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Fractional Calculus
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Book Synopsis Generalized Fractional Calculus and Applications by : Virginia S Kiryakova
Download or read book Generalized Fractional Calculus and Applications written by Virginia S Kiryakova and published by CRC Press. This book was released on 1993-12-27 with total page 412 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this volume various applications are discussed, in particular to the hyper-Bessel differential operators and equations, Dzrbashjan-Gelfond-Leontiev operators and Borel type transforms, convolutions, new representations of hypergeometric functions, solutions to classes of differential and integral equations, transmutation method, and generalized integral transforms. Some open problems are also posed. This book is intended for graduate and post-graduate students, lecturers, researchers and others working in applied mathematical analysis, mathematical physics and related disciplines.
Book Synopsis Advances in Fractional Calculus by : J. Sabatier
Download or read book Advances in Fractional Calculus written by J. Sabatier and published by Springer Science & Business Media. This book was released on 2007-07-28 with total page 550 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Generalized Fractional Calculus by : George A. Anastassiou
Download or read book Generalized Fractional Calculus written by George A. Anastassiou and published by Springer Nature. This book was released on 2020-11-25 with total page 501 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book applies generalized fractional differentiation techniques of Caputo, Canavati and Conformable types to a great variety of integral inequalities e.g. of Ostrowski and Opial types, etc. Some of these are extended to Banach space valued functions. These inequalities have also great impact in numerical analysis, stochastics and fractional differential equations. The book continues with generalized fractional approximations by positive sublinear operators which derive from the presented Korovkin type inequalities and also includes abstract cases. It presents also multivariate complex Korovkin quantitative approximation theory. It follows M-fractional integral inequalities of Ostrowski and Polya types. The results are weighted so they provide a great variety of cases and applications. The second part of the book deals with the quantitative fractional Korovkin type approximation of stochastic processes and lays there the foundations of stochastic fractional calculus. The book considers both Caputo and Conformable fractional directions and derives regular and trigonometric results. The positive linear operators can be expectation operator commutative or not. This book results are expected to find applications in many areas of pure and applied mathematics and stochastics. As such this monograph is suitable for researchers, graduate students, and seminars of the above disciplines, also to be in all science and engineering libraries.
Book Synopsis An Introduction to the Fractional Calculus and Fractional Differential Equations by : Kenneth S. Miller
Download or read book An Introduction to the Fractional Calculus and Fractional Differential Equations written by Kenneth S. Miller and published by Wiley-Interscience. This book was released on 1993-06-02 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: Commences with the historical development of fractional calculus, its mathematical theory—particularly the Riemann-Liouville version. Numerous examples and theoretical applications of the theory are presented. Features topics associated with fractional differential equations. Discusses Weyl fractional calculus and some of its uses. Includes selected physical problems which lead to fractional differential or integral equations.
Book Synopsis Discrete Fractional Calculus by : Christopher Goodrich
Download or read book Discrete Fractional Calculus written by Christopher Goodrich and published by Springer. This book was released on 2016-02-09 with total page 556 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Functional Fractional Calculus by : Shantanu Das
Download or read book Functional Fractional Calculus written by Shantanu Das and published by Springer Science & Business Media. This book was released on 2011-06-01 with total page 612 pages. Available in PDF, EPUB and Kindle. Book excerpt: When a new extraordinary and outstanding theory is stated, it has to face criticism and skeptism, because it is beyond the usual concept. The fractional calculus though not new, was not discussed or developed for a long time, particularly for lack of its application to real life problems. It is extraordinary because it does not deal with ‘ordinary’ differential calculus. It is outstanding because it can now be applied to situations where existing theories fail to give satisfactory results. In this book not only mathematical abstractions are discussed in a lucid manner, with physical mathematical and geometrical explanations, but also several practical applications are given particularly for system identification, description and then efficient controls. The normal physical laws like, transport theory, electrodynamics, equation of motions, elasticity, viscosity, and several others of are based on ‘ordinary’ calculus. In this book these physical laws are generalized in fractional calculus contexts; taking, heterogeneity effect in transport background, the space having traps or islands, irregular distribution of charges, non-ideal spring with mass connected to a pointless-mass ball, material behaving with viscous as well as elastic properties, system relaxation with and without memory, physics of random delay in computer network; and several others; mapping the reality of nature closely. The concept of fractional and complex order differentiation and integration are elaborated mathematically, physically and geometrically with examples. The practical utility of local fractional differentiation for enhancing the character of singularity at phase transition or characterizing the irregularity measure of response function is deliberated. Practical results of viscoelastic experiments, fractional order controls experiments, design of fractional controller and practical circuit synthesis for fractional order elements are elaborated in this book. The book also maps theory of classical integer order differential equations to fractional calculus contexts, and deals in details with conflicting and demanding initialization issues, required in classical techniques. The book presents a modern approach to solve the ‘solvable’ system of fractional and other differential equations, linear, non-linear; without perturbation or transformations, but by applying physical principle of action-and-opposite-reaction, giving ‘approximately exact’ series solutions. Historically, Sir Isaac Newton and Gottfried Wihelm Leibniz independently discovered calculus in the middle of the 17th century. In recognition to this remarkable discovery, J.von Neumann remarked, “...the calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more equivocally than anything else the inception of modern mathematical analysis which is logical development, still constitute the greatest technical advance in exact thinking.” This XXI century has thus started to ‘think-exactly’ for advancement in science & technology by growing application of fractional calculus, and this century has started speaking the language which nature understands the best.
