Convergence of Adomian’s method applied to integral equations

Badredine, Toufik; Abbaoui, K.; Cherruault, Y.

doi:10.1108/03684929910277779

Cited by 19 publications

(6 citation statements)

References 6 publications

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“…He proposed a new definition of the method and then he insisted that it will become possible to prove the convergence of the decomposition method. Badredine et al (1999) presented a new proof of convergence of Adomian's method applied to nonlinear integral equations. K 34,7/8…”

Section: Adapting Adomian's Methods To Equations (1) and (2)mentioning

confidence: 99%

A reliable method for obtaining approximate solutions of linear and nonlinear Volterra‐Fredholm integro‐differential equations

Cherruault

2005

Kybernetes

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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.

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Section: Adapting Adomian's Methods To Equations (1) and (2)mentioning

confidence: 99%

A reliable method for obtaining approximate solutions of linear and nonlinear Volterra‐Fredholm integro‐differential equations

Cherruault

2005

Kybernetes

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“…The algorithm can be considered as an implementation of the theoretical aspects of convergence study in [8,9]. It simply regards the ADM in terms of the series summation rather in terms of the series component to write the recursive form…”

Section: And In Generalmentioning

confidence: 99%

A new algorithm for the decomposition solution of nonlinear differential equations

Behiry

Hashish

El‐Kalla

et al. 2007

Computers & Mathematics with Applications

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In this paper, we introduce a new algorithm for applying the Adomian decomposition method to nonlinear differential and partial differential equations. The proposed algorithm does not require the explicit evaluation of Adomian polynomials to approximate the nonlinearity term. The numerical experiments show that the proposed algorithm is more accurate at larger values for time, though at the cost of more computations.

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“…However, in practice all the terms of the series Equation 15 cannot be determined, and we use an approximation of the solution u from the new series: Equation 16 The convergence of the decomposition series have been investigated by several authors. The theoretical treatment of convergence of the decomposition method has been considered by Cherruault (Akbaoui and Cherruault, 1994; Cherruault et al , 1995; Badredine et al , 1999). This paper is organized in the following way: after obtaining its geometric KdV equation as defined: Equation 17 It is analysed in the Section 2.…”

Section: Remarkmentioning

confidence: 99%

Geometrical interpretation and approximate solution of non‐linear KdV equation

Bektaş

Cherruault²

2005

Kybernetes

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PurposeThe purpose is to study an analytical solution of non‐linear Korteweg‐de Vries (KdV) equation by using the Adomian decomposition method (ADM).Design/methodology/approachThe 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.FindingsThis 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/implicationsThis 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 implicationsUsing 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/valueThis 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.

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Convergence of Adomian’s method applied to integral equations

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.

Cited by 19 publications

References 6 publications

A reliable method for obtaining approximate solutions of linear and nonlinear Volterra‐Fredholm integro‐differential equations

A reliable method for obtaining approximate solutions of linear and nonlinear Volterra‐Fredholm integro‐differential equations

A new algorithm for the decomposition solution of nonlinear differential equations

Geometrical interpretation and approximate solution of non‐linear KdV equation

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