## General Fractional Derivatives with Applications in Viscoelasticity

A Book

#### byXiao-Jun Yang,Feng Gao,Yang Ju

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

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

Theory, Methods and Applications

#### byXiao-Jun Yang

• Publisher : CRC Press
• Release : 2019-05-10
• Pages : 364
• ISBN : 0429811535
• Language : En, Es, Fr & De

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.

## Applications of Viscoelasticity

Bituminous Materials Characterization and Modeling

• Publisher : Elsevier
• Release : 2021-04-27
• Pages : 244
• ISBN : 012821211X
• Language : En, Es, Fr & De

Applications of Viscoelasticity: Bituminous Materials Characterization and Modeling starts with an introduction to the theory of viscoelasticity, emphasizing its importance to various applications in material characterization and modeling. It next looks at constitutive viscoelastic functions, outlines basic equations for different loading conditions, and introduces the Boltzmann superposition principle, relaxation modulus, and creep compliance. Mechanical models, including integer-order and fractional-order are studied next, featuring real experimentation data alongside the benefits and drawbacks of using each model in various real-world scenarios. The book then covers the correspondence principle, followed by time–temperature superposition, featuring a simple procedure to construct a real master curve and challenges that might be encountered. The concluding chapters cover the Hopkins and Hamming, Park and Kim, and General Power law methods for interconversion of constitutive viscoelastic functions, applications of viscoelasticity for experimental tests, and incremental form of viscoelastic relations for numerical modeling. The book also includes supplementary codes that users can duplicate and use in their own work. Takes an applied approach to material viscoelasticity, explaining complicated viscoelastic equations and principles Presents examples of those equations and principles being applied to common problems in realworld settings Covers constitutive viscoelastic functions, including relaxation modulus and creep compliance Outlines the construction of a master curve of viscoelastic material considering time–temperature superposition Couples the correspondence principle with common viscoelastic experiments, such as threepoint bending beam, axial and torsional bar, and dynamic shear rheometer Provides supplementary codes

## Methods of Mathematical Modelling and Computation for Complex Systems

A Book

#### byJagdev Singh,Hemen Dutta,Devendra Kumar,Dumitru Baleanu,Jordan Hristov

• Publisher : Springer Nature
• Release : 2021-08-26
• Pages : 433
• ISBN : 3030771695
• Language : En, Es, Fr & De

This book contains several contemporary topics in the areas of mathematical modelling and computation for complex systems. The readers find several new mathematical methods, mathematical models and computational techniques having significant relevance in studying various complex systems. The chapters aim to enrich the understanding of topics presented by carefully discussing the associated problems and issues, possible solutions and their applications or relevance in other scientific areas of study and research. The book is a valuable resource for graduate students, researchers and educators in understanding and studying various new aspects associated with complex systems. Key Feature • The chapters include theory and application in a mix and balanced way. • Readers find reasonable details of developments concerning a topic included in this book. • The text is emphasized to present in self-contained manner with inclusion of new research problems and questions.

## An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions

A Book

#### byXiao-Jun Yang

• Release : 2021-01-23
• Pages : 502
• ISBN : 0323852823
• Language : En, Es, Fr & De

An Introduction to Hypergeometric, Supertigonometric, and Superhyperbolic Functions gives a basic introduction to the newly established hypergeometric, supertrigonometric, and superhyperbolic functions from the special functions viewpoint. The special functions, such as the Euler Gamma function, the Euler Beta function, the Clausen hypergeometric series, and the Gauss hypergeometric have been successfully applied to describe the real-world phenomena that involve complex behaviors arising in mathematics, physics, chemistry, and engineering. Presents a collection of the most up-to-date research, providing a complete overview of Multi-Objective Combinatorial Optimization problems and applications Includes a logical investigation of a family of the hypergeometric series Provides an historical overview for a family of the special polynomials Proposes a family of the hypergeometric supertrigonometric functions Covers a family of the hypergeometric superhyperbolic functions

## Fractional Calculus and Waves in Linear Viscoelasticity

An Introduction to Mathematical Models

#### byFrancesco Mainardi

• Publisher : World Scientific
• Release : 2010
• Pages : 368
• ISBN : 1848163304
• Language : En, Es, Fr & De

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 Integrals and Derivatives: “True” versus “False”

