Solving the Kuramoto-Sivashinsky Equation via a Modified Adomian Decomposition Method

Abstract

The Kuramoto-Sivashinsky equation (KSE) is a well-known nonlinear partial differential equation (PDE) that plays a significant role in various scientific fields, particularly in fluid dynamics and reaction-diffusion systems. In this study, we employ the Modified Adomian Decomposition Method (MADM) to derive an analytical solution to the KSE over the domain $t \in \mathbb{R}$ and $0 \le x < 1$. The research provides a comprehensive analysis of the applicability of MADM in generating explicit series solutions for the KSE. The equation is formulated under proper initial conditions, and the systematic implementation of MADM is demonstrated involving the decomposition of nonlinear terms and successive approximation. The convergence and stability of the resulting series solutions are investigated, highlighting the method's efficiency in capturing the dynamics described by the KSE. Numerical simulations are presented to validate the analytical results, illustrating the effectiveness of the MADM in solving the KSE within the given parameters. This study not only enhances the theoretical understanding of the KSE but also serves as a practical guide for applying analytical techniques to complex nonlinear PDEs. Moreover, the proposed method represents a significant advancement in solving highly non-linear models by providing a straightforward recursive scheme that avoids linearization or small-parameter assumptions. The physical relevance of the obtained solutions is reflected in their ability to capture key features of instability and chaotic behavior characteristic of the KSE. These findings underscore the potential of MADM as a powerful and versatile tool in the study of spatiotemporal phenomena governed by nonlinear PDEs.