Introduction To The Fractional Calculus Of Variations

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Introduction to the Fractional Calculus of Variations

Author : Agnieszka B Malinowska
Publisher : World Scientific Publishing Company
ISBN 13 : 184816968X
Total Pages : 292 pages
Book Rating : 4.8/5 (481 download)

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Book Synopsis Introduction to the Fractional Calculus of Variations by : Agnieszka B Malinowska

Download or read book Introduction to the Fractional Calculus of Variations written by Agnieszka B Malinowska and published by World Scientific Publishing Company. This book was released on 2012-09-14 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Computational Methods In The Fractional Calculus Of Variations

Author : Ricardo Almeida
Publisher : World Scientific Publishing Company
ISBN 13 : 1783266422
Total Pages : 280 pages
Book Rating : 4.7/5 (832 download)

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Book Synopsis Computational Methods In The Fractional Calculus Of Variations by : Ricardo Almeida

Download or read book Computational Methods In The Fractional Calculus Of Variations written by Ricardo Almeida and published by World Scientific Publishing Company. This book was released on 2015-03-19 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

The Variable-Order Fractional Calculus of Variations

Author : Ricardo Almeida
Publisher : Springer
ISBN 13 : 3319940066
Total Pages : 124 pages
Book Rating : 4.3/5 (199 download)

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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.

Fractional Calculus and Waves in Linear Viscoelasticity

Author : Francesco Mainardi
Publisher : World Scientific
ISBN 13 : 1848163290
Total Pages : 368 pages
Book Rating : 4.8/5 (481 download)

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Book Synopsis Fractional Calculus and Waves in Linear Viscoelasticity by : Francesco Mainardi

Download or read book Fractional Calculus and Waves in Linear Viscoelasticity written by Francesco Mainardi and published by World Scientific. This book was released on 2010 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is likely to be of interest to applied scientists and engineers. --Book Jacket.

An Introduction to the Fractional Calculus and Fractional Differential Equations

Author : Kenneth S. Miller
Publisher : Wiley-Interscience
ISBN 13 : 9780471588849
Total Pages : 384 pages
Book Rating : 4.5/5 (888 download)

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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.

Advanced Methods in the Fractional Calculus of Variations

Author : Agnieszka B. Malinowska
Publisher : Springer
ISBN 13 : 3319147560
Total Pages : 135 pages
Book Rating : 4.3/5 (191 download)

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Book Synopsis Advanced Methods in the Fractional Calculus of Variations by : Agnieszka B. Malinowska

Download or read book Advanced Methods in the Fractional Calculus of Variations written by Agnieszka B. Malinowska and published by Springer. This book was released on 2015-02-05 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: This brief presents a general unifying perspective on the fractional calculus. It brings together results of several recent approaches in generalizing the least action principle and the Euler–Lagrange equations to include fractional derivatives. The dependence of Lagrangians on generalized fractional operators as well as on classical derivatives is considered along with still more general problems in which integer-order integrals are replaced by fractional integrals. General theorems are obtained for several types of variational problems for which recent results developed in the literature can be obtained as special cases. In particular, the authors offer necessary optimality conditions of Euler–Lagrange type for the fundamental and isoperimetric problems, transversality conditions, and Noether symmetry theorems. The existence of solutions is demonstrated under Tonelli type conditions. The results are used to prove the existence of eigenvalues and corresponding orthogonal eigenfunctions of fractional Sturm–Liouville problems. Advanced Methods in the Fractional Calculus of Variations is a self-contained text which will be useful for graduate students wishing to learn about fractional-order systems. The detailed explanations will interest researchers with backgrounds in applied mathematics, control and optimization as well as in certain areas of physics and engineering.

Introduction to Fractional Differential Equations

Author : Constantin Milici
Publisher : Springer
ISBN 13 : 3030008959
Total Pages : 188 pages
Book Rating : 4.0/5 (3 download)

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Book Synopsis Introduction to Fractional Differential Equations by : Constantin Milici

Download or read book Introduction to Fractional Differential Equations written by Constantin Milici and published by Springer. This book was released on 2018-10-28 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Introduction to the Calculus of Variations

Author : Bernard Dacorogna
Publisher : World Scientific Publishing Company
ISBN 13 : 178326554X
Total Pages : 324 pages
Book Rating : 4.7/5 (832 download)

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Book Synopsis Introduction to the Calculus of Variations by : Bernard Dacorogna

Download or read book Introduction to the Calculus of Variations written by Bernard Dacorogna and published by World Scientific Publishing Company. This book was released on 2014-08-13 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: The calculus of variations is one of the oldest subjects in mathematics, and it is very much alive and still evolving. Besides its mathematical importance and its links to other branches of mathematics, such as geometry or differential equations, it is widely used in physics, engineering, economics and biology. This book serves both as a guide to the expansive existing literature and as an aid to the non-specialist — mathematicians, physicists, engineers, students or researchers — in discovering the subject's most important problems, results and techniques. Despite the aim of addressing non-specialists, mathematical rigor has not been sacrificed; most of the theorems are either fully proved or proved under more stringent conditions. In this new edition, several new exercises have been added. The book, containing a total of 119 exercises with detailed solutions, is well designed for a course at both undergraduate and graduate levels.