A Book

#### byYuri Luchko

• Publisher : MDPI
• Release : 2021-03-16
• Pages : 280
• ISBN : 303650494X
• Language : En, Es, Fr & De

This Special Issue is devoted to some serious problems that the Fractional Calculus (FC) is currently confronted with and aims at providing some answers to the questions like “What are the fractional integrals and derivatives?”, “What are their decisive mathematical properties?”, “What fractional operators make sense in applications and why?’’, etc. In particular, the “new fractional derivatives and integrals” and the models with these fractional order operators are critically addressed. The Special Issue contains both the surveys and the research contributions. A part of the articles deals with foundations of FC that are considered from the viewpoints of the pure and applied mathematics, and the system theory. Another part of the Special issue addresses the applications of the FC operators and the fractional differential equations. Several articles devoted to the numerical treatment of the FC operators and the fractional differential equations complete the Special Issue.

## Challenges in Mechanics of Time-Dependent Materials, Volume 2

Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics

#### byH. Jerry Qi,Bonnie Antoun,Richard Hall,Hongbing Lu,Alex Arzoumanidis,Meredith Silberstein,Jevan Furmanski,Alireza Amirkhizi,Joamin Gonzalez-Gutierrez

• Publisher : Springer
• Release : 2014-07-25
• Pages : 196
• ISBN : 3319069802
• Language : En, Es, Fr & De

Challenges in Mechanics of Time-Dependent Materials, Volume 2: Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics, the second volume of eight from the Conference, brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Experimental Mechanics, including papers in the following general technical research areas: Metallic, Polymeric and Composite Materials o Effects of Extreme Environments including Radiation Resistance, Damage, and Aging o Challenges in Time-dependent Behavior Modeling of Low, Moderate and High Strain Rates o Effects of Inhomogeneities on the Time-Dependent Behavior o Time dependent granular materials · Composite, Hybrid and Multifunctional Materials o Challenges in Time-dependent Behavior Modeling Viscoelastoplasticity and Damage o Effects of Interfaces and Interphases on the Time-Dependent Behavior · Mechanics of materials from advanced manufacturing, such as additive manufacturing o Property characterization from AM o Process modeling and simulations of AM o Material design using AM · Time-dependent and Small-scale Effects in Micro/Nano-scale Testing

## Boundary Element Methods in Applied Mechanics

Proceedings of the First Joint Japan/US Symposium on Boundary Element Methods, University of Tokyo, Tokyo, Japan, 3-6 October 1988

#### byMasataka Tanaka

• Publisher : Elsevier
• Release : 2013-10-22
• Pages : 571
• ISBN : 1483286967
• Language : En, Es, Fr & De

This Proceedings features a broad range of computational mechanics papers on both solid and fluid mechanics as well as electromagnetics, acoustics, heat transfer and other interdisciplinary problems. Topics covered include theoretical developments, numerical analysis, intelligent and adaptive solution strategies and practical applications.

## Mittag-Leffler Functions, Related Topics and Applications

A Book

#### byRudolf Gorenflo,Anatoly A. Kilbas,Francesco Mainardi,Sergei Rogosin

• Publisher : Springer Nature
• Release : 2020-10-27
• Pages : 540
• ISBN : 3662615509
• Language : En, Es, Fr & De

The 2nd edition of this book is essentially an extended version of the 1st and provides a very sound overview of the most important special functions of Fractional Calculus. It has been updated with material from many recent papers and includes several surveys of important results known before the publication of the 1st edition, but not covered there. As a result of researchers’ and scientists’ increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have caught the interest of the scientific community. Focusing on the theory of Mittag-Leffler functions, this volume offers a self-contained, comprehensive treatment, ranging from rather elementary matters to the latest research results. In addition to the theory the authors devote some sections of the work to applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular, the Mittag-Leffler functions make it possible to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and related special functions. The book is intended for a broad audience, comprising graduate students, university instructors and scientists in the field of pure and applied mathematics, as well as researchers in applied sciences like mathematical physics, theoretical chemistry, bio-mathematics, control theory and several other related areas.

