Fractional Calculus and Fractional Differential Equations: Trends in Mathematics

Fractional calculus and fractional differential equations have emerged as a fascinating and rapidly growing field of research in mathematics, offering a powerful tool to study complex and non-integer order phenomena in various scientific and engineering applications. This article aims to provide an overview of this intriguing subject, highlighting its key concepts, recent trends, and potential applications.

Fractional Calculus and Fractional Differential Equations (Trends in Mathematics)

by Nathan Van Coops

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Language	:	English
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Fractional Calculus: A Brief Overview

Fractional calculus is a generalization of classical calculus that deals with functions with non-integer orders of derivatives and integrals. It was first introduced by Leonhard Euler in 1736 and has since gained significant attention due to its ability to model real-world phenomena more accurately than traditional integer-order calculus.

In fractional calculus, the fractional derivative of order $\alpha$ of a function $f(x)$ is defined as:

$$_aD_x^\alpha f(x) = \frac{d^{\alpha}}{dx^{\alpha}}f(x) = \frac{1}{\Gamma(n-\alpha)}\int_a^x (x-t)^{n-\alpha - 1}f^{(n)}(t) \, dt$$

where $n$ is the smallest integer greater than $\alpha$, $\Gamma(\cdot)$ is the Gamma function, and $a$ is the lower limit of integration.

Similarly, the fractional integral of order $\alpha$ is given by:

$$_a^{\ast}D_x^{-\alpha}f(x) = \frac{1}{\Gamma(\alpha)}\int_a^x (x-t)^{\alpha - 1}f(t) \, dt$$

Fractional Differential Equations

Fractional differential equations (FDEs) are differential equations that involve fractional derivatives or integrals. They have attracted significant interest due to their ability to capture complex behaviors and memory effects in various physical, biological, and engineering systems.

FDEs take the general form:

$$_aD_x^{\alpha}u(x) = f(x, u(x),D^{\beta}u(x),\dots)$$

where $\alpha$ is the order of the fractional derivative, $u(x)$ is the unknown function, and $f$ is a given function.

Recent Trends in Fractional Calculus and FDEs

In recent years, fractional calculus and FDEs have witnessed a surge of research activities, leading to significant advancements in both theoretical foundations and applications. Some of the key trends include:

1. Novel Fractional Operators

Researchers have proposed novel fractional operators beyond the classical Riemann-Liouville and Caputo derivatives, such as the Erdelyi-Kober fractional derivative, the Katugampola fractional derivative, and the Atangana-Baleanu fractional derivative. These operators offer different advantages and have been found to be more suitable for certain applications.

2. Fractional Order Control Theory

Fractional calculus has been applied to develop fractional order control systems, which exhibit better performance and robustness compared to integer-order systems. This field has applications in various areas, including robotics, automatic control, and process control.

3. Numerical Methods for FDEs

Solving FDEs analytically can be challenging, hence efficient numerical methods have been developed. These methods include finite difference schemes, spectral methods, and fractional Adams-Bashforth-Moulton methods.

4. Applications in Science and Engineering

FDEs have found applications in diverse fields such as:

- Fluid mechanics: Modeling non-Newtonian fluids and anomalous diffusion - Electrochemistry: Describing electrochemical processes and battery dynamics - Biology: Studying biological rhythms, circadian clocks, and epidemics - Finance: Modeling financial market fluctuations and risk assessment

Fractional calculus and fractional differential equations have emerged as powerful tools for studying and modeling complex phenomena in various fields of mathematics and science. Recent trends in this domain include the development of novel fractional operators, applications in fractional order control theory, the advancement of numerical methods for FDEs, and a wide range of applications in diverse scientific and engineering disciplines. As research continues, we can expect further advancements and breakthroughs in this exciting and rapidly growing field.

Fractional Calculus and Fractional Differential Equations (Trends in Mathematics)

by Nathan Van Coops

5 out of 5

Language	:	English
File size	:	5815 KB
Screen Reader	:	Supported
Print length	:	191 pages

FREE