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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.

The purpose of this paper is to propose a new modification of the Adomian decomposition method for resolution of higher‐order inhomogeneous nonlinear initial value problems.

First the authors review the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. Next, the advantages of Duan's new algorithms and subroutines for fast generation of the Adomian polynomials to high orders are discussed. Then algorithms are considered for the solution of a sequence of first‐, second‐, third‐ and fourth‐order inhomogeneous nonlinear initial value problems with constant system coefficients by the new modified recursion scheme in order to derive a systematic algorithm for the general case of higher‐order inhomogeneous nonlinear initial value problems.

The authors investigate seven expository examples of inhomogeneous nonlinear initial value problems: the exact solution was known in advance, in order to demonstrate the rapid convergence of the new approach, including first‐ through sixth‐order derivatives and quadratic, cubic, quartic and exponential nonlinear terms in the solution and a sextic nonlinearity in the first‐order derivative. The key difference between the various modified recursion schemes is the choice of the initial solution component, using different choices to partition and delay the subsequent parts through the recursion steps. The authors' new approach extends this concept.

The new modified decomposition method provides a significant advantage for computing the solution's Taylor expansion series, both systematically and rapidly, as demonstrated in the various expository examples.

The purpose of this paper is to provide a comparison of the Adomian decomposition method (ADM) with the variational iteration method (VIM) for solving the Lane‐Emden equations of the first and second kinds.

The paper examines the theoretical framework of the Adomian decomposition method and compares it with the variational iteration method. The paper seeks to determine the relative merits and computational benefits of both the Adomian decomposition method and the variational iteration method in the context of the important physical models of the Lane‐Emden equations of the first and second kinds.

The Adomian decomposition method is shown to readily solve the Lane‐Emden equations of both the first and second kinds for all positive real values of the system coefficient α and for all positive real values of the nonlinear exponent m. The decomposition series solution of these nonlinear differential equations requires the calculation of the Adomian polynomials appropriate to the particular system nonlinearity. The paper shows that the variational iteration method works effectively to solve the Lane‐Emden equation of the first kind for system coefficient values α=1, 2, 3, 4 but only for positive integer values of the nonlinear exponent m. The successive approximations of the solution of these nonlinear differential equations require the determination of the appropriate Lagrange multipliers, which are established in this paper. These two methodologies overcome the singular behavior at the origin x=0. The paper shows that the variational iteration method is impractical for solving either the Lane‐Emden equation of the first kind for non‐integer values of the system exponent m or the Lane‐Emden equations of the second kind. Indeed the Adomian decomposition method is shown to solve even the generalized Lane‐Emden equation for any analytic nonlinearity and all positive values of the system coefficient α in a practical and straightforward manner. The conclusions are supported by several numerical examples.

This paper presents an accurate comparison of the Adomian decomposition method with the variational iteration method for solving the Lane‐Emden equations of the first and second kinds. The paper presents a new solution algorithm for the generalized Lane‐Emden equation for any analytic system nonlinearity and for any model geometry as characterized by all possible positive real values of the system coefficient α.

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.

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.

The purpose of this paper is to investigate the possibility of extension to the variational iteration and the Adomian decomposition methods for solving nonlinear Huxley equation with time-fractional derivative.

Objectives achieved the main methods: the fractional derivative of f (x) in the Caputo sense is first stated. Second, the time-fractional Huxley equation is written in a differential operator form where the differential operator is in Caputo sense. After acting on both sides by the inverse operator of the fractional differential operator in Caputo sense, the Adomian's decomposition is then used to get the power series solution of the resulted time-fractional Huxley equation. Also, a second objective is achieved by applying the variational iteration method to get approximate solutions for the time-fractional Huxley equation.

There are some important findings to state and summarize here. First, the variational iteration method and the decomposition method provide the solutions in terms of convergent series with easily computable components for this considered problem. Second, it seems that the approximate solution of time-fractional Huxley equation using the decomposition method converges faster than the approximate solution using the variational iteration method. Third, the variational iteration method handles nonlinear equations without any need for the so-called Adomian polynomials. However, Adomian decomposition method provides the components of the exact solution, where these components should follow the summation given in Equation (21).

This paper presents new materials in terms of employing the variational iteration and the Adomian decomposition methods to solve the problem under consideration. It is expected that the results will give some insightful conclusions for the used techniques to handle similar fractional differential equations. This emphasizes the fact that the two methods are applicable to a broad class of nonlinear problems in fractional differential equations.

The purpose of this paper is to use the Adomian decomposition method (ADM) for solving boundary value problems with dual solutions.

The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific applications. In this work, the authors seek to determine the relative merits of the ADM in the context of several important nonlinear boundary value models characterized by the existence of dual solutions.

The ADM is shown to readily solve specific nonlinear BVPs possessing more than one solution. The decomposition series solution of these models requires the calculation of the Adomian polynomials appropriate to the particular system nonlinearity. The authors show that the ADM solves these models for any analytic nonlinearity in a practical and straightforward manner. The conclusions are supported by several numerical examples arising in various scientific applications which admit dual solutions.

This paper presents an accurate work for solving nonlinear BVPs that possess dual solutions. The authors have demonstrated the widespread applicability of the ADM for solving various forms of these nonlinear equations.

Aims to propose methodology to solve adaptive control problems by the use of the Adomian decomposition method (ADM).

The approach to this problem is through the ADM which consists in finding a solution of nonlinear state equations as a convergent series that depends on the unknown parameters of the system.

It was shown that a first approach for solving adaptive control problems could be tackled by using the ADM. The solution of the systems was obtained in a series forms as a function of the unknown parameters. The objective function became a function that explicitly depends on parameters. This, it was shown, can be minimised by classical or nonclassical global optimization methods.

The methods presented depends on ADM. It is proposed that future work involves the study of an adaptive control problem associated to a nonlinear compartmental systems of Michaels‐Menten type.

Presents an applications of the developed ADM using it, as adapted to control problems. It contributes to mathematical modelling studied concerning most biological, physical phenomena which are described by nonlinear systems: differential, partial differential in form and for which there is little hope of finding an exact solution.

To find methods for solving non‐linear partial differential equations. The decomposition method may be applied, but a difficulty arises when applied to non‐linear partial differential equations with initial and boundary conditions. In this work, two methods are described that take into account the boundary conditions.

The decomposition method whilst being a powerful tool for solving non‐linear functional equations encounters difficulties in finding solutions of partial differential equations with boundary conditions. In this paper, two methods are introduced which consist of setting boundary conditions to the equations so that the decomposition methods can be applied.

By using the two proposed methods the decomposition method can then be easily used. In this work the two methods taking account of the boundary conditions were found to be efficient and allows a solution to be found using the Adomian decomposition method.

The two new methods provide solutions by the application of the decomposition method of George Adomian as extended by other researchers. Both are efficient: the first giving interesting results for linear and non‐linear problems; the second one is also efficient, but difficulties could arise from the calculations of the required series.

The research provides two efficient methods. The first method gives the demonstrated results for linear and non‐linear problems due to the use of symbolic software such as Mathematica or Maple.

Both methods illustrate the powerful use of the decomposition techniques pioneered by Adomian and as a result of their application may be applied to the solving of non‐linear functional equations of any kind. This paper tackles the problems by introducing new methods of applying the Adomian techniques to partial differential equations with boundary conditions.