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.

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.

In this paper, Adomian decomposition method has been adopted to resolve the non-linear and non-autonomous ordinary differential equations. It has been proved that this technique permits to give new expressions for the Adomian's polynomials (??) and (??).

In this work, we adapted another time the Adomian decomposition method for solving nonlinear and non-autonomous ODEs systems. Therefore, our expressions of the Adomian polynomials are determined for a several-variable differential operators. The solution series is shown that it stay coincide with the Taylor series. Thus new conditions of convergence have been established, some systemes has been solved by ADM using Maple 2020. keywords Adomian decomposition method, Adomian polynomials, ODEs systems, initial value problems, several-variables differential operators. Classification 26B12, 34L30, 47E05, 65B10, 65L05, 65L80