Book Synopsis Fractional Calculus and Its Applications by : B. Ross
Download or read book Fractional Calculus and Its Applications written by B. Ross and published by Springer. This book was released on 2006-11-15 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Variable-Order Fractional Calculus of Variations by : Ricardo Almeida
Download or read book The Variable-Order Fractional Calculus of Variations written by Ricardo Almeida and published by Springer. This book was released on 2018-06-29 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Variable-Order Fractional Calculus of Variations is devoted to the study of fractional operators with variable order and, in particular, variational problems involving variable-order operators. This brief presents a new numerical tool for the solution of differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one, an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. The contributors consider variational problems that may be subject to one or more constraints, where the functional depends on a combined Caputo derivative of variable fractional order. In particular, they establish necessary optimality conditions of Euler–Lagrange type. As the terminal point in the cost integral is free, as is the terminal state, transversality conditions are also obtained. The Variable-Order Fractional Calculus of Variations is a valuable source of information for researchers in mathematics, physics, engineering, control and optimization; it provides both analytical and numerical methods to deal with variational problems. It is also of interest to academics and postgraduates in these fields, as it solves multiple variational problems subject to one or more constraints in a single brief.
Book Synopsis Stochastic Models for Fractional Calculus by : Mark M. Meerschaert
Download or read book Stochastic Models for Fractional Calculus written by Mark M. Meerschaert and published by Walter de Gruyter GmbH & Co KG. This book was released on 2019-10-21 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fractional calculus is a rapidly growing field of research, at the interface between probability, differential equations, and mathematical physics. It is used to model anomalous diffusion, in which a cloud of particles spreads in a different manner than traditional diffusion. This monograph develops the basic theory of fractional calculus and anomalous diffusion, from the point of view of probability. In this book, we will see how fractional calculus and anomalous diffusion can be understood at a deep and intuitive level, using ideas from probability. It covers basic limit theorems for random variables and random vectors with heavy tails. This includes regular variation, triangular arrays, infinitely divisible laws, random walks, and stochastic process convergence in the Skorokhod topology. The basic ideas of fractional calculus and anomalous diffusion are closely connected with heavy tail limit theorems. Heavy tails are applied in finance, insurance, physics, geophysics, cell biology, ecology, medicine, and computer engineering. The goal of this book is to prepare graduate students in probability for research in the area of fractional calculus, anomalous diffusion, and heavy tails. Many interesting problems in this area remain open. This book will guide the motivated reader to understand the essential background needed to read and unerstand current research papers, and to gain the insights and techniques needed to begin making their own contributions to this rapidly growing field.
Book Synopsis Fractional Dynamics by : Vasily E. Tarasov
Download or read book Fractional Dynamics written by Vasily E. Tarasov and published by Springer Science & Business Media. This book was released on 2011-01-04 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media" presents applications of fractional calculus, integral and differential equations of non-integer orders in describing systems with long-time memory, non-local spatial and fractal properties. Mathematical models of fractal media and distributions, generalized dynamical systems and discrete maps, non-local statistical mechanics and kinetics, dynamics of open quantum systems, the hydrodynamics and electrodynamics of complex media with non-local properties and memory are considered. This book is intended to meet the needs of scientists and graduate students in physics, mechanics and applied mathematics who are interested in electrodynamics, statistical and condensed matter physics, quantum dynamics, complex media theories and kinetics, discrete maps and lattice models, and nonlinear dynamics and chaos. Dr. Vasily E. Tarasov is a Senior Research Associate at Nuclear Physics Institute of Moscow State University and an Associate Professor at Applied Mathematics and Physics Department of Moscow Aviation Institute.
