The Variable Order Fractional Calculus Of Variations

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The Variable-Order Fractional Calculus of Variations

Ricardo Almeida,Dina Tavares,Delfim F. M. Torres
The Variable Order Fractional Calculus of Variations Book PDF

Author : Ricardo Almeida,Dina Tavares,Delfim F. M. Torres
Publisher : Springer
Page : 124 pages
File Size : 53,7 Mb
Release : 2018-06-29
Category : Technology & Engineering
ISBN : 9783319940069

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The Variable-Order Fractional Calculus of Variations by Ricardo Almeida,Dina Tavares,Delfim F. M. Torres Pdf

​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

Agnieszka B Malinowska,Delfim F M Torres
Introduction to the Fractional Calculus of Variations Book PDF

Author : Agnieszka B Malinowska,Delfim F M Torres
Publisher : World Scientific Publishing Company
Page : 292 pages
File Size : 51,7 Mb
Release : 2012-09-14
Category : Mathematics
ISBN : 9781848169685

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Introduction to the Fractional Calculus of Variations by Agnieszka B Malinowska,Delfim F M Torres Pdf

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

Ricardo Almeida,Shakoor Pooseh,Delfim F M Torres
Computational Methods In The Fractional Calculus Of Variations Book PDF

Author : Ricardo Almeida,Shakoor Pooseh,Delfim F M Torres
Publisher : World Scientific Publishing Company
Page : 280 pages
File Size : 41,6 Mb
Release : 2015-03-19
Category : Mathematics
ISBN : 9781783266425

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Computational Methods In The Fractional Calculus Of Variations by Ricardo Almeida,Shakoor Pooseh,Delfim F M Torres Pdf

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

Agnieszka B. Malinowska,Tatiana Odzijewicz,Delfim F.M. Torres
Advanced Methods in the Fractional Calculus of Variations Book PDF

Author : Agnieszka B. Malinowska,Tatiana Odzijewicz,Delfim F.M. Torres
Publisher : Springer
Page : 135 pages
File Size : 54,9 Mb
Release : 2015-02-05
Category : Mathematics
ISBN : 9783319147567

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Advanced Methods in the Fractional Calculus of Variations by Agnieszka B. Malinowska,Tatiana Odzijewicz,Delfim F.M. Torres Pdf

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 Derivatives with Mittag-Leffler Kernel

José Francisco Gómez,Lizeth Torres,Ricardo Fabricio Escobar
Fractional Derivatives with Mittag Leffler Kernel Book PDF

Author : José Francisco Gómez,Lizeth Torres,Ricardo Fabricio Escobar
Publisher : Springer
Page : 341 pages
File Size : 44,6 Mb
Release : 2019-02-13
Category : Technology & Engineering
ISBN : 9783030116620

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Fractional Derivatives with Mittag-Leffler Kernel by José Francisco Gómez,Lizeth Torres,Ricardo Fabricio Escobar Pdf

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 Dynamics and Control

Dumitru Baleanu,José António Tenreiro Machado,Albert C. J. Luo
Fractional Dynamics and Control Book PDF

Author : Dumitru Baleanu,José António Tenreiro Machado,Albert C. J. Luo
Publisher : Springer Science & Business Media
Page : 302 pages
File Size : 50,8 Mb
Release : 2011-11-19
Category : Technology & Engineering
ISBN : 9781461404576

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Fractional Dynamics and Control by Dumitru Baleanu,José António Tenreiro Machado,Albert C. J. Luo Pdf

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.

Advances in the Theory and Applications of Non-integer Order Systems

Wojciech Mitkowski,Janusz Kacprzyk,Jerzy Baranowski
Advances in the Theory and Applications of Non integer Order Systems Book PDF

Author : Wojciech Mitkowski,Janusz Kacprzyk,Jerzy Baranowski
Publisher : Springer Science & Business Media
Page : 325 pages
File Size : 54,9 Mb
Release : 2013-06-03
Category : Technology & Engineering
ISBN : 9783319009339

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Advances in the Theory and Applications of Non-integer Order Systems by Wojciech Mitkowski,Janusz Kacprzyk,Jerzy Baranowski Pdf

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.

