Application of a Modified Adomian Decomposition Method for Solving Linear and Nonlinear Partial Differential Equations

Abstract

Partial Differential Equations (PDEs) are fundamental tools for modeling dynamic behaviors in physical, chemical, and engineering systems. However, solving nonlinear PDEs poses significant challenges due to the lack of closed-form solutions and the computational limitations of classical numerical approaches. This study introduces the Modified Adomian Decomposition Method (MADM) as an effective semi-analytical technique for solving both linear and nonlinear PDEs, with applications to the Advection, Burgers’, and Sine-Gordon equations. MADM enhances the classical Adomian Decomposition Method by incorporating refined recursive structures and inverse operators, which improve the convergence rate and simplify the solution process. The results demonstrate that MADM provides highly accurate solutions, often matching known exact solutions, and exhibits faster convergence compared to existing methods. Comparative analysis with the Variational Iteration Method (VIM) and the New Iteration Method (NIM) further highlights MADM’s computational efficiency and precision. These findings establish MADM as a robust and reliable tool for addressing complex PDEs across various scientific domains.