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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.

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.

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.

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.

Adomian has developed a numerical technique using special kinds of polynomials for solving non‐linear functional equations. General conditions and a new formulation are proposed for proving the convergence of Adomian's method for the numerical resolution of non‐linear functional equations depending on one or several variables. The methods proposed are applicable to a very wide class of problems.

Aim's to show how to approach the solution for a class of first order p.d.e. using Adomian decomposition method. Discusses the generalities of the method and α‐dense curves. Outlines the new approach and provides applications of its use.

In this paper, our aim is to determine the fundamental eigenfunction of a two nonlinear differential complex equation, which arises in microchip laser theory, using Adomian decompositon method.

In this work new results about the Adomian method are presented. Also we prove a new and general result of convergence of the Adomain method, and give two results of convergence of this method applied to ordinary differential equations. Finally, we generalize the Adomian method and prove two new results of convergence with one of them applied to the modified method.

New results about convergence of Adomian’s method are presented. This method was developed by G. Adomian for solving non‐linear functional equations of different kinds. New conditions for obtaining convergence of the decomposition series are given. In a similar way, the convergence of a regularisation method which can, for example, be applied to Fredholm integral equations of the first kind, is proved.

A new approach of the decomposition method (Adomian) in which the Adomian scheme is obtained in a more natural way than in the classical presentation, is given. A new condition for obtaining convergence of the decomposition series is also included.

To compute an optimal control of non‐linear reaction diffusion equations that are modelling inhibitor problems in the brain.

A new numerical approach that combines a spectral method in time and the Adomian decomposition method in space. The coupling of these two methods is used to solve an optimal control problem in cancer research.

The main conclusion is that the numerical approach we have developed leads to a new way for solving such problems.

Focused research on computing control optimal in non‐linear diffusion reaction equations. The main idea that is developed lies in the approximation of the control space in view of the spectral expansion in the Legendre basis.

Through this work we are convinced that one way to derive efficient numerical optimal control is to associate the Legendre expansion in time and Runge Kutta approximation. We expect to obtain general results from optimal control associated with non‐linear parabolic problem in higher dimension.

Coupling of methods provides a numerical solution of an optical control problem in Cancer research.

In this paper our objective is to show how to approach the solution of any first order partial differential equation (p.d.e.), using Adomian decompositon method and α‐dense curves. Indicates that this method is perfectly adapted to the solution of such equations and especially in the nonlinear case. The results are of particular importance in biomedicine and biocybernetics.