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## Fractional Evolution Equations and Inclusions

- Author : Yong Zhou
- Publisher :Unknown
- Release Date :2016-02-05
- Total pages :294
- ISBN : 9780128047750

**Summary :** Fractional evolution inclusions are an important form of differential inclusions within nonlinear mathematical analysis. They are generalizations of the much more widely developed fractional evolution equations (such as time-fractional diffusion equations) seen through the lens of multivariate analysis. Compared to fractional evolution equations, research on the theory of fractional differential inclusions is however only in its initial stage of development. This is important because differential models with the fractional derivative providing an excellent instrument for the description of memory and hereditary properties, and have recently been proved valuable tools in the modeling of many physical phenomena. The fractional order models of real systems are always more adequate than the classical integer order models, since the description of some systems is more accurate when the fractional derivative is used. The advantages of fractional derivatization become evident in modeling mechanical and electrical properties of real materials, description of rheological properties of rocks and in various other fields. Such models are interesting for engineers and physicists as well as so-called pure mathematicians. Phenomena investigated in hybrid systems with dry friction, processes of controlled heat transfer, obstacle problems and others can be described with the help of various differential inclusions, both linear and nonlinear. Fractional Evolution Equations and Inclusions is devoted to a rapidly developing area of the research for fractional evolution equations & inclusions and their applications to control theory. It studies Cauchy problems for fractional evolution equations, and fractional evolution inclusions with Hille-Yosida operators. It discusses control problems for systems governed by fractional evolution equations. Finally it provides an investigation of fractional stochastic evolution inclusions in Hilbert spaces. Systematic analysis of existence theory and topological structure of solution sets for fractional evolution inclusions and control systems Differential models with fractional derivative provide an excellent instrument for the description of memory and hereditary properties, and their description and working will provide valuable insights into the modelling of many physical phenomena suitable for engineers and physicists The book provides the necessary background material required to go further into the subject and explore the rich research literature

## Fractional-Order Equations and Inclusions

- Author : Michal Fečkan,JinRong Wang,Michal Pospíšil
- Publisher :Unknown
- Release Date :2017-11-07
- Total pages :383
- ISBN : 9783110521559

**Summary :** This book presents fractional difference, integral, differential, evolution equations and inclusions, and discusses existence and asymptotic behavior of their solutions. Controllability and relaxed control results are obtained. Combining rigorous deduction with abundant examples, it is of interest to nonlinear science researchers using fractional equations as a tool, and physicists, mechanics researchers and engineers studying relevant topics. Contents Fractional Difference Equations Fractional Integral Equations Fractional Differential Equations Fractional Evolution Equations: Continued Fractional Differential Inclusions

## Fractional Differential Equations, Inclusions and Inequalities with Applications

- Author : Sotiris K. Ntouyas
- Publisher :Unknown
- Release Date :2020-11-09
- Total pages :518
- ISBN : 9783039432189

**Summary :** During the last decade, there has been an increased interest in fractional differential equations, inclusions, and inequalities, as they play a fundamental role in the modeling of numerous phenomena, in particular, in physics, biomathematics, blood flow phenomena, ecology, environmental issues, viscoelasticity, aerodynamics, electrodynamics of complex medium, electrical circuits, electron-analytical chemistry, control theory, etc. This book presents collective works published in the recent Special Issue (SI) entitled "Fractional Differential Equation, Inclusions and Inequalities with Applications" of the journal Mathematics. This Special Issue presents recent developments in the theory of fractional differential equations and inequalities. Topics include but are not limited to the existence and uniqueness results for boundary value problems for different types of fractional differential equations, a variety of fractional inequalities, impulsive fractional differential equations, and applications in sciences and engineering.

## Advanced Functional Evolution Equations and Inclusions

- Author : Saïd Abbas,Mouffak Benchohra
- Publisher :Unknown
- Release Date :2015-06-30
- Total pages :408
- ISBN : 9783319177687

**Summary :** This book presents up-to-date results on abstract evolution equations and differential inclusions in infinite dimensional spaces. It covers equations with time delay and with impulses, and complements the existing literature in functional differential equations and inclusions. The exposition is devoted to both local and global mild solutions for some classes of functional differential evolution equations and inclusions, and other densely and non-densely defined functional differential equations and inclusions in separable Banach spaces or in Fréchet spaces. The tools used include classical fixed points theorems and the measure-of non-compactness, and each chapter concludes with a section devoted to notes and bibliographical remarks. This monograph is particularly useful for researchers and graduate students studying pure and applied mathematics, engineering, biology and all other applied sciences.

