A Comprehensive Review of Fractional Differential Equations and Their Role in Modeling Complex Dynamical Systems

Abstract

Fractional Differential Equations (FDEs) have emerged as a powerful mathematical framework for modeling complex dynamical systems that exhibit memory effects, non-local interactions, and anomalous transport phenomena. Unlike traditional integer-order models, which often oversimplify real-world behaviors, FDEs extend differentiation and integration to non-integer orders, providing an elegant means of describing hereditary and multiscale processes. This review paper presents a comprehensive analysis of the theoretical foundations, computational approaches, and interdisciplinary applications of fractional differential equations in the modeling of complex systems. It discusses the evolution of fractional calculus from its classical roots to modern formulations, including the Riemann–Liouville, Caputo, and Caputo–Fabrizio derivatives, emphasizing their respective strengths in physical interpretation and mathematical tractability. Additionally, the paper evaluates advanced numerical techniques—such as spectral, wavelet, and hybrid methods—that have enhanced the efficiency and scalability of FDE solutions. Applications spanning physics, biology, engineering, and finance are examined to illustrate the versatility of fractional models in capturing real-world dynamics with higher precision. The review also highlights ongoing challenges, including parameter estimation, computational complexity, and model validation, while identifying promising future directions such as data-driven fractional modeling and machine learning integration. Overall, this study underscores the pivotal role of FDEs in bridging theory and computation for the accurate representation of complex dynamical phenomena.