The Rise of Fractional Calculus: Novel Applications in Engineering and Biological Systems - Prashant Roy (Recent Trends in Mathematics, 2025)
Fractional calculus (FC) is an advanced mathematical framework that generalizes the classical concepts of differentiation and integration to non-integer, or fractional, orders. This extension of traditional calculus allows for the modeling of complex dynamic systems that exhibit behavior not easily captured by integer-order differential equations. Over the last few decades, fractional calculus has seen a rapid rise in popularity, particularly in applied mathematics, engineering, and biological sciences, due to its ability to model systems with memory effects, hereditary properties, and non-local interactions. These features make fractional calculus an invaluable tool for understanding and describing complex, real-world phenomena that involve long-range dependencies, such as those found in mechanical systems, biological processes, and environmental systems.In the field of engineering, fractional calculus has been applied to improve the design of control systems, such as fractional-order controllers (FOCs), which offer enhanced performance over traditional integer-order controllers. It has also found applications in vibration analysis, structural dynamics, signal processing, and the modeling of viscoelastic materials. These applications benefit from the ability of fractional models to more accurately represent the dynamics of materials and systems that exhibit memory effects or complex, non-linear behaviors. In biological systems, fractional calculus has been used to model population dynamics, neural networks, and pharmacokinetics, where interactions over time or between different components exhibit non-local effects. This has led to more accurate predictions and better understanding of processes like disease spread, drug absorption, and neural signaling. This article provides a comprehensive review of fractional calculus, exploring its theoretical foundation, various solution techniques, and a wide array of novel applications. The paper emphasizes the growing importance of fractional calculus as a tool for both researchers and practitioners in mathematics, engineering, and biological sciences, highlighting its potential to address complex problems that traditional methods cannot adequately solve.
Keywords: Biological Systems Modeling; Fractional Calculus; Fractional differential equations; Non-local Interactions; control systems; viscoelasticity