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General Fractional Derivatives with Applications in Viscoelasticity

General Fractional Derivatives with Applications in Viscoelasticity
A Book

by Xiao-Jun Yang,Feng Gao,Yang Ju

  • Publisher : Academic Press
  • Release : 2020-04-03
  • Pages : 454
  • ISBN : 0128172096
  • Language : En, Es, Fr & De
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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

General Fractional Derivatives

General Fractional Derivatives
Theory, Methods and Applications

by Xiao-Jun Yang

  • Publisher : CRC Press
  • Release : 2019-05-10
  • Pages : 364
  • ISBN : 0429811527
  • Language : En, Es, Fr & De
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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.

Fractional Calculus and Waves in Linear Viscoelasticity

Fractional Calculus and Waves in Linear Viscoelasticity
An Introduction to Mathematical Models

by Francesco Mainardi

  • Publisher : World Scientific
  • Release : 2010
  • Pages : 368
  • ISBN : 1848163304
  • Language : En, Es, Fr & De
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This monograph provides a comprehensive overview of the author's work on the fields of fractional calculus and waves in linear viscoelastic media, which includes his pioneering contributions on the applications of special functions of the Mittag-Leffler and Wright types. It is intended to serve as a general introduction to the above-mentioned areas of mathematical modeling. The explanations in the book are detailed enough to capture the interest of the curious reader, and complete enough to provide the necessary background material needed to delve further into the subject and explore the research literature given in the huge general bibliography. This book is likely to be of interest to applied scientists and engineers.

Fractional Calculus with Applications in Mechanics

Fractional Calculus with Applications in Mechanics
Vibrations and Diffusion Processes

by Teodor M. Atanackovic,Stevan Pilipovic,Bogoljub Stankovic,Dusan Zorica

  • Publisher : John Wiley & Sons
  • Release : 2014-02-19
  • Pages : 336
  • ISBN : 1118577469
  • Language : En, Es, Fr & De
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This book contains mathematical preliminaries in which basic definitions of fractional derivatives and spaces are presented. The central part of the book contains various applications in classical mechanics including fields such as: viscoelasticity, heat conduction, wave propagation and variational Hamilton–type principles. Mathematical rigor will be observed in the applications. The authors provide some problems formulated in the classical setting and some in the distributional setting. The solutions to these problems are presented in analytical form and these solutions are then analyzed numerically. Theorems on the existence of solutions will be presented for all examples discussed. In using various constitutive equations the restrictions following from the second law of thermodynamics will be implemented. Finally, the physical implications of obtained solutions will be discussed in detail.

Fractional Differential Equations

Fractional Differential Equations
An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications

by Igor Podlubny

  • Publisher : Elsevier
  • Release : 1998-10-27
  • Pages : 340
  • ISBN : 9780080531984
  • Language : En, Es, Fr & De
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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

Advances in Fractional Calculus

Advances in Fractional Calculus
Theoretical Developments and Applications in Physics and Engineering

by J. Sabatier,O. P. Agrawal,J. A. Tenreiro Machado

  • Publisher : Springer Science & Business Media
  • Release : 2007-07-28
  • Pages : 552
  • ISBN : 1402060424
  • Language : En, Es, Fr & De
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In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal and image processing. For example, in the last three fields, some important considerations such as modelling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability and robustness are now linked to long-range dependence phenomena. Similar progress has been made in other fields listed here. The scope of the book is thus to present the state of the art in the study of fractional systems and the application of fractional differentiation. As this volume covers recent applications of fractional calculus, it will be of interest to engineers, scientists, and applied mathematicians.

