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The purpose of this paper is to introduce a new modified version of the homotopy perturbation method (NMHPM) and Adomian decomposition method (ADM) for solving the nonlinear ordinary differential equation arising in MHD non-Newtonian fluid flow over a linear stretching sheet.

The governing equation is solved analytically by applying a newly developed optimal homotopy perturbation approach and ADM. This optimal approach contains convergence-control parameter and is computationally rather efficient. The results of numerical example are presented and only a few terms are required to obtain accurate solutions.

A new modified optimal and ADM methods accelerate the rapid convergence of the series solution. These methods dramatically reduce the size of work. The obtained series solution is combined with the diagonal Padé approximants to handle the boundary condition at infinity. Results derived from these methods are shown graphically and in tabulated forms to study the efficiency and accuracy.

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. The formulation of mathematical model for these processes can be based on the equations of non-Newtonian fluid mechanics. The flow of an electrically conducting fluid in the presence of a magnetic field is of importance in various areas of technology and engineering such as MHD power generation, MHD flow meters, MHD pumps, etc. It is generally admitted that a number of astronomical bodies (e.g. the sun, Earth, Jupiter, Magnetic stars, Pulsars) posses fluid interiors and (or least surface) magnetic fields.

The present results are original and new for the MHD non-Newtonian fluid flow over a linear stretching sheet. The results attained in this paper confirm the idea that NMHPM and ADM are powerful mathematical tools and that can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

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.

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.

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.

The study of the convergence of Adomian's method presents some difficulties when applied to real problems. Proposes a convergence proof of this technique adapted to non‐linear partial differential equations. Solves some real examples numerically.

Considers the convergence of Adomian’s method applied to algebraic equations. Presents directly some conditions of convergence which only depend on the equations’ coefficients and also gives an estimation of the truncation error together with some numerical applications.

The purpose of this paper is to consider a steady two-dimensional magneto-hydrodynamic (MHD) Falkner-Skan boundary layer flow of an incompressible viscous electrically fluid over a permeable wall in the presence of a magnetic field.

The governing equations of MHD Falkner-Skan flow are transformed into an initial values problem of an ordinary differential equation using the Lie symmetry method which are then solved by He's variational iteration method with He's polynomials.

The approximate solution is compared with the known solution using the diagonal Pad’e approximants and the geometrical behavior for the values of various parameters. The results reveal the reliability and validity of the present work, and this combinational method can be applied to other nonlinear boundary layer flow problems.

In this paper, an approximate analytical solution of the MHD Falkner-Skan flow problem is obtained by combining the Lie symmetry method with the variational iteration method and He's polynomials.

We show that Padé approximants considerably improve convergence of Adomian's decomposition. The power of the method proposed is demonstrated through two illustrative examples from the field of nonlinear optics.

The decomposition method is used for solving differential systems in biology and medicine. A comparison is given between the Runge‐Kutta method and the decomposition technique. New relationships for calculating Adomian’s polynomials are used for solving the differential systems governing the competition between species and based on the Lotka‐Volterra model.