Approximate Analytical Approaches to Nonlinear Differential Equations: A Review of Perturbation, Decomposition and Coefficient Methods in Engineering

This paper reviews modern approximate analytical methods for solving symmetric and non-symmetric dynamical problems, including the Perturbation Method using the Green function, the Regular Perturbation Method, the Adomian Decomposition Method, the Undetermined Coefficient Method, the Poincaré-Lindstedt Method, and Multiple-Scale Analysis. The applicability of each method is assessed based on its purpose, constraints, mathematical domain, and accessibility. Example applications demonstrate the solution process and the effectiveness of each method, with analytical solutions verified against numerical results for accuracy and stability. A comparison of the advantages, disadvantages, and suitable applications is presented in tabular form to aid in selecting the appropriate method for specific problems. Finally, this evaluation highlights future trends and potential applications in engineering and applied sciences.