Advancing the Modeling of Complex Systems through Fractional Differential Equations: Analytical Insights and Computational Innovations

Abstract

The advancement of Fractional Differential Equations (FDEs) has transformed the modeling of complex systems by providing a mathematical framework capable of representing memory effects, non-local interactions, and multiscale dependencies inherent in natural and engineered processes. Unlike traditional integer-order models, FDEs generalize differentiation and integration to non-integer orders, allowing for more realistic and flexible descriptions of dynamical systems exhibiting anomalous diffusion, hereditary behavior, and power-law relaxation. This study presents a comprehensive exploration of analytical insights and computational innovations that enhance the applicability and efficiency of FDE-based modeling. It examines recent developments in fractional operators, including Caputo–Fabrizio and Atangana–Baleanu formulations, and evaluates their impact on the stability and physical interpretability of dynamic systems. Furthermore, advanced computational techniques—such as spectral, wavelet, and hybrid Radial Basis Function (RBF) approaches—are assessed for their ability to achieve high accuracy with reduced computational complexity. The integration of fractional calculus with data-driven methods and machine learning is also discussed as an emerging direction for adaptive and predictive modeling. Collectively, these developments establish FDEs as a foundational tool in understanding and simulating nonlinear, multiscale, and memory-dependent phenomena across scientific and engineering disciplines.