Applied Mechanics Reviews

Applied Mechanics Reviews
A Book

by Anonim

  • Publisher : Unknown Publisher
  • Release : 1997
  • Pages : 329
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, Part 2

Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, Part 2
A Book

by Anonim

  • Publisher : Unknown Publisher
  • Release : 1991
  • Pages : 464
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Fractals and Fractional Calculus in Continuum Mechanics

Fractals and Fractional Calculus in Continuum Mechanics
A Book

by Alberto Carpinteri,Francesco Mainardi

  • Publisher : Springer
  • Release : 2014-05-04
  • Pages : 348
  • ISBN : 3709126649
  • Language : En, Es, Fr & De
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The book is characterized by the illustration of cases of fractal, self-similar and multi-scale structures taken from the mechanics of solid and porous materials, which have a technical interest. In addition, an accessible and self-consistent treatment of the mathematical technique of fractional calculus is provided, avoiding useless complications.

Fractional Calculus with Applications in Mechanics

Fractional Calculus with Applications in Mechanics
Vibrations and Diffusion Processes

by Teodor M. Atanackovic,Stevan Pilipovic,Bogoljub Stankovic,Dusan Zorica

  • Publisher : John Wiley & Sons
  • Release : 2014-03-17
  • Pages : 336
  • ISBN : 1848214170
  • Language : En, Es, Fr & De
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Fractional Calculus with Applications in Mechanics is the first complete compilation of fractional calculus applications to mechanics. It examines classical mechanics topics, such as viscoelasticity, heat conduction, wave propagation, and variational principles of Hamilton?s type. Author Teodor Atanackovic presents students and researchers in physics, mechanical engineering, and civil engineering with a systematic description of mathematical solutions to mechanical problems.

Applications of Fractional Calculus in Physics

Applications of Fractional Calculus in Physics
A Book

by R Hilfer

  • Publisher : World Scientific
  • Release : 2000-03-02
  • Pages : 472
  • ISBN : 9814496200
  • Language : En, Es, Fr & De
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Fractional calculus is a collection of relatively little-known mathematical results concerning generalizations of differentiation and integration to noninteger orders. While these results have been accumulated over centuries in various branches of mathematics, they have until recently found little appreciation or application in physics and other mathematically oriented sciences. This situation is beginning to change, and there are now a growing number of research areas in physics which employ fractional calculus. This volume provides an introduction to fractional calculus for physicists, and collects easily accessible review articles surveying those areas of physics in which applications of fractional calculus have recently become prominent. Contents:An Introduction to Fractional Calculus (P L Butzer & U Westphal)Fractional Time Evolution (R Hilfer)Fractional Powers of Infinitesimal Generators of Semigroups (U Westphal)Fractional Differences, Derivatives and Fractal Time Series (B J West & P Grigolini)Fractional Kinetics of Hamiltonian Chaotic Systems (G M Zaslavsky)Polymer Science Applications of Path-Integration, Integral Equations, and Fractional Calculus (J F Douglas)Applications to Problems in Polymer Physics and Rheology (H Schiessel et al.)Applications of Fractional Calculus Techniques to Problems in Biophysics (T F Nonnenmacher & R Metzler)Fractional Calculus and Regular Variation in Thermodynamics (R Hilfer) Readership: Statistical, theoretical and mathematical physicists. Keywords:Fractional Calculus in PhysicsReviews: “This monograph provides a systematic treatment of the theory and applications of fractional calculus for physicists. It contains nine review articles surveying those areas in which fractional calculus has become important. All the chapters are self-contained.” Mathematics Abstracts

Fractional Dynamics

Fractional Dynamics
Applications of Fractional Calculus to Dynamics of Particles, Fields and Media

by Vasily E. Tarasov

  • Publisher : Springer Science & Business Media
  • Release : 2011-01-04
  • Pages : 505
  • ISBN : 3642140033
  • Language : En, Es, Fr & De
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"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 Operators with Constant and Variable Order with Application to Geo-hydrology

