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
In recent years, the number of sensor and actuator nodes in the Internet of Things (IoT) networks has increased, generating a large amount of data. Most research techniques are based on dividing target data into subsets. On a large scale, this volume increases exponentially, which will affect search algorithms. This problem is caused by the inherent deficiencies of space partitioning. This paper introduces a new and efficient indexing structure to index massive IoT data called BCCF‐tree (Binary tree based on containers at the cloud‐fog computing level). This structure is based on recursive partitioning of space using the k‐means clustering algorithm to effectively separate space into nonoverlapping subspace to improve the quality of search and discovery algorithm results. A good topology should avoid a biased allocation of objects for separable sets and should not influence the structure of the index. BCCF‐tree structure benefits to the emerging cloud‐fog computing system, which represents the most powerful real‐time processing capacity provided by fog computing due to its proximity to sensors and the largest storage capacity provided by cloud computing. The paper also discusses the effectiveness of construction and search algorithms, as well as the quality of the index compared to other recent indexing data structures. The experimental results showed good performance.
In the present paper, we consider large-scale continuous-time differential matrix Riccati equations. To the authors' knowledge, the two main approaches proposed in the litterature are based on a splitting scheme or on a Rosenbrock / Backward Differentiation Formula (BDF) methods. The approach we propose is based on the reduction of the problem dimension prior to integration. We project the initial problem onto an extended block Krylov subspace and obtain a low-dimensional differential matrix Riccati equation. The latter matrix differential problem is then solved by a Backward Differentiation Formula (BDF) method and the obtained solution is used to reconstruct an approximate solution of the original problem. This process is repeated, increasing the dimension of the projection subspace until achieving a chosen accuracy. We give some theoretical results and a simple expression of the residual allowing the implementation of a stop test in order to limit the dimension of the projection space. Some numerical experiments will be given.
In this paper, the homotopy analysis method is applied to obtain the solution of nonlinear fractional partial differential equations. The method has been successively provided for finding approximate analytical solutions of the fractional nonlinear Klein-Gordon equation. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter . The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.
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