A Comprehensive Review of Fractional Calculus in Modeling Real-World Phenomena

Abstract

Fractional calculus, an extension of classical calculus to non-integer orders of differentiation and integration, has garnered significant attention for its remarkable ability to model complex real-world systems. Unlike traditional integer-order models that can oversimplify or neglect certain memory and hereditary properties, fractional calculus provides a robust framework to capture long-range dependence and fractal-like behavior across diverse disciplines. Recent advances in computation have further propelled the application of fractional derivatives and integrals in fields such as viscoelasticity, anomalous diffusion, signal processing, and control theory, among others.

In this paper, we present a comprehensive review of the theoretical underpinnings of fractional calculus, highlighting key definitions, properties, and numerical techniques. We also examine the broad array of real-world applications where fractional derivatives offer enhanced predictive accuracy. Moreover, this review compiles and critically discusses recent research data, offering insights into how fractional models outperform their integer-order counterparts in capturing the complexity of natural and engineered systems. By integrating both theoretical and applied perspectives, this work aims to provide researchers, educators, and industry professionals a thorough resource to understand the role of fractional calculus in addressing emerging challenges in science and engineering.

Central to this discussion is the incorporation of newly published findings and illustrative examples. These include empirical measurements, theoretical predictions, and graphical representations that demonstrate the efficacy of fractional-order models. Ultimately, this review underscores the growing consensus that fractional calculus is not merely a mathematical curiosity but a potent tool in developing more accurate and efficient models for real-world phenomena.