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Article
Publication date: 1 December 1999

244

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Kybernetes, vol. 28 no. 9
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 1 July 1999

T. Badredine, K. Abbaoui and Y. Cherruault

This paper deals with a new proof of convergence of Adomian’s method applied to nonlinear integral equations. By using a new formulation of Adomian’s polynomials, we give the…

687

Abstract

This paper deals with a new proof of convergence of Adomian’s method applied to nonlinear integral equations. By using a new formulation of Adomian’s polynomials, we give the relation between the Picard method and Adomian’s technique.

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Kybernetes, vol. 28 no. 5
Type: Research Article
ISSN: 0368-492X

Keywords

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2358

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Kybernetes, vol. 41 no. 7/8
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 1 August 2005

M. Inc and Y. Cherruault

Based on the original methods of Adomian a decomposition method has been developed to find the analytic approximation of the linear and nonlinear Volterra‐Fredholm (V‐F…

348

Abstract

Purpose

Based on the original methods of Adomian a decomposition method has been developed to find the analytic approximation of the linear and nonlinear Volterra‐Fredholm (V‐F) integro‐differential equations under the initial or boundary conditions.

Design/methodology/approach

Designed around the methods of Adomian and later researchers. The methodology to obtain numerical solutions of the V‐F integro‐differential equations is one whose essential features is its rapid convergence and high degree of accuracy which it approximates. This is achieved in only a few terms of its iterative scheme which is devised to avoid linearization, perturbation and any transformation in order to find solutions to given problems.

Findings

The scheme was shown to have many advantages over the traditional methods. In particular it provided discretization and provided an efficient numerical solution with high accuracy, minimal calculations as well as an avoidance of physical unrealistic assumptions.

Research limitations/implications

A reliable method for obtaining approximate solutions of linear and nonlinear V‐F integro‐differential using the decomposition method which avoids the tedious work needed by traditional techniques has been developed. Exact solutions were easily obtained.

Practical implications

The new method had most of its symbolic and numerical computations performed using the Computer Algebra Systems‐Mathematica. Numerical results from selected examples were presented.

Originality/value

A new effective and accurate methodology has been developed and demonstrated.

Details

Kybernetes, vol. 34 no. 7/8
Type: Research Article
ISSN: 0368-492X

Keywords

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Article
Publication date: 1 August 2005

M. Bektas, M. Inc and Y. Cherruault

The purpose is to study an analytical solution of non‐linear Korteweg‐de Vries (KdV) equation by using the Adomian decomposition method (ADM).

227

Abstract

Purpose

The purpose is to study an analytical solution of non‐linear Korteweg‐de Vries (KdV) equation by using the Adomian decomposition method (ADM).

Design/methodology/approach

The solution is calculated in the form of a series with easily computable components. The non‐linear KdV equation has been considered and the analytic solution is compared with its numerical solution by using the ADM and Mathematica software program.

Findings

This approach to the non‐linear evolution equation was found to be valuable as a tool for scientists and applied mathematicians, because it provides immediate and visible symbolic terms of analytical solution as well as its numerical approximate solution to both linear and non‐linear problems without linearization or discretization.

Research limitations/implications

This geometrical interpretation and the produced approximate solution of the non‐linear KdV equation illustrates the use of the ADM. Research using ADM is ongoing but already the numerical results obtained in this paper justify the advantages of this methodology, even in a few terms of approximation.

Practical implications

Using the Mathematica software package the ADM was implemented for homogenous KdV equation as an illustrative example which has distinct applications for scientists and applied mathematicians.

Originality/value

This is an original study of the use of ADM for the solution of the non‐linear KdV equation. It also shows how the Mathematica software package can be used in such studies.

Details

Kybernetes, vol. 34 no. 7/8
Type: Research Article
ISSN: 0368-492X

Keywords

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Article
Publication date: 1 December 2003

M. Inç and Y. Cherruault

A decomposition method is implemented for solving travelling wave solutions of a fourth‐order semilinear diffusion equation.

305

Abstract

A decomposition method is implemented for solving travelling wave solutions of a fourth‐order semilinear diffusion equation.

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Kybernetes, vol. 32 no. 9/10
Type: Research Article
ISSN: 0368-492X

Keywords

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Article
Publication date: 1 February 2002

Yves Cherruault, Gaspar Mora and Yves Tourbier

Gives a new method for defining and calculating multiple integrals. More precisely proposes that it is possible to define a multiple integral by means of a simple integral. This…

303

Abstract

Gives a new method for defining and calculating multiple integrals. More precisely proposes that it is possible to define a multiple integral by means of a simple integral. This can be performed by using α‐dense curves in Rn, already introduced for global optimization using the ALIENOR method.

