Discrete Fractional Calculus

Discrete Fractional Calculus PDF Author: Christopher Goodrich
Publisher: Springer
ISBN: 3319255622
Category : Mathematics
Languages : en
Pages : 556
Book Description
This text provides the first comprehensive treatment of the discrete fractional calculus. Experienced researchers will find the text useful as a reference for discrete fractional calculus and topics of current interest. Students who are interested in learning about discrete fractional calculus will find this text to provide a useful starting point. Several exercises are offered at the end of each chapter and select answers have been provided at the end of the book. The presentation of the content is designed to give ample flexibility for potential use in a myriad of courses and for independent study. The novel approach taken by the authors includes a simultaneous treatment of the fractional- and integer-order difference calculus (on a variety of time scales, including both the usual forward and backwards difference operators). The reader will acquire a solid foundation in the classical topics of the discrete calculus while being introduced to exciting recent developments, bringing them to the frontiers of the subject. Most chapters may be covered or omitted, depending upon the background of the student. For example, the text may be used as a primary reference in an introductory course for difference equations which also includes discrete fractional calculus. Chapters 1—2 provide a basic introduction to the delta calculus including fractional calculus on the set of integers. For courses where students already have background in elementary real analysis, Chapters 1—2 may be covered quickly and readers may then skip to Chapters 6—7 which present some basic results in fractional boundary value problems (FBVPs). Chapters 6—7 in conjunction with some of the current literature listed in the Bibliography can provide a basis for a seminar in the current theory of FBVPs. For a two-semester course, Chapters 1—5 may be covered in depth, providing a very thorough introduction to both the discrete fractional calculus as well as the integer-order calculus.

Discrete Fractional Calculus and Fractional Difference Equations

Discrete Fractional Calculus and Fractional Difference Equations PDF Author: Rui A. C. Ferreira
Publisher: Springer Nature
ISBN: 3030927245
Category :
Languages : en
Pages :
Book Description


Discrete Fractional Calculus

Discrete Fractional Calculus PDF Author: Piotr Ostalczyk
Publisher: World Scientific
ISBN: 9814725676
Category : Mathematics
Languages : en
Pages : 396
Book Description
The main subject of the monograph is the fractional calculus in the discrete version. The volume is divided into three main parts. Part one contains a theoretical introduction to the classical and fractional-order discrete calculus where the fundamental role is played by the backward difference and sum. In the second part, selected applications of the discrete fractional calculus in the discrete system control theory are presented. In the discrete system identification, analysis and synthesis, one can consider integer or fractional models based on the fractional-order difference equations. The third part of the book is devoted to digital image processing.

Computational Methods In The Fractional Calculus Of Variations

Computational Methods In The Fractional Calculus Of Variations PDF Author: Ricardo Almeida
Publisher: World Scientific Publishing Company
ISBN: 1783266422
Category : Mathematics
Languages : en
Pages : 280
Book Description
This book fills a gap in the literature by introducing numerical techniques to solve problems of fractional calculus of variations (FCV). In most cases, finding the analytic solution to such problems is extremely difficult or even impossible, and numerical methods need to be used.The authors are well-known researchers in the area of FCV and the book contains some of their recent results, serving as a companion volume to Introduction to the Fractional Calculus of Variations by A B Malinowska and D F M Torres, where analytical methods are presented to solve FCV problems. After some preliminaries on the subject, different techniques are presented in detail with numerous examples to help the reader to better understand the methods. The techniques presented may be used not only to deal with FCV problems but also in other contexts of fractional calculus, such as fractional differential equations and fractional optimal control. It is suitable as an advanced book for graduate students in mathematics, physics and engineering, as well as for researchers interested in fractional calculus.

