Fractional Calculus and Its Applications in Applied Mathematics and Other Sciences

Abstract

The subject of fractional calculus has applications in diverse and widespread fields of engineering and science such as electromagnetics, viscoelasticity, fluid mechanics, electrochemistry, biological population models, optics, and signals processing. It has been used to model physical and engineering processes that are found to be best described by fractional differential equations. The fractional derivative models are used for accurate modelling of those systems that require accurate modelling of damping. In these fields, various analytical and numerical methods including their applications to new problems have been proposed in recent years. This special issue on "Fractional Calculus and its Applications in Applied Mathematics and Other Sciences" is devoted to study the recent works in the above fields of fractional calculus done by the leading researchers. The papers for this special issue were selected after a careful and studious peer-review process.Mathematical modelling of real-life problems usually results in fractional differential equations and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one or more variables. In addition, most physical phenomena of fluid dynamics, quantum mechanics, electricity, ecological systems, and many other models are controlled within their domain of validity by fractional order PDEs. Therefore, it becomes increasingly important to be familiar with all traditional and recently developed methods for solving fractional order PDEs and the implementations of these methods.The aim of this special issue is to bring together the leading researchers of diverse fields of engineering including applied mathematicians and allow them to share their innovative research work. Analytical and numerical methods with advanced mathematical modelling and recent developments of differential and integral equations of arbitrary order arising in physical systems are included in the main focus of the issue.Accordingly, various papers on fractional differential equations have been included in this special issue after completing a heedful, rigorous, and peer-review process. The issue contains eight research papers. The issue of robust stability for fractional order Hopfield neural networks with parameter uncertainties is rigorously investigated. Based on the fractional order Lyapunov direct method, the sufficient condition of the existence, uniqueness, and globally robust stability of the equilibrium point is presented. Moreover, the sufficient condition of the robust synchronization between such neural systems with the same parameter uncertainties is proposed owing to the robust stability analysis of its synchronization error system. In addition, for different Hindawi Publishing Corporation

How Memory Regulates Drug Resistant Pathogenic Bacteria? A Mathematical Study

“…A similar mode of presentation of fractional order derivatives to model viscoelastic materials, genetics related modelling and the power law stress-strain relationship for modeling of biomaterials was reviewed by Magin et al [70]. The subject of fractional calculus had been applied in diverse applications in various fields of engineering and science such as electromagnetics, viscoelasticity, fluid mechanics, electrochemistry, biological population models, optics, and signals processing [71]. Here we, introduced the term alpha as the index of memory operator to derive the contribution of the history dependent response to release of the amount of beta lactam drug hydrolyzing enzyme beta lactamase which is realistic to our assumptions of model development process.…”

The Fractional View Analysis of Polytropic Gas, Unsteady Flow System

“…In recent years, nonlinear fractional partial differential equations (FPDEs) have attracted researchers because of their useful applications in science and engineering [1][2][3]. e analysis of exact solutions to these nonlinear PDEs plays a very significant role in the Soliton theory since much of the information are provided on the description of the physical models, in the transmission of electrical signals, as a standard diffusion-wave equation, the transfer of neutrons by nuclear reactor, the theory of random walks, and so on [4][5][6][7][8][9][10][11][12][13][14].…”

Unraveling the combined actions of a Holling type III predator–prey model incorporating Allee response and memory effects

“…Entire population keeps in mind where food is available as well as where they can get attacked. To include previous effects in biological models, currently authors are using fractional order prey–predator models 6‐8 …”