General Fractional Derivatives

Xiao-Jun Yang
General Fractional Derivatives Book PDF

Author : Xiao-Jun Yang
Publisher : CRC Press
Page : 306 pages
File Size : 48,6 Mb
Release : 2019-05-10
Category : Mathematics
ISBN : 9780429811524

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General Fractional Derivatives by Xiao-Jun Yang Pdf

General Fractional Derivatives: Theory, Methods and Applications provides knowledge of the special functions with respect to another function, and the integro-differential operators where the integrals are of the convolution type and exist the singular, weakly singular and nonsingular kernels, which exhibit the fractional derivatives, fractional integrals, general fractional derivatives, and general fractional integrals of the constant and variable order without and with respect to another function due to the appearance of the power-law and complex herbivores to figure out the modern developments in theoretical and applied science. Features: Give some new results for fractional calculus of constant and variable orders. Discuss some new definitions for fractional calculus with respect to another function. Provide definitions for general fractional calculus of constant and variable orders. Report new results of general fractional calculus with respect to another function. Propose news special functions with respect to another function and their applications. Present new models for the anomalous relaxation and rheological behaviors. This book serves as a reference book and textbook for scientists and engineers in the fields of mathematics, physics, chemistry and engineering, senior undergraduate and graduate students. Dr. Xiao-Jun Yang is a full professor of Applied Mathematics and Mechanics, at China University of Mining and Technology, China. He is currently an editor of several scientific journals, such as Fractals, Applied Numerical Mathematics, Mathematical Modelling and Analysis, International Journal of Numerical Methods for Heat & Fluid Flow, and Thermal Science.

Advances in Harmonic Analysis and Operator Theory

Alexandre Almeida,Luís Castro,Frank-Olme Speck
Advances in Harmonic Analysis and Operator Theory Book PDF

Author : Alexandre Almeida,Luís Castro,Frank-Olme Speck
Publisher : Springer Science & Business Media
Page : 389 pages
File Size : 52,9 Mb
Release : 2013-01-31
Category : Mathematics
ISBN : 9783034805162

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Advances in Harmonic Analysis and Operator Theory by Alexandre Almeida,Luís Castro,Frank-Olme Speck Pdf

This volume is dedicated to Professor Stefan Samko on the occasion of his seventieth birthday. The contributions display the range of his scientific interests in harmonic analysis and operator theory. Particular attention is paid to fractional integrals and derivatives, singular, hypersingular and potential operators in variable exponent spaces, pseudodifferential operators in various modern function and distribution spaces, as well as related applications, to mention but a few. Most contributions were firstly presented in two conferences at Lisbon and Aveiro, Portugal, in June‒July 2011.

Fractional Calculus

Dumitru Baleanu,Kai Diethelm,Enrico Scalas,Juan J Trujillo
Fractional Calculus Book PDF

Author : Dumitru Baleanu,Kai Diethelm,Enrico Scalas,Juan J Trujillo
Publisher : World Scientific
Page : 476 pages
File Size : 48,7 Mb
Release : 2016-09-15
Category : Mathematics
ISBN : 9789813140059

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Fractional Calculus by Dumitru Baleanu,Kai Diethelm,Enrico Scalas,Juan J Trujillo Pdf

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 Calculus and Fractional Differential Equations

Varsha Daftardar-Gejji
Fractional Calculus and Fractional Differential Equations Book PDF

Author : Varsha Daftardar-Gejji
Publisher : Springer
Page : 180 pages
File Size : 48,7 Mb
Release : 2019-08-10
Category : Mathematics
ISBN : 9789811392276

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Fractional Calculus and Fractional Differential Equations by Varsha Daftardar-Gejji Pdf

This book provides a broad overview of the latest developments in fractional calculus and fractional differential equations (FDEs) with an aim to motivate the readers to venture into these areas. It also presents original research describing the fractional operators of variable order, fractional-order delay differential equations, chaos and related phenomena in detail. Selected results on the stability of solutions of nonlinear dynamical systems of the non-commensurate fractional order have also been included. Furthermore, artificial neural network and fractional differential equations are elaborated on; and new transform methods (for example, Sumudu methods) and how they can be employed to solve fractional partial differential equations are discussed. The book covers the latest research on a variety of topics, including: comparison of various numerical methods for solving FDEs, the Adomian decomposition method and its applications to fractional versions of the classical Poisson processes, variable-order fractional operators, fractional variational principles, fractional delay differential equations, fractional-order dynamical systems and stability analysis, inequalities and comparison theorems in FDEs, artificial neural network approximation for fractional operators, and new transform methods for solving partial FDEs. Given its scope and level of detail, the book will be an invaluable asset for researchers working in these areas.