An Introduction to Fractional Calculus

Author : A. M. Mathai
Publisher : Nova Science Publishers
ISBN 13 : 9781536120424
Total Pages : 0 pages
Book Rating : 4.1/5 (24 download)

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Book Synopsis An Introduction to Fractional Calculus by : A. M. Mathai

Download or read book An Introduction to Fractional Calculus written by A. M. Mathai and published by Nova Science Publishers. This book was released on 2017 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Generalized Fractional Calculus and Applications

Author : Virginia S Kiryakova
Publisher : CRC Press
ISBN 13 : 9780582219779
Total Pages : 412 pages
Book Rating : 4.2/5 (197 download)

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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.

Fractional Differential Equations

Author : Anatoly Kochubei
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110571668
Total Pages : 528 pages
Book Rating : 4.1/5 (15 download)

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Book Synopsis Fractional Differential Equations by : Anatoly Kochubei

Download or read book Fractional Differential Equations written by Anatoly Kochubei and published by Walter de Gruyter GmbH & Co KG. This book was released on 2019-02-19 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This second volume collects authoritative chapters covering the mathematical theory of fractional calculus, including ordinary and partial differential equations of fractional order, inverse problems, and evolution equations.

Fractional Calculus

Author : Dumitru Baleanu
Publisher : World Scientific
ISBN 13 : 9813140054
Total Pages : 476 pages
Book Rating : 4.8/5 (131 download)

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Book Synopsis Fractional Calculus by : Dumitru Baleanu

Download or read book Fractional Calculus written by Dumitru Baleanu and published by World Scientific. This book was released on 2016-09-15 with total page 476 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 will keep in mind the trade-off 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. The second edition of the book has been expanded and now includes a discussion of additional, newly developed numerical methods for fractional calculus and a chapter on the application of fractional calculus for modeling processes in the life sciences.

The Analysis of Fractional Differential Equations

Author : Kai Diethelm
Publisher : Springer
ISBN 13 : 3642145744
Total Pages : 247 pages
Book Rating : 4.6/5 (421 download)

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Book Synopsis The Analysis of Fractional Differential Equations by : Kai Diethelm

Download or read book The Analysis of Fractional Differential Equations written by Kai Diethelm and published by Springer. This book was released on 2010-08-18 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations.

Mathematical Economics

Author : Vasily E. Tarasov
Publisher : MDPI
ISBN 13 : 303936118X
Total Pages : 278 pages
Book Rating : 4.0/5 (393 download)

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Book Synopsis Mathematical Economics by : Vasily E. Tarasov

Download or read book Mathematical Economics written by Vasily E. Tarasov and published by MDPI. This book was released on 2020-06-03 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the application of fractional calculus in economics to describe processes with memory and non-locality. Fractional calculus is a branch of mathematics that studies the properties of differential and integral operators that are characterized by real or complex orders. Fractional calculus methods are powerful tools for describing the processes and systems with memory and nonlocality. Recently, fractional integro-differential equations have been used to describe a wide class of economical processes with power law memory and spatial nonlocality. Generalizations of basic economic concepts and notions the economic processes with memory were proposed. New mathematical models with continuous time are proposed to describe economic dynamics with long memory. This book is a collection of articles reflecting the latest mathematical and conceptual developments in mathematical economics with memory and non-locality based on applications of fractional calculus.

Fractional Differential Equations

Author : Igor Podlubny
Publisher : Elsevier
ISBN 13 : 0080531989
Total Pages : 366 pages
Book Rating : 4.0/5 (85 download)

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Book Synopsis Fractional Differential Equations by : Igor Podlubny

Download or read book Fractional Differential Equations written by Igor Podlubny and published by Elsevier. This book was released on 1998-10-27 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Fractional Calculus

Author : Dumitru Baleanu
Publisher : World Scientific
ISBN 13 : 9814355208
Total Pages : 426 pages
Book Rating : 4.8/5 (143 download)

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Book Synopsis Fractional Calculus by : Dumitru Baleanu

Download or read book Fractional Calculus written by Dumitru Baleanu and published by World Scientific. This book was released on 2012 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: This title 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.

The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order

Author :
Publisher : Elsevier
ISBN 13 : 9780080956206
Total Pages : 322 pages
Book Rating : 4.9/5 (562 download)

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Book Synopsis The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order by :

Download or read book The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order written by and published by Elsevier. This book was released on 1974-09-05 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation; methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory. As a result, the book represents a blend of new methods in general computational analysis, and specific, but also generic, techniques for study of systems theory ant its particular branches, such as optimal filtering and information compression. - Best operator approximation, - Non-Lagrange interpolation, - Generic Karhunen-Loeve transform - Generalised low-rank matrix approximation - Optimal data compression - Optimal nonlinear filtering