Advanced Methods In The Fractional Calculus Of Variations

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Advanced Methods in the Fractional Calculus of Variations

Author : Agnieszka B. Malinowska
Publisher : Springer
ISBN 13 : 3319147560
Total Pages : 142 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 142 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.

Computational Methods In The Fractional Calculus Of Variations

Author : Ricardo Almeida
Publisher : World Scientific Publishing Company
ISBN 13 : 1783266422
Total Pages : 279 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 279 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.

Introduction To The Fractional Calculus Of Variations

Author : Delfim F M Torres
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 : Delfim F M Torres

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

Fractional Dynamics and Control

Author : Dumitru Baleanu
Publisher : Springer Science & Business Media
ISBN 13 : 1461404576
Total Pages : 302 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Fractional Dynamics and Control by : Dumitru Baleanu

Download or read book Fractional Dynamics and Control written by Dumitru Baleanu and published by Springer Science & Business Media. This book was released on 2011-11-19 with total page 302 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

The Variable-Order Fractional Calculus of Variations

Author : Ricardo Almeida
Publisher : Springer
ISBN 13 : 3319940066
Total Pages : 135 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 135 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.

Computational Methods In The Fractional Calculus Of Variations

Author : Ricardo Almeida (Mathematics professor)
Publisher :
ISBN 13 : 9781783266418
Total Pages : 279 pages
Book Rating : 4.2/5 (664 download)

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Book Synopsis Computational Methods In The Fractional Calculus Of Variations by : Ricardo Almeida (Mathematics professor)

Download or read book Computational Methods In The Fractional Calculus Of Variations written by Ricardo Almeida (Mathematics professor) and published by . This book was released on 2015 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Fractional Calculus

Author : Dumitru Baleanu
Publisher : World Scientific
ISBN 13 : 9814355216
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: The subject of fractional calculus and its applications (that is, convolution-type pseudo-differential operators including integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, mainly due to its applications in diverse fields of science and engineering. These operators have been used to model problems with anomalous dynamics, however, they also are an effective tool as filters and controllers, and they can be applied to write complicated functions in terms of fractional integrals or derivatives of elementary functions, and so on. 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 was written with a trade-off in mind 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. Numerical code is also provided.

Fractional Order Processes

Author : Seshu Kumar Damarla
Publisher : CRC Press
ISBN 13 : 0429996896
Total Pages : 340 pages
Book Rating : 4.4/5 (299 download)

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Book Synopsis Fractional Order Processes by : Seshu Kumar Damarla

Download or read book Fractional Order Processes written by Seshu Kumar Damarla and published by CRC Press. This book was released on 2018-09-03 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book presents efficient numerical methods for simulation and analysis of physical processes exhibiting fractional order (FO) dynamics. The book introduces FO system identification method to estimate parameters of a mathematical model under consideration from experimental or simulated data. A simple tuning technique, which aims to produce a robust FO PID controller exhibiting iso-damping property during re-parameterization of a plant, is devised in the book. A new numerical method to find an equivalent finite dimensional integer order system for an infinite dimensional FO system is developed in the book. The book also introduces a numerical method to solve FO optimal control problems. Key features Proposes generalized triangular function operational matrices. Shows significant applications of triangular orthogonal functions as well as triangular strip operational matrices in simulation, identification and control of fractional order processes. Provides numerical methods for simulation of physical problems involving different types of weakly singular integral equations, Abel’s integral equation, fractional order integro-differential equations, fractional order differential and differential-algebraic equations, and fractional order partial differential equations. Suggests alternative way to do numerical computation of fractional order signals and systems and control. Provides source codes developed in MATLAB for each chapter, allowing the interested reader to take advantage of these codes for broadening and enhancing the scope of the book itself and developing new results.

Basic Theory

Author : Anatoly Kochubei
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110571625
Total Pages : 489 pages
Book Rating : 4.1/5 (15 download)

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Book Synopsis Basic Theory by : Anatoly Kochubei

Download or read book Basic Theory written by Anatoly Kochubei and published by Walter de Gruyter GmbH & Co KG. This book was released on 2019-02-19 with total page 489 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 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.

