Abstract
In this paper, we work out the theoretical basis of the Adomian method. This powerful method that G. Adomian has developed in the beginning of the 1980's supplies analytical approximations to the solution of many kinds of equations. Thanks to our decomposition theory, we justify the practical method and we generalize the convergence results obtained by Y. Cherruault. Afterward, we explain how to solve rigorously equations with linear, nonlinear or composed operator and also equations with several terms or with a second member.
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