The Variable Order Fractional Calculus Of Variations
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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.
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.
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.
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.
Fractional Dynamics And Control
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Author : Dumitru Baleanu
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-11-19
Fractional Dynamics And Control written by Dumitru Baleanu 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 2011-11-19 with Technology & Engineering categories.
Fractional Dynamics and Control provides a comprehensive overview of recent advances in the areas of nonlinear dynamics, vibration and control with analytical, numerical, and experimental results. This book provides an overview of recent discoveries in fractional control, delves into fractional variational principles and differential equations, and applies advanced techniques in fractional calculus to solving complicated mathematical and physical problems.Finally, this book also discusses the role that fractional order modeling can play in complex systems for engineering and science.
Fractional Calculus
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Author : Dumitru Baleanu
language : en
Publisher: World Scientific
Release Date : 2016-09-15
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 2016-09-15 with Mathematics categories.
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.
Fractional Derivatives With Mittag Leffler Kernel
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Author : José Francisco Gómez
language : en
Publisher: Springer
Release Date : 2019-02-13
Fractional Derivatives With Mittag Leffler Kernel written by José Francisco Gómez and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-02-13 with Technology & Engineering categories.
This book offers a timely overview of fractional calculus applications, with a special emphasis on fractional derivatives with Mittag-Leffler kernel. The different contributions, written by applied mathematicians, physicists and engineers, offers a snapshot of recent research in the field, highlighting the current methodological frameworks together with applications in different fields of science and engineering, such as chemistry, mechanics, epidemiology and more. It is intended as a timely guide and source of inspiration for graduate students and researchers in the above-mentioned areas.
Fractional Calculus Theory And Applications
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Author : Francesco Mainardi
language : en
Publisher: MDPI
Release Date : 2018-09-20
Fractional Calculus Theory And Applications written by Francesco Mainardi and has been published by MDPI this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-20 with Electronic books categories.
This book is a printed edition of the Special Issue "Fractional Calculus: Theory and Applications" that was published in Mathematics
Advances In The Theory And Applications Of Non Integer Order Systems
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Author : Wojciech Mitkowski
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-03
Advances In The Theory And Applications Of Non Integer Order Systems written by Wojciech Mitkowski 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 2013-06-03 with Technology & Engineering categories.
This volume presents various aspects of non-integer order systems, also known as fractional systems, which have recently attracted an increasing attention in the scientific community of systems science, applied mathematics, control theory. Non-integer systems have become relevant for many fields of science and technology exemplified by the modeling of signal transmission, electric noise, dielectric polarization, heat transfer, electrochemical reactions, thermal processes, acoustics, etc. The content is divided into six parts, every of which considers one of the currently relevant problems. In the first part the Realization problem is discussed, with a special focus on positive systems. The second part considers stability of certain classes of non-integer order systems with and without delays. The third part is focused on such important aspects as controllability, observability and optimization especially in discrete time. The fourth part is focused on distributed systems where non-integer calculus leads to new and interesting results. The next part considers problems of solutions and approximations of non-integer order equations and systems. The final and most extensive part is devoted to applications. Problems from mechatronics, biomedical engineering, robotics and others are all analyzed and solved with tools from fractional systems. This volume came to fruition thanks to high level of talks and interesting discussions at RRNR 2013 - 5th Conference on Non-integer Order Calculus and its Applications that took place at AGH University of Science and Technology in Kraków, Poland, which was organized by the Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering.
Fractional Calculus In Analysis Dynamics And Optimal Control
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Author : Jacky Cresson
language : en
Publisher: Nova Science Publishers
Release Date : 2014
Fractional Calculus In Analysis Dynamics And Optimal Control written by Jacky Cresson 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 2014 with Fractional calculus categories.
This book is devoted to applications of fractional calculus in classical fields of mathematics like analysis, dynamics, partial differential equations and optimal control. The first chapter deals with the notion of local fractional derivatives and its applications to the study of regularity and geometry of curves. The second chapter develops the notion of fractional embedding and fractional assymetric calculus of variations in order to find fractional Lagrangian variational structures for classical dissipative partial differential equations. In continuation of this chapter, a fractional analogue of the classical Pontryagin maximum principle is proved for discrete and continuous fractional optimal control problems. The fourth chapter gives a first mathematical model that allows a rigorous connection to be made between the dynamics of chaotic Hamiltonian systems and fractional dynamics, mixing the previous approaches of G Zaslavsky and R Hilfer. Finally, numerical methods to deal with fractional optimal control problems are discussed and implemented. All the chapters are self-contained and complete proofs are given.