A successful and easy to use modification of the classical Adomian decomposition method (ADM) was implemented by Duan and Rach  by simply adding a predetermined parameter into the ADM formulation. We provide a mathematical rigor in this paper to approve that the formulation of Duan and Rach indeed improves the classical Adomian method by both preventing its divergence and speeding up its convergence. Instead of prescribing it, the best suitable value of the added parameter is determined within the global squared residual approximation to ensure the quickest convergence of the recursive scheme. Within the presence of a least change interval of the approximate series, an optimum value of the added parameter leading to the accelerated convergence is shown to exist. Moreover, via the improved Adomian Decomposition Method, the convergence region of the series approximation is found to be enlarged to a bigger physical domain. The most important outcome of the present approach is that the results generated by the ADM series are no longer have to be validated via other numerical means. (C) 2018 Elsevier B.V. All rights reserved.