Mathematical Economics

Vasily E. Tarasov
Mathematical Economics Book PDF

Author : Vasily E. Tarasov
Publisher : MDPI
Page : 278 pages
File Size : 41,6 Mb
Release : 2020-06-03
Category : Business & Economics
ISBN : 9783039361182

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Mathematical Economics by Vasily E. Tarasov Pdf

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.

General Fractional Derivatives with Applications in Viscoelasticity

Xiao-Jun Yang,Feng Gao,Yang Ju
General Fractional Derivatives with Applications in Viscoelasticity Book PDF

Author : Xiao-Jun Yang,Feng Gao,Yang Ju
Publisher : Academic Press
Page : 456 pages
File Size : 53,7 Mb
Release : 2020-04-03
Category : Mathematics
ISBN : 9780128172094

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General Fractional Derivatives with Applications in Viscoelasticity by Xiao-Jun Yang,Feng Gao,Yang Ju Pdf

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

Advances in Modelling and Control of Non-integer-Order Systems

Krzysztof J. Latawiec,Marian Łukaniszyn,Rafał Stanisławski
Advances in Modelling and Control of Non integer Order Systems Book PDF

Author : Krzysztof J. Latawiec,Marian Łukaniszyn,Rafał Stanisławski
Publisher : Springer
Page : 296 pages
File Size : 51,9 Mb
Release : 2014-08-16
Category : Technology & Engineering
ISBN : 9783319099002

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Advances in Modelling and Control of Non-integer-Order Systems by Krzysztof J. Latawiec,Marian Łukaniszyn,Rafał Stanisławski Pdf

This volume presents selected aspects of non-integer, or fractional order systems, whose analysis, synthesis and applications have increasingly become a real challenge for various research communities, ranging from science to engineering. The spectrum of applications of the fractional order calculus has incredibly expanded, in fact it would be hard to find a science/engineering-related subject area where the fractional calculus had not been incorporated. The content of the fractional calculus is ranged from pure mathematics to engineering implementations and so is the content of this volume. The volume is subdivided into six parts, reflecting particular aspects of the fractional order calculus. The first part contains a single invited paper on a new formulation of fractional-order descriptor observers for fractional-order descriptor continous LTI systems. The second part provides new elements to the mathematical theory of fractional-order systems. In the third part of this volume, a bunch of new results in approximation, modeling and simulations of fractional-order systems is given. The fourth part presents new solutions to some problems in controllability and control of non-integer order systems, in particular fractional PID-like control. The fifth part analyzes the stability of non-integer order systems and some new results are offered in this important respect, in particular for discrete-time systems. The final, sixth part of this volume presents a spectrum of applications of the noninteger order calculus, ranging from bi-fractional filtering, in particular of electromyographic signals, through the thermal diffusion and advection diffusion processes to the SIEMENS platform implementation. This volume's papers were all subjected to stimulating comments and discussions from the active audience of the RRNR'2014, the 6th Conference on Non-integer Order Calculus and Its Applications that was organized by the Department of Electrical, Control and Computer Engineering, Opole University of Technology, Opole, Poland.

Theoretical Developments and Applications of Non-Integer Order Systems

Stefan Domek,Paweł Dworak
Theoretical Developments and Applications of Non Integer Order Systems Book PDF

Author : Stefan Domek,Paweł Dworak
Publisher : Springer
Page : 293 pages
File Size : 47,5 Mb
Release : 2015-08-20
Category : Technology & Engineering
ISBN : 9783319230399

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Theoretical Developments and Applications of Non-Integer Order Systems by Stefan Domek,Paweł Dworak Pdf

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.