An Introduction To Fractional Calculus
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Introduction To The Fractional Calculus Of Variations
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Author : Agnieszka B Malinowska
language : en
Publisher: World Scientific Publishing Company
Release Date : 2012-09-14
Introduction To The Fractional Calculus Of Variations written by Agnieszka B Malinowska and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-09-14 with Mathematics categories.
This invaluable book provides a broad introduction to the fascinating and beautiful subject of Fractional Calculus of Variations (FCV). In 1996, FVC evolved in order to better describe non-conservative systems in mechanics. The inclusion of non-conservatism is extremely important from the point of view of applications. Forces that do not store energy are always present in real systems. They remove energy from the systems and, as a consequence, Noether's conservation laws cease to be valid. However, it is still possible to obtain the validity of Noether's principle using FCV. The new theory provides a more realistic approach to physics, allowing us to consider non-conservative systems in a natural way. The authors prove the necessary Euler–Lagrange conditions and corresponding Noether theorems for several types of fractional variational problems, with and without constraints, using Lagrangian and Hamiltonian formalisms. Sufficient optimality conditions are also obtained under convexity, and Leitmann's direct method is discussed within the framework of FCV. The book is self-contained and unified in presentation. It may be used as an advanced textbook by graduate students and ambitious undergraduates in mathematics and mechanics. It provides an opportunity for an introduction to FCV for experienced researchers. The explanations in the book are detailed, in order to capture the interest of the curious reader, and the book provides the necessary background material required to go further into the subject and explore the rich research literature.
Fractional Calculus
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Author : Herrmann Richard
language : en
Publisher: World Scientific
Release Date : 2001-02-12
Fractional Calculus written by Herrmann Richard and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-02-12 with Science categories.
The book presents a concise introduction to the basic methods and strategies in fractional calculus which enables the reader to catch up with the state-of-the-art in this field and to participate and contribute in the development of this exciting research area. This book is devoted to the application of fractional calculus on physical problems. The fractional concept is applied to subjects in classical mechanics, image processing, folded potentials in cluster physics, infrared spectroscopy, group theory, quantum mechanics, nuclear physics, hadron spectroscopy up to quantum field theory and will surprise the reader with new intriguing insights. This new, extended edition includes additional chapters about numerical solution of the fractional Schrödinger equation, self-similarity and the geometric interpretation of non-isotropic fractional differential operators. Motivated by the positive response, new exercises with elaborated solutions are added, which significantly support a deeper understanding of the general aspects of the theory. Besides students as well as researchers in this field, this book will also be useful as a supporting medium for teachers teaching courses devoted to this subject.
Fractional Differential Equations
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Author : Igor Podlubny
language : en
Publisher: Elsevier
Release Date : 1998-10-27
Fractional Differential Equations written by Igor Podlubny and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-10-27 with Mathematics categories.
This book is a landmark title in the continuous move from integer to non-integer in mathematics: from integer numbers to real numbers, from factorials to the gamma function, from integer-order models to models of an arbitrary order. For historical reasons, the word 'fractional' is used instead of the word 'arbitrary'. This book is written for readers who are new to the fields of fractional derivatives and fractional-order mathematical models, and feel that they need them for developing more adequate mathematical models. In this book, not only applied scientists, but also pure mathematicians will find fresh motivation for developing new methods and approaches in their fields of research. A reader will find in this book everything necessary for the initial study and immediate application of fractional derivatives fractional differential equations, including several necessary special functions, basic theory of fractional differentiation, uniqueness and existence theorems, analytical numerical methods of solution of fractional differential equations, and many inspiring examples of applications. A unique survey of many applications of fractional calculus Presents basic theory Includes a unified presentation of selected classical results, which are important for applications Provides many examples Contains a separate chapter of fractional order control systems, which opens new perspectives in control theory The first systematic consideration of Caputo's fractional derivative in comparison with other selected approaches Includes tables of fractional derivatives, which can be used for evaluation of all considered types of fractional derivatives
Introduction To Fractional Differential Equations
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Author : Constantin Milici
language : en
Publisher: Springer
Release Date : 2018-10-28
Introduction To Fractional Differential Equations written by Constantin Milici and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-10-28 with Technology & Engineering categories.
This book introduces a series of problems and methods insufficiently discussed in the field of Fractional Calculus – a major, emerging tool relevant to all areas of scientific inquiry. The authors present examples based on symbolic computation, written in Maple and Mathematica, and address both mathematical and computational areas in the context of mathematical modeling and the generalization of classical integer-order methods. Distinct from most books, the present volume fills the gap between mathematics and computer fields, and the transition from integer- to fractional-order methods.
An Introduction To The Fractional Calculus And Fractional Differential Equations
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Author : Kenneth S. Miller
language : en
Publisher: Wiley-Interscience
Release Date : 1993-06-02
An Introduction To The Fractional Calculus And Fractional Differential Equations written by Kenneth S. Miller and has been published by Wiley-Interscience this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-06-02 with Mathematics categories.
