Theory And Numerical Approximations Of Fractional Integrals And Derivatives
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Theory and Numerical Approximations of Fractional Integrals and Derivatives
Author | : Changpin Li (Mathematics professor),Min Cai (Mathematician) |
Publsiher | : Unknown |
Total Pages | : 312 |
Release | : 2019 |
Genre | : Fractional calculus |
ISBN | : 1611975875 |
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"This book provides a comprehensive review of fractional calculus, covering both theory and numerical methods, and presents recent results on the subject"--
Theory and Numerical Approximations of Fractional Integrals and Derivatives
Author | : Changpin Li,Min Cai |
Publsiher | : SIAM |
Total Pages | : 326 |
Release | : 2019-10-31 |
Genre | : Mathematics |
ISBN | : 9781611975888 |
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Due to its ubiquity across a variety of fields in science and engineering, fractional calculus has gained momentum in industry and academia. While a number of books and papers introduce either fractional calculus or numerical approximations, no current literature provides a comprehensive collection of both topics. This monograph introduces fundamental information on fractional calculus, provides a detailed treatment of existing numerical approximations, and presents an inclusive review of fractional calculus in terms of theory and numerical methods and systematically examines almost all existing numerical approximations for fractional integrals and derivatives. The authors consider the relationship between the fractional Laplacian and the Riesz derivative, a key component absent from other related texts, and highlight recent developments, including their own research and results. The core audience spans several fractional communities, including those interested in fractional partial differential equations, the fractional Laplacian, and applied and computational mathematics. Advanced undergraduate and graduate students will find the material suitable as a primary or supplementary resource for their studies.
Fractional Calculus
Author | : Dumitru Baleanu |
Publsiher | : World Scientific |
Total Pages | : 426 |
Release | : 2012 |
Genre | : Mathematics |
ISBN | : 9789814355216 |
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The subject of fractional calculus and its applications (that is, convolution-type pseudo-differential operators including integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, mainly due to its applications in diverse fields of science and engineering. These operators have been used to model problems with anomalous dynamics, however, they also are an effective tool as filters and controllers, and they can be applied to write complicated functions in terms of fractional integrals or derivatives of elementary functions, and so on. This book will give readers the possibility of finding very important mathematical tools for working with fractional models and solving fractional differential equations, such as a generalization of Stirling numbers in the framework of fractional calculus and a set of efficient numerical methods. Moreover, we will introduce some applied topics, in particular fractional variational methods which are used in physics, engineering or economics. We will also discuss the relationship between semi-Markov continuous-time random walks and the space-time fractional diffusion equation, which generalizes the usual theory relating random walks to the diffusion equation. These methods can be applied in finance, to model tick-by-tick (log)-price fluctuations, in insurance theory, to study ruin, as well as in macroeconomics as prototypical growth models. All these topics are complementary to what is dealt with in existing books on fractional calculus and its applications. This book was written with a trade-off in mind between full mathematical rigor and the needs of readers coming from different applied areas of science and engineering. In particular, the numerical methods listed in the book are presented in a readily accessible way that immediately allows the readers to implement them on a computer in a programming language of their choice. Numerical code is also provided.
Theory of Fractional Engineering Vibrations
Author | : Ming Li |
Publsiher | : Walter de Gruyter GmbH & Co KG |
Total Pages | : 334 |
Release | : 2021-03-08 |
Genre | : Mathematics |
ISBN | : 9783110726152 |
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Vibration is important subject in many fields, ranging from mechanical engineering to electronic one. This book aims at giving a combination of conventional linear vibrations with recent fractional ones from a view of engineering. It consists of two parts. One is for conventional linear vibrations in Chapters 1 - 6 based on the authors lectures on the course of ship hull vibrations for undergraduates and postgraduates in Ocean College, Zhejiang University, China. The other, Chapters 7 - 15, contains his research in fractional vibrations. the book is suitable for researchers and graduate students in science and engieering. Preferred preliminaries are calculus, university physics, theoretic mechanics, and material mechanics for readers.
