An Iterative Fractional Operational Matrix Method for Solving Fractional Undamped Duffing Equation with High Nonlinearity

Abstract

This study presents a novel iterative fractional operational matrix method for solving the highly nonlinear fractional undamped Duffing equation. The proposed method efficiently approximates the solution by transforming the original fractional differential equation into a system of algebraic equations using modified operational matrices. The accuracy and effectiveness of this approach are validated through comparisons with established numerical methods and alternative analytical techniques from the literature. The results demonstrate that the proposed method provides a highly accurate approximation with rapid convergence. Furthermore, a rigorous convergence analysis is conducted to establish the existence and uniqueness of the solution. Notably, the study explores the impact of varying the fractional order  revealing its significant influence on the system's dynamic behavior. As the fractional order approaches unity, the fractional model converges to its classical counterpart, highlighting the role of fractional derivatives in capturing memory effects and hereditary properties in physical systems.