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.
An Introduction To Fractional Calculus
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Author : A. M. Mathai
language : en
Publisher: Nova Science Publishers
Release Date : 2017
An Introduction To Fractional Calculus written by A. M. Mathai and has been published by Nova Science Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with Fractional calculus categories.
This is a modified version of Module 10 of the Centre for Mathematical and Statistical Sciences (CMSS). CMSS modules are notes prepared on various topics with many examples from real-life situations and exercises so that the subject matter becomes interesting to students. These modules are used for undergraduate level courses and graduate level training in various topics at CMSS. Aside from Module 8, these modules were developed by Dr A M Mathai, Director of CMSS and Emeritus Professor of Mathematics and Statistics, McGill University, Canada. Module 8 is based on the lecture notes of Professor W J Anderson of McGill University, developed for his undergraduate course (Mathematics 447). Professor Dr Hans J Haubold has been a research collaborator of Dr A M Mathais since 1984, mainly in the areas of astrophysics, special functions and statistical distribution theory. He is also a lifetime member of CMSS and a Professor at CMSS. A large number of papers have been published jointly in these areas since 1984. The following monographs and books have been brought out in conjunction with this joint research: Modern Problems in Nuclear and Neutrino Astrophysics (A M Mathai and H J Haubold, 1988, Akademie-Verlag, Berlin); Special Functions for Applied Scientists (A MMathai and H J Haubold, 2008, Springer, New York); and The H-Function: Theory and Applications (A M Mathai, R K Saxena and H J Haubold, 2010, Springer, New York). These CMSS modules are printed at CMSS Press and published by CMSS. Copies are made available to students free of charge, and to researchers and others at production cost. For the preparation of the initial drafts of all these modules, financial assistance was made available from the Department of Science and Technology, the Government of India (DST), New Delhi under project number SR/S4/MS:287/05. Hence, the authors would like to express their thanks and gratitude to DST, the Government of India, for its financial assistance.
Applications Of Fractional Calculus In Physics
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Author : Rudolf Hilfer
language : en
Publisher: World Scientific
Release Date : 2000-03-02
Applications Of Fractional Calculus In Physics written by Rudolf Hilfer and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-03-02 with Science categories.
Fractional calculus is a collection of relatively little-known mathematical results concerning generalizations of differentiation and integration to noninteger orders. While these results have been accumulated over centuries in various branches of mathematics, they have until recently found little appreciation or application in physics and other mathematically oriented sciences. This situation is beginning to change, and there are now a growing number of research areas in physics which employ fractional calculus.This volume provides an introduction to fractional calculus for physicists, and collects easily accessible review articles surveying those areas of physics in which applications of fractional calculus have recently become prominent.
Fractional Calculus
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Author : Varsha Daftardar-Gejji
language : en
Publisher: ALPHA SCIENCE INTERNATIONAL LIMITED
Release Date : 2013-07-26
Fractional Calculus written by Varsha Daftardar-Gejji and has been published by ALPHA SCIENCE INTERNATIONAL LIMITED this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-07-26 with Mathematics categories.
FRACTIONAL CALCULUS: Theory and Applications deals with differentiation and integration of arbitrary order. The origin of this subject can be traced back to the end of seventeenth century, the time when Newton and Leibniz developed foundations of differential and integral calculus. Nonetheless, utility and applicability of FC to various branches of science and engineering have been realized only in last few decades. Recent years have witnessed tremendous upsurge in research activities related to the applications of FC in modeling of real-world systems. Unlike the derivatives of integral order, the non-local nature of fractional derivatives correctly models many natural phenomena containing long memory and give more accurate description than their integer counterparts.The present book comprises of contributions from academicians and leading researchers and gives a panoramic overview of various aspects of this subject: Introduction to Fractional Calculus Fractional Differential Equations Fractional Ordered Dynamical Systems Fractional Operators on Fractals Local Fractional Derivatives Fractional Control Systems Fractional Operators and Statistical Distributions Applications to Engineering
Functional Fractional Calculus
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Author : Shantanu Das
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-06-01
Functional Fractional Calculus written by Shantanu Das and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-06-01 with Technology & Engineering categories.
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.
An Introduction To Fractional Differential Equations
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Author : K. Balachandran
language : en
Publisher: Springer Nature
Release Date : 2023-12-25
An Introduction To Fractional Differential Equations written by K. Balachandran and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-12-25 with Mathematics categories.
This is an introductory-level text on fractional calculus and fractional differential equations. Targeted to graduate students of mathematics and researchers, it contains several new definitions of fractional integrals and fractional derivatives. With interesting applications of the subject in several areas of physical sciences, life sciences, engineering, and technology, the book helps the students understand the importance and developments of this topic. The book is enriched with a list of useful references to published literature, and the presentation of the book is entirely new and easily comprehensible to the students. Some of the topics are refined, and new examples are included to supplement theories to help students understand the concepts easily and clearly.