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Article
Publication date: 2 August 2013

María José Pujol, Francisco A. Pujol, Fidel Aznar, Mar Pujol and Ramón Rizo

In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that…

Abstract

Purpose

In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium.

Design/methodology/approach

In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known.

Findings

Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal.

Originality/value

The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 26 September 2019

Razan Alchikh and Suheil Khuri

The purpose of this paper is to apply an efficient semi-analytical method for the approximate solution of Lienard’s equation of fractional order.

Abstract

Purpose

The purpose of this paper is to apply an efficient semi-analytical method for the approximate solution of Lienard’s equation of fractional order.

Design/methodology/approach

A Laplace decomposition method (LDM) is implemented for the nonlinear fractional Lienard’s equation that is complemented with initial conditions. The nonlinear term is decomposed and then a recursive algorithm is constructed for the determination of the proposed infinite series solution.

Findings

A number of examples are tested to explicate the efficiency of the proposed technique. The results confirm that this approach is convergent and highly accurate by using only few iterations of the proposed scheme.

Originality/value

The approach is original and is of value because it is the first time that this approach is used successfully to tackle fractional differential equations, which are of great interest for authors in the recent years.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 25 July 2019

Michael Chapwanya, Robert Dozva and Gift Muchatibaya

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…

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.

Details

Engineering Computations, vol. 36 no. 5
Type: Research Article
ISSN: 0264-4401

Keywords

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