## Untitled

A Book

#### byAnonim

• Publisher : Springer Nature
• Release : 2022
• Pages : 129
• ISBN : 3031045483
• Language : En, Es, Fr & De

## Fractional Derivatives for Physicists and Engineers

Volume I Background and Theory Volume II Applications

• Publisher : Springer Science & Business Media
• Release : 2013-07-09
• Pages : 385
• ISBN : 3642339115
• Language : En, Es, Fr & De

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.

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

A Book

#### byAnonim

• Publisher : Unknown Publisher
• Release : 1991
• Pages : 464
• ISBN : 9876543210XXX
• Language : En, Es, Fr & De

## Fractional Calculus and its Applications in Physics

A Book

#### byDumitru Baleanu,Devendra Kumar

• Publisher : Frontiers Media SA
• Release : 2019-11-15
• Pages : 93
• ISBN : 2889459586
• Language : En, Es, Fr & De

## Applications of Fractional Calculus in Physics

A Book

#### byR Hilfer

• Publisher : World Scientific
• Release : 2000-03-02
• Pages : 472
• ISBN : 9814496200
• Language : En, Es, Fr & De

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

## The Human Respiratory System

An Analysis of the Interplay between Anatomy, Structure, Breathing and Fractal Dynamics

#### byClara Mihaela Ionescu

• Publisher : Springer Science & Business Media
• Release : 2013-08-19
• Pages : 217
• ISBN : 144715388X
• Language : En, Es, Fr & De

The Human Respiratory System combines emerging ideas from biology and mathematics to show the reader how to produce models for the development of biomedical engineering applications associated with the lungs and airways. Mathematically mature but in its infancy as far as engineering uses are concerned, fractional calculus is the basis of the methods chosen for system analysis and modelling. This reflects two decades’ worth of conceptual development which is now suitable for bringing to bear in biomedical engineering. The text reveals the latest trends in modelling and identification of human respiratory parameters with a view to developing diagnosis and monitoring technologies. Of special interest is the notion of fractal structure which is indicative of the large-scale biological efficiency of the pulmonary system. The related idea of fractal dimension represents the adaptations in fractal structure caused by environmental factors, notably including disease. These basics are linked to model the dynamical patterns of breathing as a whole. The ideas presented in the book are validated using real data generated from healthy subjects and respiratory patients and rest on non-invasive measurement methods. The Human Respiratory System will be of interest to applied mathematicians studying the modelling of biological systems, to clinicians with interests outside the traditional borders of medicine, and to engineers working with technologies of either direct medical significance or for mitigating changes in the respiratory system caused by, for example, high-altitude or deep-sea environments.

## Fractional Differential Equations

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

#### byIgor Podlubny

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

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

## Fractional Order Systems

An Overview of Mathematics, Design, and Applications for Engineers

• Release : 2021-10-13
• Pages : 612
• ISBN : 0128243341
• Language : En, Es, Fr & De

Fractional Order Systems: An Overview of Mathematics, Design, and Applications for Engineers introduces applications from a design perspective, helping readers plan and design their own applications. The book includes the different techniques employed to design fractional-order systems/devices comprehensively and straightforwardly. Furthermore, mathematics is available in the literature on how to solve fractional-order calculus for system applications. This book introduces the mathematics that has been employed explicitly for fractional-order systems. It will prove an excellent material for students and scholars who want to quickly understand the field of fractional-order systems and contribute to its different domains and applications. Fractional-order systems are believed to play an essential role in our day-to-day activities. Therefore, several researchers around the globe endeavor to work in the different domains of fractional-order systems. The efforts include developing the mathematics to solve fractional-order calculus/systems and to achieve the feasible designs for various applications of fractional-order systems. Presents a simple and comprehensive understanding of the field of fractional-order systems Offers practical knowledge on the design of fractional-order systems for different applications Exposes users to possible new applications for fractional-order systems

## Fractional Dynamical Systems

Methods, Algorithms and Applications

#### byPiotr Kulczycki

• Publisher : Springer Nature
• Release : 2022
• Pages : 398
• ISBN : 3030899721
• Language : En, Es, Fr & De

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, becausein generalproperties 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, andfinallysome examples of relevant and successful practical applications.

## Mathematical Economics

Application of Fractional Calculus

#### byVasily E. Tarasov

• Publisher : MDPI
• Release : 2020-06-03
• Pages : 278
• ISBN : 303936118X
• Language : En, Es, Fr & De

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.