Stability of fractional order systems

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled

The Wiener–Hopf technique, its generalizations and applications: constructive and approximate methods

From The Royal Society via Jisc Publications RouterHistory: received 2021-07-01, accepted 2021-09-10, pub-electronic 2021-10-20, pub-print 2021-10-27Article version: VoRPublication status: PublishedFunder: Royal Society; Id: http://dx.doi.org/10.13039/501100000288; Grant(s): Dorothy Hodgkin Research Fellowship, Wolfson Research Merit Award and Ser Cymru FutureFunder: Belarusian Republican Foundation for Fundamental Research; Id: http://dx.doi.org/10.13039/100007595; Grant(s): F20MS-083Funder: Engineering and Physical Sciences Research Council; Id: http://dx.doi.org/10.13039/501100000266; Grant(s): EP/R014604/1This paper reviews the modern state of the Wiener–Hopf factorization method and its generalizations. The main constructive results for matrix Wiener–Hopf problems are presented, approximate methods are outlined and the main areas of applications are mentioned. The aim of the paper is to offer an overview of the development of this method, and demonstrate the importance of bringing together pure and applied analysis to effectively employ the Wiener–Hopf technique

An Optimal Design Problem for Two-dimensional Composite Materials. A Constructive Approach

Светлана Ф. Макарук (Lebed) (Лебедь), В. Митюшев, С. В. Рогозин. Задача оптимального проектирования для двумерных композитных материалов. Конструктивный подходWe consider an optimal design problem, when it is necessary to locate circular inclusions of fixed size having a given conductivity in a material of fixed shape and a given conductivity in such a way that with prescribed concentration of the inclusions and prescribed external field the whole composite material has extremal conductivity in a given direction. The problem is reformulated in terms of a boundary value problem for analytic functions with unknown geometrical parameters and constraints. The boundary value problem in special cases is solved in closed form and the optimization problem is reduced to the classical problem of the extrema of a continuous function on a compact set

Mittag-Leffler functions, related topics and applications: theory and applications

As a result of researchers’ and scientists’ increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have recently caught the interest of the scientific community. Focusing on the theory of the Mittag-Leffler functions, the present volume offers a self-contained, comprehensive treatment, ranging from rather elementary matters to the latest research results. In addition to the theory the authors devote some sections of the work to the applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular the Mittag-Leffler functions allow us to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and its successors. The book is intended for a broad audience, comprising graduate students, university instructors and scientists in the field of pure and applied mathematics, as well as researchers in applied sciences like mathematical physics, theoretical chemistry, bio-mathematics, theory of control, and several other related areas

The Wiener–Hopf technique, its generalizations and applications: constructive and approximate methods

Peer reviewed: TrueThis paper reviews the modern state of the Wiener–Hopf factorization method and its generalizations. The main constructive results for matrix Wiener–Hopf problems are presented, approximate methods are outlined and the main areas of applications are mentioned. The aim of the paper is to offer an overview of the development of this method, and demonstrate the importance of bringing together pure and applied analysis to effectively employ the Wiener–Hopf technique

Mittag-Leffler Functions, Related Topics and ApplicationsTheory and Applications /

XIV, 443 p. 7 illus.online resource