Advancements and Applications of the Adomian Decomposition Method in Solving Nonlinear Differential Equations

The Adomian Decomposition Method (ADM), introduced by George Adomian in the 1980s, stands out as a revolutionary technique for solving linear and nonlinear differential equations. ADM's compelling simplicity and remarkable computational efficiency have propelled its adoption across diverse scientific and engineering disciplines. This paper offers an in-depth exploration of ADM, delving into its robust theoretical foundations and versatile practical applications. By presenting detailed examples, we showcase ADM's adaptability and efficacy in addressing complex challenges. We highlight significant advancements that have enhanced the method's capabilities, tackling contemporary obstacles and unveiling innovative solutions. Through meticulous simulations and real-world case studies, we demonstrate ADM's exceptional prowess in optimizing renewable energy systems, modeling turbulent flows, and analyzing structural dynamics under seismic forces. Our findings underscore ADM's critical role in advancing computational approaches for differential equations, emphasizing its practical advantages. This comprehensive evaluation not only attests to the current effectiveness of ADM but also charts future research pathways poised to make substantial contributions to applied mathematics and engineering.