Numerical Methods for Fractional Differentiation

Author : Kolade M. Owolabi
Publisher : Springer Nature
ISBN 13 : 9811500983
Total Pages : 328 pages
Book Rating : 4.8/5 (115 download)

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Book Synopsis Numerical Methods for Fractional Differentiation by : Kolade M. Owolabi

Download or read book Numerical Methods for Fractional Differentiation written by Kolade M. Owolabi and published by Springer Nature. This book was released on 2019-10-14 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses numerical methods for solving partial differential and integral equations, as well as ordinary differential and integral equations, involving fractional differential and integral operators. Differential and integral operators presented in the book include those with exponential decay law, known as Caputo–Fabrizio differential and integral operators, those with power law, known as Riemann–Liouville fractional operators, and those for the generalized Mittag–Leffler function, known as the Atangana–Baleanu fractional operators. The book reviews existing numerical schemes associated with fractional operators including those with power law, while also highlighting new trends in numerical schemes for recently introduced differential and integral operators. In addition, the initial chapters address useful properties of each differential and integral fractional operator. Methods discussed in the book are subsequently used to solved problems arising in many fields of science, technology, and engineering, including epidemiology, chaos, solitons, fractals, diffusion, groundwater, and fluid mechanics. Given its scope, the book offers a valuable resource for graduate students of mathematics and engineering, and researchers in virtually all fields of science, technology, and engineering, as well as an excellent addition to libraries.

Theoretical Developments and Applications of Non-Integer Order Systems

Author : Stefan Domek
Publisher : Springer
ISBN 13 : 3319230395
Total Pages : 294 pages
Book Rating : 4.3/5 (192 download)

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Book Synopsis Theoretical Developments and Applications of Non-Integer Order Systems by : Stefan Domek

Download or read book Theoretical Developments and Applications of Non-Integer Order Systems written by Stefan Domek and published by Springer. This book was released on 2015-08-20 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

General Fractional Derivatives with Applications in Viscoelasticity

Author : Xiao-Jun Yang
Publisher : Academic Press
ISBN 13 : 0128172096
Total Pages : 456 pages
Book Rating : 4.1/5 (281 download)

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Book Synopsis General Fractional Derivatives with Applications in Viscoelasticity by : Xiao-Jun Yang

Download or read book General Fractional Derivatives with Applications in Viscoelasticity written by Xiao-Jun Yang and published by Academic Press. This book was released on 2020-04-03 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: General Fractional Derivatives with Applications in Viscoelasticity introduces the newly established fractional-order calculus operators involving singular and non-singular kernels with applications to fractional-order viscoelastic models from the calculus operator viewpoint. Fractional calculus and its applications have gained considerable popularity and importance because of their applicability to many seemingly diverse and widespread fields in science and engineering. Many operations in physics and engineering can be defined accurately by using fractional derivatives to model complex phenomena. Viscoelasticity is chief among them, as the general fractional calculus approach to viscoelasticity has evolved as an empirical method of describing the properties of viscoelastic materials. General Fractional Derivatives with Applications in Viscoelasticity makes a concise presentation of general fractional calculus. Presents a comprehensive overview of the fractional derivatives and their applications in viscoelasticity Provides help in handling the power-law functions Introduces and explores the questions about general fractional derivatives and its applications

Fractional Dynamical Systems: Methods, Algorithms and Applications

Author : Piotr Kulczycki
Publisher : Springer Nature
ISBN 13 : 3030899721
Total Pages : 398 pages
Book Rating : 4.0/5 (38 download)

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Book Synopsis Fractional Dynamical Systems: Methods, Algorithms and Applications by : Piotr Kulczycki

Download or read book Fractional Dynamical Systems: Methods, Algorithms and Applications written by Piotr Kulczycki and published by Springer Nature. This book was released on 2022-01-04 with total page 398 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a wide and comprehensive spectrum of issues and problems related to fractional-order dynamical systems. It is meant to be a full-fledge, comprehensive presentation of many aspects related to the broadly perceived fractional-order dynamical systems which constitute an extension of the traditional integer-order-type descriptions. This implies far-reaching consequences, both analytic and algorithmic, because—in general—properties of the traditional integer-order systems cannot be directly extended by a straightforward generalization to fractional-order systems, modeled by fractional-order differential equations involving derivatives of an non-integer order. This can be useful for describing and analyzing, for instance, anomalies in the behavior of various systems, chaotic behavior, etc. The book contains both analytic contributions with state-of-the-art and theoretical foundations, algorithmic implementation of tools and techniques, and—finally—some examples of relevant and successful practical applications.

Introduction to Fractional Differential Equations

Author : Constantin Milici
Publisher : Springer
ISBN 13 : 3030008959
Total Pages : 199 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 199 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.