Fractional Operators with Constant and Variable Order with Application to Geo-hydrology
A Book

by Abdon Atangana

  • Publisher : Academic Press
  • Release : 2017-09-19
  • Pages : 414
  • ISBN : 0128097965
  • Language : En, Es, Fr & De
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Fractional Operators with Constant and Variable Order with Application to Geo-hydrology provides a physical review of fractional operators, fractional variable order operators, and uncertain derivatives to groundwater flow and environmental remediation. It presents a formal set of mathematical equations for the description of groundwater flow and pollution problems using the concept of non-integer order derivative. Both advantages and disadvantages of models with fractional operators are discussed. Based on the author’s analyses, the book proposes new techniques for groundwater remediation, including guidelines on how chemical companies can be positioned in any city to avoid groundwater pollution. Proposes new aquifer derivatives for leaky, confined and unconfined formations Presents useful aids for applied scientists and engineers seeking to solve complex problems that cannot be handled using constant fractional order derivatives Provides a real physical interpretation of operators relevant to groundwater flow problems Models both fractional and variable order derivatives, presented together with uncertainties analysis

ASME Technical Papers

ASME Technical Papers
A Book

by Anonim

  • Publisher : Unknown Publisher
  • Release : 1984
  • Pages : 329
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Chaos in a Fractional Order Chua System

Chaos in a Fractional Order Chua System
A Book

by Anonim

  • Publisher : Unknown Publisher
  • Release : 1996
  • Pages : 28
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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This report studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. Varying the total system order incrementally from 2.6 to 3.7 demonstrates that systems of "order" less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable.

On the Analysis of Structures with Viscoelastic Dampers

On the Analysis of Structures with Viscoelastic Dampers
A Book

by Jose Antonio Inaudi,Alessandra Zambrano,James M. Kelly,University of California, Berkeley. Earthquake Engineering Research Center

  • Publisher : Unknown Publisher
  • Release : 1994
  • Pages : 126
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Fractional Derivatives for Physicists and Engineers

Fractional Derivatives for Physicists and Engineers
Volume I Background and Theory Volume II Applications

by Vladimir V. Uchaikin

  • Publisher : Springer Science & Business Media
  • Release : 2013-07-09
  • Pages : 385
  • ISBN : 3642339115
  • Language : En, Es, Fr & De
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The first derivative of a particle coordinate means its velocity, the second means its acceleration, but what does a fractional order derivative mean? Where does it come from, how does it work, where does it lead to? The two-volume book written on high didactic level answers these questions. Fractional Derivatives for Physicists and Engineers— The first volume contains a clear introduction into such a modern branch of analysis as the fractional calculus. The second develops a wide panorama of applications of the fractional calculus to various physical problems. This book recovers new perspectives in front of the reader dealing with turbulence and semiconductors, plasma and thermodynamics, mechanics and quantum optics, nanophysics and astrophysics. The book is addressed to students, engineers and physicists, specialists in theory of probability and statistics, in mathematical modeling and numerical simulations, to everybody who doesn't wish to stay apart from the new mathematical methods becoming more and more popular. Prof. Vladimir V. UCHAIKIN is a known Russian scientist and pedagogue, a Honored Worker of Russian High School, a member of the Russian Academy of Natural Sciences. He is the author of about three hundreds articles and more than a dozen books (mostly in Russian) in Cosmic ray physics, Mathematical physics, Levy stable statistics, Monte Carlo methods with applications to anomalous processes in complex systems of various levels: from quantum dots to the Milky Way galaxy.

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

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

by Anonim

  • Publisher : Elsevier
  • Release : 1974-09-05
  • Pages : 322
  • ISBN : 9780080956206
  • Language : En, Es, Fr & De
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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

Report

Report
A Book

by Anonim

  • Publisher : Unknown Publisher
  • Release : 1993
  • Pages : 329
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Applications in Engineering, Life and Social Sciences

Applications in Engineering, Life and Social Sciences
A Book

by Dumitru Bǎleanu,António Mendes Lopes

  • Publisher : Walter de Gruyter GmbH & Co KG
  • Release : 2019-04-01
  • Pages : 292
  • ISBN : 3110571927
  • Language : En, Es, Fr & De
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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 eighth volume collects authoritative chapters covering several applications of fractional calculus in engineering, life and social sciences, including applications in signal and image analysis, and chaos.