New Trends in Nanotechnology and Fractional Calculus Applications

New Trends in Nanotechnology and Fractional Calculus Applications PDF Author: Dumitru Baleanu
Publisher: Springer Science & Business Media
ISBN: 9048132932
Category : Technology & Engineering
Languages : en
Pages : 531
Book Description
In recent years fractional calculus has played an important role in various fields such as mechanics, electricity, chemistry, biology, economics, modeling, identification, control theory and signal processing. The scope of this book is to present the state of the art in the study of fractional systems and the application of fractional differentiation. Furthermore, the manufacture of nanowires is important for the design of nanosensors and the development of high-yield thin films is vital in procuring clean solar energy. This wide range of applications is of interest to engineers, physicists and mathematicians.

Fractional Calculus in Medical and Health Science

Fractional Calculus in Medical and Health Science PDF Author: Devendra Kumar
Publisher: CRC Press
ISBN: 1000081850
Category : Technology & Engineering
Languages : en
Pages : 153
Book Description
This book covers applications of fractional calculus used for medical and health science. It offers a collection of research articles built into chapters on classical and modern dynamical systems formulated by fractional differential equations describing human diseases and how to control them. The mathematical results included in the book will be helpful to mathematicians and doctors by enabling them to explain real-life problems accurately. The book will also offer case studies of real-life situations with an emphasis on describing the mathematical results and showing how to apply the results to medical and health science, and at the same time highlighting modeling strategies. The book will be useful to graduate level students, educators and researchers interested in mathematics and medical science.

Progress on Difference Equations and Discrete Dynamical Systems

Progress on Difference Equations and Discrete Dynamical Systems PDF Author: Steve Baigent
Publisher: Springer Nature
ISBN: 3030601072
Category : Mathematics
Languages : en
Pages : 448
Book Description
This book comprises selected papers of the 25th International Conference on Difference Equations and Applications, ICDEA 2019, held at UCL, London, UK, in June 2019. The volume details the latest research on difference equations and discrete dynamical systems, and their application to areas such as biology, economics, and the social sciences. Some chapters have a tutorial style and cover the history and more recent developments for a particular topic, such as chaos, bifurcation theory, monotone dynamics, and global stability. Other chapters cover the latest personal research contributions of the author(s) in their particular area of expertise and range from the more technical articles on abstract systems to those that discuss the application of difference equations to real-world problems. The book is of interest to both Ph.D. students and researchers alike who wish to keep abreast of the latest developments in difference equations and discrete dynamical systems.

Fractional Calculus

Fractional Calculus PDF Author: Dumitru Baleanu
Publisher: World Scientific
ISBN: 9814355216
Category : Mathematics
Languages : en
Pages : 426
Book Description
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.

Robust Discrete-Time Flight Control of UAV with External Disturbances

Robust Discrete-Time Flight Control of UAV with External Disturbances PDF Author: Shuyi Shao
Publisher: Springer Nature
ISBN: 3030579573
Category : Technology & Engineering
Languages : en
Pages : 207
Book Description
This book studies selected discrete-time flight control schemes for fixed-wing unmanned aerial vehicle (UAV) systems in the presence of system uncertainties, external disturbances and input saturation. The main contributions of this book for UAV systems are as follows: (i) the proposed integer-order discrete-time control schemes are based on the designed discrete-time disturbance observers (DTDOs) and the neural network (NN); and (ii) the fractional-order discrete-time control schemes are developed by using the fractional-order calculus theory, the NN and the DTDOs. The book offers readers a good understanding of how to establish discrete-time tracking control schemes for fixed-wing UAV systems subject to system uncertainties, external wind disturbances and input saturation. It represents a valuable reference guide for academic research on uncertain UAV systems, and can also support advanced / Ph.D. studies on control theory and engineering.

Bifurcation and Chaos in Fractional-Order Systems

Bifurcation and Chaos in Fractional-Order Systems PDF Author: Marius-F. Danca
Publisher: MDPI
ISBN: 3036500928
Category : Technology & Engineering
Languages : en
Pages : 108
Book Description
This book presents a collection of seven technical papers on fractional-order complex systems, especially chaotic systems with hidden attractors and symmetries, in the research front of the field, which will be beneficial for scientific researchers, graduate students, and technical professionals to study and apply. It is also suitable for teaching lectures and for seminars to use as a reference on related topics.