Differential Equations on Measures and Functional Spaces

Author : Vassili Kolokoltsov
Publisher : Springer
ISBN 13 : 3030033775
Total Pages : 525 pages
Book Rating : 4.0/5 (3 download)

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Book Synopsis Differential Equations on Measures and Functional Spaces by : Vassili Kolokoltsov

Download or read book Differential Equations on Measures and Functional Spaces written by Vassili Kolokoltsov and published by Springer. This book was released on 2019-06-20 with total page 525 pages. Available in PDF, EPUB and Kindle. Book excerpt: This advanced book focuses on ordinary differential equations (ODEs) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It briefly discusses the fundamentals before moving on to the cutting edge research in linear and nonlinear partial and pseudo-differential equations, general kinetic equations and fractional evolutions. The level of generality chosen is suitable for the study of the most important nonlinear equations of mathematical physics, such as Boltzmann, Smoluchovskii, Vlasov, Landau-Fokker-Planck, Cahn-Hilliard, Hamilton-Jacobi-Bellman, nonlinear Schroedinger, McKean-Vlasov diffusions and their nonlocal extensions, mass-action-law kinetics from chemistry. It also covers nonlinear evolutions arising in evolutionary biology and mean-field games, optimization theory, epidemics and system biology, in general models of interacting particles or agents describing splitting and merging, collisions and breakage, mutations and the preferential-attachment growth on networks. The book is intended mainly for upper undergraduate and graduate students, but is also of use to researchers in differential equations and their applications. It particularly highlights the interconnections between various topics revealing where and how a particular result is used in other chapters or may be used in other contexts, and also clarifies the links between the languages of pseudo-differential operators, generalized functions, operator theory, abstract linear spaces, fractional calculus and path integrals.

Advances in Non-Integer Order Calculus and Its Applications

Author : Agnieszka B. Malinowska
Publisher : Springer
ISBN 13 : 3030173445
Total Pages : 307 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Advances in Non-Integer Order Calculus and Its Applications by : Agnieszka B. Malinowska

Download or read book Advances in Non-Integer Order Calculus and Its Applications written by Agnieszka B. Malinowska and published by Springer. This book was released on 2019-04-17 with total page 307 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an overview of some recent findings in the theory and applications of non-integer order systems. Discussing topics ranging from the mathematical foundations to technical applications of continuous-time and discrete-time fractional calculus, it includes 22 original research papers and is subdivided into four parts: • Mathematical Foundations • Approximation, Modeling and Simulations • Fractional Systems Analysis and Control • Applications The papers were selected from those presented at the 10th International Conference of Non-integer Order Calculus and its Applications, which was held at the Bialystok University of Technology, Poland, September 20–21, 2018. Thanks to the broad spectrum of topics covered, the book is suitable for researchers from applied mathematics and engineering. It is also a valuable resource for graduate students, as well as for scholars looking for new mathematical tools.

Nonlinear Analysis and Boundary Value Problems

Author : Iván Area
Publisher : Springer Nature
ISBN 13 : 3030269876
Total Pages : 295 pages
Book Rating : 4.0/5 (32 download)

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Book Synopsis Nonlinear Analysis and Boundary Value Problems by : Iván Area

Download or read book Nonlinear Analysis and Boundary Value Problems written by Iván Area and published by Springer Nature. This book was released on 2019-09-19 with total page 295 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to Prof. Juan J. Nieto, on the occasion of his 60th birthday. Juan José Nieto Roig (born 1958, A Coruña) is a Spanish mathematician, who has been a Professor of Mathematical Analysis at the University of Santiago de Compostela since 1991. His most influential contributions to date are in the area of differential equations. Nieto received his degree in Mathematics from the University of Santiago de Compostela in 1980. He was then awarded a Fulbright scholarship and moved to the University of Texas at Arlington where he worked with Professor V. Lakshmikantham. He received his Ph.D. in Mathematics from the University of Santiago de Compostela in 1983. Nieto's work may be considered to fall within the ambit of differential equations, and his research interests include fractional calculus, fuzzy equations and epidemiological models. He is one of the world’s most cited mathematicians according to Web of Knowledge, and appears in the Thompson Reuters Highly Cited Researchers list. Nieto has also occupied different positions at the University of Santiago de Compostela, such as Dean of Mathematics and Director of the Mathematical Institute. He has also served as an editor for various mathematical journals, and was the editor-in-chief of the journal Nonlinear Analysis: Real World Applications from 2009 to 2012. In 2016, Nieto was admitted as a Fellow of the Royal Galician Academy of Sciences. This book consists of contributions presented at the International Conference on Nonlinear Analysis and Boundary Value Problems, held in Santiago de Compostela, Spain, 4th-7th September 2018. Covering a variety of topics linked to Nieto’s scientific work, ranging from differential, difference and fractional equations to epidemiological models and dynamical systems and their applications, it is primarily intended for researchers involved in nonlinear analysis and boundary value problems in a broad sense.