOI
10.1108/ec-08-2018-0344
Publisher site
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Abstract

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

Journal

Engineering Computations – Emerald Publishing

Published: Aug 15, 2019

Keywords: Nonstandard finite difference; Exact schemes; Lane–Emden equation

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