Analysis of Convergence and Noise Terms in the Adomian Decomposition Method for Nonlinear PDEs

Abstract

In this study, we enter the field of nonlinear partial differential equations (PDEs) and propose a novel approach to their solution. We employ the iterative Adomian decomposition method (ADM) to address a distinct subclass of nonlinear PDEs. The main objective of this paper is to evaluate the efficiency and reliability of the ADM in handling complex mathematical models. To this end, we explore the convergence characteristics of the method when applied to the aforementioned class of equations. Our analysis sheds light on the intricate interplay between the iterative steps of the method and the underlying nonlinear dynamics, providing valuable insights into the behavior of the solution process. Through this investigation, we find noise terms in the context of inhomogeneous equations. We reveal that the presence of noise is intrinsically linked to the inherent heterogeneity of the equations, and its impact on convergence behavior is systematically elucidated. To validate the robustness and accuracy of the proposed method, we present a series of numerical results. These results not only underscore the reliability of the method in producing accurate solutions but also highlight its potential to overcome computational challenges associated with nonlinear PDEs. In summary, our study contributes to the broader picture of nonlinear PDE solvers by showing the potential of the Adomian decomposition method, especially in models involving noise terms and inhomogeneous equations. The efficiency of this method is illustrated by investigating the convergence results for this equation. We show that the noise terms are conditional for nonhomogeneous equations, and the numerical results demonstrate the reliability and accuracy of the ADM. Keywords: the Adomian method, noise terms, iterative method. https://doi.org/10.55463/issn.1674-2974.50.7.22