Fractional Differential Equations
Author | : Zhi-Zhong Sun,Guang-hua Gao |
Publsiher | : Walter de Gruyter GmbH & Co KG |
Total Pages | : 396 |
Release | : 2020-08-24 |
Genre | : Mathematics |
ISBN | : 9783110616064 |
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Starting with an introduction to fractional derivatives and numerical approximations, this book presents finite difference methods for fractional differential equations, including time-fractional sub-diffusion equations, time-fractional wave equations, and space-fractional differential equations, among others. Approximation methods for fractional derivatives are developed and approximate accuracies are analyzed in detail.
Advances in Fractional Calculus
Author | : J. Sabatier,O. P. Agrawal,J. A. Tenreiro Machado |
Publsiher | : Springer Science & Business Media |
Total Pages | : 550 |
Release | : 2007-07-28 |
Genre | : Technology & Engineering |
ISBN | : 9781402060427 |
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In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal and image processing. For example, in the last three fields, some important considerations such as modelling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability and robustness are now linked to long-range dependence phenomena. Similar progress has been made in other fields listed here. The scope of the book is thus to present the state of the art in the study of fractional systems and the application of fractional differentiation. As this volume covers recent applications of fractional calculus, it will be of interest to engineers, scientists, and applied mathematicians.
Fractional Partial Differential Equations and Their Numerical Solutions
Author | : Boling Guo,Xueke Pu,Fenghui Huang |
Publsiher | : World Scientific |
Total Pages | : 348 |
Release | : 2015-03-09 |
Genre | : Mathematics |
ISBN | : 9789814667067 |
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This book aims to introduce some new trends and results on the study of the fractional differential equations, and to provide a good understanding of this field to beginners who are interested in this field, which is the authors' beautiful hope. This book describes theoretical and numerical aspects of the fractional partial differential equations, including the authors' researches in this field, such as the fractional Nonlinear Schrödinger equations, fractional Landau–Lifshitz equations and fractional Ginzburg–Landau equations. It also covers enough fundamental knowledge on the fractional derivatives and fractional integrals, and enough background of the fractional PDEs. Contents:Physics BackgroundFractional Calculus and Fractional Differential EquationsFractional Partial Differential EquationsNumerical Approximations in Fractional CalculusNumerical Methods for the Fractional Ordinary Differential EquationsNumerical Methods for Fractional Partial Differential Equations Readership: Graduate students and researchers in mathematical physics, numerical analysis and computational mathematics. Key Features:This book covers the fundamentals of this field, especially for the beginnersThe book covers new trends and results in this fieldThe book covers numerical results, which will be of broad interests to researchersKeywords:Fractional Partial Differential Equations;Numerical Solutions
Fractional Vibrations with Applications to Euler Bernoulli Beams
Author | : Ming Li |
Publsiher | : CRC Press |
Total Pages | : 559 |
Release | : 2024-01-15 |
Genre | : Technology & Engineering |
ISBN | : 9781003802549 |
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The book examines vibration phenomena with an emphasis on fractional vibrations using the functional form of linear vibrations with frequency-dependent mass, damping, or stiffness, covering the theoretical analysis potentially applicable to structures and, in particular, ship hulls. Covering the six classes of fractional vibrators and seven classes of fractionally damped Euler-Bernoulli beams that play a major role in hull vibrations, this book presents analytical formulas of all results with concise expressions and elementary functions that set it apart from other recondite studies. The results show that equivalent mass or damping can be negative and depends on fractional orders. Other key highlights of the book include a concise mathematical explanation of the Rayleigh damping assumption, a novel description of the nonlinearity of fractional vibrations, and a new concept of fractional motion, offering exciting additions to the field of fractional vibrations. This title will be a must-read for students, mathematicians, physicists, and engineers interested in vibration phenomena and novel vibration performances, especially fractional vibrations.