Fractional Calculus: A Modern Tool for Modeling Complex Systems

Abstract

An extension of classical calculus, fractional calculus has recently become a potent tool for the modelling of non-locality and memory effects complicated systems, which are common in both natural and artificial systems. In contrast to classical calculus, which is limited to integrals and derivatives of integer order, fractional calculus provides a more generalised and flexible framework for analysing dynamic behaviours by allowing integrals and derivatives of arbitrary (non-integer) order. how fractional calculus has evolved and what it can do in the realm of complex system modelling, especially in the scientific, mathematical, and economic domains. the theory behind integrals and fractional derivatives, demonstrating how these tools can capture anomalous diffusion, system memory effects, and long-range interactions that are hard to represent using more conventional methods. ways to represent viscoelastic materials, heat transmission, electrical circuits, and market dynamics using fractional differential equations, and the real-world issues that these methods solve. Further, we discuss the difficulties of fractional calculus, such as its computational complexity and numerical solutions, and we offer suggestions for where the field may go from here in terms of future studies to improve its theoretical and practical applications. The power and flexibility of fractional calculus to improve our comprehension and representation of complicated events are demonstrated by this all-encompassing review.