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

Using the new derivative called beta-derivative, we modelled the well-known infectious disease called break-bone fever or the dengue fever. We presented the endemic equilibrium points under certain conditions of the physical parameters included in the model. We made use of an iteration method to solve the extended model. To show the efficiency of the method used, we have presented in detail the stability and the convergence of the method for solving the system (2). We presented the uniqueness of the special solution of system (2) and finally the numerical simulations were presented for various values of beta.

Human immunodeficiency virus infection destroys the body immune system, increases the risk of certain pathologies, damages body organs such as the brain, kidney, and heart, and causes death. Unfortunately, this infectious disease currently has no cure; however, there are effective retroviral drugs for improving the patients' health conditions but excessive use of these drugs is not without harmful side effects. This study presents a mathematical model with two control variables, where the uninfected CD4+T cells follow the logistic growth function and the incidence term is saturated with free virions. We use the efficacy of drug therapies to block the infection of new cells and prevent the production of new free virions. Our aim is to apply optimal control approach to maximize the concentration of uninfected CD4+T cells in the body by using minimum drug therapies. We establish the existence of an optimal control pair and use Pontryagin's principle to characterize the optimal levels of the two controls. The resulting optimality system is solved numerically to obtain the optimal control pair. Finally, we discuss the numerical simulation results which confirm the effectiveness of the model.

Measles is a higher contagious disease that can spread in a community population depending on the number of people (children) susceptible or infected and also depending on their movement in the community. In this paper we present a fractional SEIR metapopulation system modeling the spread of measles. We restrict ourselves to the dynamics between four distinct cities (patches). We prove that the fractional metapopulation model is well posed (nonnegative solutions) and we provide the condition for the stability of the disease-free equilibrium. Numerical simulations show that infection will be proportional to the size of population in each city, but the disease will die out. This is an expected result since it is well known for measles (Bartlett (1957)) that, in communities which generate insufficient new hosts, the disease will die out.

Existence of global solutions to continuous nonlocal convection-fragmentation equations is investigated in spaces of distributions with finite higher moments. Under the assumption that the velocity field is divergence-free, we make use of the method of characteristics and Friedrichs's lemma (Mizohata, 1973) to show that the transport operator generates a stochastic dynamical system. This allows for the use of substochastic methods and Kato-Voigt perturbation theorem (Banasiak and Arlotti, 2006) to ensure that the combined transport-fragmentation operator is the infinitesimal generator of a strongly continuous semigroup. In particular, we show that the solution represented by this semigroup is conservative.

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The medium through which the groundwater moves varies in time and space. The Hantush equation describes the movement of groundwater through a leaky aquifer. To include explicitly the deformation of the leaky aquifer into the mathematical formulation, we modify the equation by replacing the partial derivative with respect to time by the time-fractional variable order derivative. The modified equation is solved numerically via the Crank-Nicolson scheme. The stability and the convergence in this case are presented in details.

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