Fractional Partial Differential Equations

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【簡介】 This monograph offers a comprehensive exposition of the theory surrounding time-fractional partial differential equations, featuring recent advancements in fundamental techniques and results. The topics covered encompass crucial aspects of the theory, such as well-posedness, regularity, approximation, and optimal control. The book delves into the intricacies of fractional Navier–Stokes equations, fractional Rayleigh–Stokes equations, fractional Fokker–Planck equations, and fractional Schrödinger equations, providing a thorough exploration of these subjects. Numerous real-world applications associated with these equations are meticulously examined, enhancing the practical relevance of the presented concepts. The content in this monograph is based on the research works carried out by the author and other excellent experts during the past five years. Rooted in the latest advancements, it not only serves as a valuable resource for understanding the theoretical foundations but also lays the groundwork for delving deeper into the subject and navigating the extensive research landscape. Geared towards researchers, graduate students, and PhD scholars specializing in differential equations, applied analysis, and related research domains, this monograph facilitates a nuanced understanding of time-fractional partial differential equations and their broader implications. 【目錄】

FRACTIONAL DISCRETE CHAOS

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In the nineteenth-century, fractional calculus had its origin in extending differentiation and integration operators from the integer-order case to the fractional-order case. Discrete fractional calculus has recently become an important research topic, useful in various science and engineering applications. The first definition of the fractional-order discrete-time/difference operator was introduced in 1974 by Diaz and Osler, where such operator was derived by discretizing the fractional-order continuous-time operator. Successfully, several types of fractional-order difference operators have then been proposed and introduced through further generalizing numerous classical operators, motivating several researchers to publish extensively on a new class of systems, viz the nonlinear fractional-order discrete-time systems (or simply, the fractional-order maps), and their chaotic behaviors. This discovery of chaos in such maps, has led to novel control methods for effectively stabilizing their chaotic dynamics. The aims of this book are as follows: Presenting the recent developments, trends, research solutions, applications and open problems related to fractional-order chaotic maps; Illustrating many interdisciplinary applications, like modulization, control, circuits, security and encryption; Including all theories associated with chaos, control and synchronization of discrete-time systems; Providing a useful reference on the topic of fractional-order chaotic maps and their applications. Sample Chapter(s) Preface Chapter 1: Discrete Fractional Calculus Contents: Discrete Fractional Calculus Chaotic Methods and Tests Chaos in 2D Discrete Fractional Systems Chaos in 3D Discrete Fractional Systems Applications of Fractional Chaotic Maps Readership: Graduate students and researchers in chaos, applied mathematics, computer science and engineering.