Nonlinear problems via a convergence accelerated decomposition method of Adomian

Abstract

The present paper is devoted to the convergence control and accelerating the traditional Decomposition Method of Adomian (ADM). By means of perturbing the initial or early terms of the Adomian iterates by adding a parameterized term, containing an embedded parameter, new modified ADM is constructed. The optimal value of this parameter is later determined via squared residual minimizing the error. The failure of the classical ADM is also prevented by a suitable value of the embedded parameter, particularly beneficial for the Duan–Rach modification of the ADM incorporating all the boundaries into the formulation. With the presented squared residual error analysis, there is no need to check out the results against the numerical ones, as usually has to be done in the traditional ADM studies to convince the readers that the results are indeed converged to the realistic solutions. Physical examples selected from the recent application of ADM demonstrate the validity, accuracy and power of the presented novel approach in this paper. Hence, the highly nonlinear equations arising from engineering applications can be safely treated by the outlined method for which the classical ADM may fail or be slow to converge.