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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.

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.

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.

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.

Considers the Adomian decomposition method to be a powerful technique that can solve efficiently a large class of linear and nonlinear differential equations. Describes a general method for approximating the solution of the Laplace equation with Dirichlet‐boundary conditions and which can be applied to a large class of problems.

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

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.

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.

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.

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.

This paper describes a non‐linear reaction‐diffusion equation, which models how a substance spreads in the surface of the cortex so as to avoid a massive destruction of neurones when cerebral tissue is not oxygenated correctly. For the explicit finite differences method, the necessary stability condition is provided by a reaction‐diffusion equation with non‐linearity given by a decreasing function. The solution to the non‐linear reaction‐diffusion equation of the model can be obtained via one of the two methods: the finite differences (explicit schema) method and the Adomian method.