Advanced Methods In The Fractional Calculus Of Variations

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Advanced Methods In The Fractional Calculus Of Variations

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Author : Agnieszka B. Malinowska
language : en
Publisher: Springer
Release Date : 2015-02-05

Advanced Methods In The Fractional Calculus Of Variations written by Agnieszka B. Malinowska and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-02-05 with Mathematics categories.

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.

Advanced Methods In The Fractional Calculus Of Variations

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Author : Agnieszka B. Malinowska
language : en
Publisher:
Release Date : 2015

Advanced Methods In The Fractional Calculus Of Variations written by Agnieszka B. Malinowska and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with categories.

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 The Fractional Calculus Of Variations

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Author : Delfim F M Torres
language : en
Publisher: World Scientific Publishing Company
Release Date : 2012-09-14

Introduction To The Fractional Calculus Of Variations written by Delfim F M Torres 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./a

Computational Methods In The Fractional Calculus Of Variations

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Author : Ricardo Almeida
language : en
Publisher: World Scientific Publishing Company
Release Date : 2015-03-19

Computational Methods In The Fractional Calculus Of Variations written by Ricardo Almeida 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 2015-03-19 with Mathematics categories.

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

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Author : Ricardo Almeida
language : en
Publisher: Springer
Release Date : 2018-06-29

The Variable Order Fractional Calculus Of Variations written by Ricardo Almeida and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-29 with Technology & Engineering categories.

​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 Dynamics

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Author : Vasily E. Tarasov
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-01-04

Fractional Dynamics written by Vasily E. Tarasov 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 2011-01-04 with Science categories.

"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.

Fractional Calculus

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Author : Dumitru Baleanu
language : en
Publisher: World Scientific
Release Date : 2012

Fractional Calculus written by Dumitru Baleanu and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.

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.

Basic Theory

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Author : Anatoly Kochubei
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2019-02-19

Basic Theory written by Anatoly Kochubei and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-02-19 with Mathematics categories.

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 first volume collects authoritative chapters covering the mathematical theory of fractional calculus, including fractional-order operators, integral transforms and equations, special functions, calculus of variations, and probabilistic and other aspects.

Direct Methods In The Calculus Of Variations

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Author : Enrico Giusti
language : en
Publisher: World Scientific
Release Date : 2003

Direct Methods In The Calculus Of Variations written by Enrico Giusti and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Mathematics categories.

This book provides a comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order. While direct methods for the existence of solutions are well known and have been widely used in the last century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this book, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations. This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The book is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory.

Theoretical Developments And Applications Of Non Integer Order Systems

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Author : Stefan Domek
language : en
Publisher: Springer
Release Date : 2015-08-20

Theoretical Developments And Applications Of Non Integer Order Systems written by Stefan Domek and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-08-20 with Technology & Engineering categories.

This volume is devoted to presentation of new results of research on systems of non-integer order, called also fractional systems. Their analysis and practical implementation have been the object of spontaneous development for a few last decades. The fractional order models can depict a physical plant better than the classical integer order ones. This covers different research fields such as insulator properties, visco-elastic materials, electrodynamic, electrothermal, electrochemical, economic processes modelling etc. On the other hand fractional controllers often outperform their integer order counterparts. This volume contains new ideas and examples of implementation, theoretical and pure practical aspects of using a non-integer order calculus. It is divided into four parts covering: mathematical fundamentals, modeling and approximations, controllability, observability and stability problems and practical applications of fractional control systems. The first part expands the base of tools and methods of the mathematical basis for non-integer order calculus. Part two focuses on new methods and developments in process modeling and fractional derivatives approximations. In the third part a bunch of papers which raise problems of controllability, observability and stability of non-integer order systems is provided. Part four is devoted to presentation of different fractional order control applications. This book was created thanks to many experts in the field of fractional calculus: authors, anonymous referees whose comments allowed us to improve the final form of the papers and active and inspiring discussion of the participants of RRNR'2015, the 7th Conference on Non-Integer Order Calculus and Its Applications that was organized by the Faculty of Electrical Engineering, West Pomeranian University of Technology, Szczecin, Poland.