Book Synopsis Fractional Calculus by : Richard Herrmann
Download or read book Fractional Calculus written by Richard Herrmann and published by World Scientific. This book was released on 2011 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fractional calculus is undergoing rapidly and ongoing development. We can already recognize, that within its framework new concepts and strategies emerge, which lead to new challenging insights and surprising correlations between different branches of physics. This book is an invitation both to the interested student and the professional researcher. It presents a thorough introduction to the basics of fractional calculus and guides the reader directly to the current state-of-the-art physical interpretation. It is also devoted to the application of fractional calculus on physical problems, in the subjects of classical mechanics, friction, damping, oscillations, group theory, quantum mechanics, nuclear physics, and hadron spectroscopy up to quantum field theory.
Book Synopsis Fractional Calculus in Medical and Health Science by : Devendra Kumar
Download or read book Fractional Calculus in Medical and Health Science written by Devendra Kumar and published by CRC Press. This book was released on 2020-07-09 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Fractional Calculus View of Complexity by : Bruce J. West
Download or read book Fractional Calculus View of Complexity written by Bruce J. West and published by CRC Press. This book was released on 2016-01-05 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is not a text devoted to a pedagogical presentation of a specialized topic nor is it a monograph focused on the author's area of research. It accomplishes both these things while providing a rationale for why the reader ought to be interested in learning about fractional calculus. This book is for researchers who has heard about many
Book Synopsis Functional Fractional Calculus for System Identification and Controls by : Shantanu Das
Download or read book Functional Fractional Calculus for System Identification and Controls written by Shantanu Das and published by Springer Science & Business Media. This book was released on 2007-09-26 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, not only are mathematical abstractions discussed in a lucid manner, but also several practical applications are given particularly for system identification, description and then efficient controls. The reader gets a feeling of the wide applicability of fractional calculus in the field of science and engineering. With this book, a starter can understand the concepts of this emerging field with a minimal effort and basic mathematics.
Book Synopsis Economic Dynamics with Memory by : Vasily E. Tarasov
Download or read book Economic Dynamics with Memory written by Vasily E. Tarasov and published by Walter de Gruyter GmbH & Co KG. This book was released on 2021-01-18 with total page 562 pages. Available in PDF, EPUB and Kindle. Book excerpt: The series is devoted to the publication of high-level monographs which cover progresses in fractional calculus research in mathematics and applications in physics, mechanics, engineering and biology etc. Methodological aspects e.g., theory, modeling and computational methods are presented from mathematical point of view, and emphases are placed in computer simulation, analysis, design and control of application-oriented issues in various scientific disciplines. It is designed for mathematicians, and researchers using fractional calculus as a tool in the field of physics, mechanics, engineering and biology. Contributions which are interdisciplinary and which stimulate further research at the crossroads of sciences and engineering are particularly welcomed. Editor-in-chief: Changpin Li, Shanghai University, China Editorial Board: Virginia Kiryakova, Bulgarian Academy of Sciences, Bulgaria Francesco Mainardi, University of Bologna, Italy Dragan Spasic, University of Novi Sad, Serbia Bruce Ian Henry, University of New South Wales, Australia YangQuan Chen, University of California, Merced, USA Please submit book proposals to Leonardo Milla, [email protected]
Book Synopsis The Fractional Calculus by : Keith B. Oldham
Download or read book The Fractional Calculus written by Keith B. Oldham and published by . This book was released on 1974 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Fractional Calculus and Fractional Differential Equations by : Varsha Daftardar-Gejji
Download or read book Fractional Calculus and Fractional Differential Equations written by Varsha Daftardar-Gejji and published by Springer. This book was released on 2019-08-10 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a broad overview of the latest developments in fractional calculus and fractional differential equations (FDEs) with an aim to motivate the readers to venture into these areas. It also presents original research describing the fractional operators of variable order, fractional-order delay differential equations, chaos and related phenomena in detail. Selected results on the stability of solutions of nonlinear dynamical systems of the non-commensurate fractional order have also been included. Furthermore, artificial neural network and fractional differential equations are elaborated on; and new transform methods (for example, Sumudu methods) and how they can be employed to solve fractional partial differential equations are discussed. The book covers the latest research on a variety of topics, including: comparison of various numerical methods for solving FDEs, the Adomian decomposition method and its applications to fractional versions of the classical Poisson processes, variable-order fractional operators, fractional variational principles, fractional delay differential equations, fractional-order dynamical systems and stability analysis, inequalities and comparison theorems in FDEs, artificial neural network approximation for fractional operators, and new transform methods for solving partial FDEs. Given its scope and level of detail, the book will be an invaluable asset for researchers working in these areas.