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Kybernetes, vol. 31 no. 1
Type: Research Article
ISSN: 0368-492X

Keywords

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Article
Publication date: 17 June 2008

Elçin Yusufoğlu and Barış Erbaş

This paper sets out to introduce a numerical method to obtain solutions of Fredholm‐Volterra type linear integral equations.

608

Abstract

Purpose

This paper sets out to introduce a numerical method to obtain solutions of Fredholm‐Volterra type linear integral equations.

Design/methodology/approach

The flow of the paper uses well‐known formulations, which are referenced at the end, and tries to construct a new approach for the numerical solutions of Fredholm‐Volterra type linear equations.

Findings

The approach and obtained method exhibit consummate efficiency in the numerical approximation to the solution. This fact is illustrated by means of examples and results are provided in tabular formats.

Research limitations/implications

Although the method is suitable for linear equations, it may be possible to extend the approach to nonlinear, even to singular, equations which are the future objectives.

Practical implications

In many areas of mathematics, mathematical physics and engineering, integral equations arise and most of these equations are only solvable in terms of numerical methods. It is believed that the method is applicable to many problems in these areas such as loads in elastic plates, contact problems of two surfaces, and similar.

Originality/value

The paper is original in its contents, extends the available work on numerical methods in the solution of certain problems, and will prove useful in real‐life problems.

Details

Kybernetes, vol. 37 no. 6
Type: Research Article
ISSN: 0368-492X

Keywords

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Article
Publication date: 17 September 2008

Randolph C. Rach

To provide a new proof of convergence of the Adomian decomposition series for solving nonlinear ordinary and partial differential equations based upon a thorough examination of…

1421

Abstract

Purpose

To provide a new proof of convergence of the Adomian decomposition series for solving nonlinear ordinary and partial differential equations based upon a thorough examination of the historical milieu preceding the Adomian decomposition method.

Design/methodology/approach

Develops a theoretical background of the Adomian decomposition method under the auspices of the Cauchy‐Kovalevskaya theorem of existence and uniqueness for solution of differential equations. Beginning from the concepts of a parametrized Taylor expansion series as previously introduced in the Murray‐Miller theorem based on analytic parameters, and the Banach‐space analog of the Taylor expansion series about a function instead of a constant as briefly discussed by Cherruault et al., the Adomian decompositions series and the series of Adomian polynomials are found to be a uniformly convergent series of analytic functions for the solution u and the nonlinear composite function f(u). To derive the unifying formula for the family of classes of Adomian polynomials, the author develops the novel notion of a sequence of parametrized partial sums as defined by truncation operators, acting upon infinite series, which induce these parametrized sums for simple discard rules and appropriate decomposition parameters. Thus, the defining algorithm of the Adomian polynomials is the difference of these consecutive parametrized partial sums.

Findings

The four classes of Adomian polynomials are shown to belong to a common family of decomposition series, which admit solution by recursion, and are derived from one unifying formula. The series of Adomian polynomials and hence the solution as computed as an Adomian decomposition series are shown to be uniformly convergent. Furthermore, the limiting value of the mth Adomian polynomial approaches zero as the index m approaches infinity for the prerequisites of the Cauchy‐Kovalevskaya theorem. The novel truncation operators as governed by discard rules are analogous to an ideal low‐pass filter, where the decomposition parameters represent the cut‐off frequency for rearranging a uniformly convergent series so as to induce the parametrized partial sums.

Originality/value

This paper unifies the notion of the family of Adomian polynomials for solving nonlinear differential equations. Further it presents the new notion of parametrized partial sums as a tool for rearranging a uniformly convergent series. It offers a deeper understanding of the elegant and powerful Adomian decomposition method for solving nonlinear ordinary and partial differential equations, which are of paramount importance in modeling natural phenomena and man‐made device performance parameters.

Details

Kybernetes, vol. 37 no. 7
Type: Research Article
ISSN: 0368-492X

Keywords

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Article
Publication date: 12 June 2009

Feyed Ben Zitoun and Yves Cherruault

The purpose of this paper is to present a method for solving nonlinear integral equations of the second and third kind.

276

Abstract

Purpose

The purpose of this paper is to present a method for solving nonlinear integral equations of the second and third kind.

Design/methodology/approach

The method converts the nonlinear integral equation into a system of nonlinear equations. By solving the system, the solution can be determined. Comparing the methodology with some known techniques shows that the present approach is simple, easy to use, and highly accurate.

Findings

The proposed technique allows the authors to obtain an approximate solution in a series form. Test problems are given to illustrate the pertinent features of the method. The accuracy of the numerical results indicates that the technique is efficient and well‐suited for solving nonlinear integral equations.

Originality/value

The present approach provides a reliable technique that avoids the difficulties and massive computational work if compared with the traditional techniques and does not require discretization in order to find solutions to the given problems.

Details

Kybernetes, vol. 38 no. 5
Type: Research Article
ISSN: 0368-492X

Keywords

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