## Topological Structure of the Solution Set for Evolution Inclusions

- Author : Yong Zhou,Rong-Nian Wang,Li Peng
- Publisher :Unknown
- Release Date :2017-10-31
- Total pages :269
- ISBN : 9789811066566

**Summary :** This book systematically presents the topological structure of solution sets and attractability for nonlinear evolution inclusions, together with its relevant applications in control problems and partial differential equations. It provides readers the background material needed to delve deeper into the subject and explore the rich research literature. In addition, the book addresses many of the basic techniques and results recently developed in connection with this theory, including the structure of solution sets for evolution inclusions with m-dissipative operators; quasi-autonomous and non-autonomous evolution inclusions and control systems; evolution inclusions with the Hille-Yosida operator; functional evolution inclusions; impulsive evolution inclusions; and stochastic evolution inclusions. Several applications of evolution inclusions and control systems are also discussed in detail. Based on extensive research work conducted by the authors and other experts over the past four years, the information presented is cutting-edge and comprehensive. As such, the book fills an important gap in the body of literature on the structure of evolution inclusions and its applications.

## Topics in Fractional Differential Equations

- Author : Saïd Abbas,Mouffak Benchohra,Gaston M. N'Guérékata
- Publisher :Unknown
- Release Date :2012-08-17
- Total pages :398
- ISBN : 9781461440369

**Summary :** Topics in Fractional Differential Equations is devoted to the existence and uniqueness of solutions for various classes of Darboux problems for hyperbolic differential equations or inclusions involving the Caputo fractional derivative. Fractional calculus generalizes the integrals and derivatives to non-integer orders. During the last decade, fractional calculus was found to play a fundamental role in the modeling of a considerable number of phenomena; in particular the modeling of memory-dependent and complex media such as porous media. It has emerged as an important tool for the study of dynamical systems where classical methods reveal strong limitations. Some equations present delays which may be finite, infinite, or state-dependent. Others are subject to an impulsive effect. The above problems are studied using the fixed point approach, the method of upper and lower solution, and the Kuratowski measure of noncompactness. This book is addressed to a wide audience of specialists such as mathematicians, engineers, biologists, and physicists.

## Discrete Fractional Calculus

- Author : Christopher Goodrich,Allan C. Peterson
- Publisher :Unknown
- Release Date :2016-02-09
- Total pages :556
- ISBN : 9783319255620

**Summary :** This text provides the first comprehensive treatment of the discrete fractional calculus. Experienced researchers will find the text useful as a reference for discrete fractional calculus and topics of current interest. Students who are interested in learning about discrete fractional calculus will find this text to provide a useful starting point. Several exercises are offered at the end of each chapter and select answers have been provided at the end of the book. The presentation of the content is designed to give ample flexibility for potential use in a myriad of courses and for independent study. The novel approach taken by the authors includes a simultaneous treatment of the fractional- and integer-order difference calculus (on a variety of time scales, including both the usual forward and backwards difference operators). The reader will acquire a solid foundation in the classical topics of the discrete calculus while being introduced to exciting recent developments, bringing them to the frontiers of the subject. Most chapters may be covered or omitted, depending upon the background of the student. For example, the text may be used as a primary reference in an introductory course for difference equations which also includes discrete fractional calculus. Chapters 1—2 provide a basic introduction to the delta calculus including fractional calculus on the set of integers. For courses where students already have background in elementary real analysis, Chapters 1—2 may be covered quickly and readers may then skip to Chapters 6—7 which present some basic results in fractional boundary value problems (FBVPs). Chapters 6—7 in conjunction with some of the current literature listed in the Bibliography can provide a basis for a seminar in the current theory of FBVPs. For a two-semester course, Chapters 1—5 may be covered in depth, providing a very thorough introduction to both the discrete fractional calculus as well as the integer-order calculus.

## Functional Fractional Calculus

- Author : Shantanu Das
- Publisher :Unknown
- Release Date :2011-06-01
- Total pages :612
- ISBN : 9783642205453

**Summary :** When a new extraordinary and outstanding theory is stated, it has to face criticism and skeptism, because it is beyond the usual concept. The fractional calculus though not new, was not discussed or developed for a long time, particularly for lack of its application to real life problems. It is extraordinary because it does not deal with ‘ordinary’ differential calculus. It is outstanding because it can now be applied to situations where existing theories fail to give satisfactory results. In this book not only mathematical abstractions are discussed in a lucid manner, with physical mathematical and geometrical explanations, but also several practical applications are given particularly for system identification, description and then efficient controls. The normal physical laws like, transport theory, electrodynamics, equation of motions, elasticity, viscosity, and several others of are based on ‘ordinary’ calculus. In this book these physical laws are generalized in fractional calculus contexts; taking, heterogeneity effect in transport background, the space having traps or islands, irregular distribution of charges, non-ideal spring with mass connected to a pointless-mass ball, material behaving with viscous as well as elastic properties, system relaxation with and without memory, physics of random delay in computer network; and several others; mapping the reality of nature closely. The concept of fractional and complex order differentiation and integration are elaborated mathematically, physically and geometrically with examples. The practical utility of local fractional differentiation for enhancing the character of singularity at phase transition or characterizing the irregularity measure of response function is deliberated. Practical results of viscoelastic experiments, fractional order controls experiments, design of fractional controller and practical circuit synthesis for fractional order elements are elaborated in this book. The book also maps theory of classical integer order differential equations to fractional calculus contexts, and deals in details with conflicting and demanding initialization issues, required in classical techniques. The book presents a modern approach to solve the ‘solvable’ system of fractional and other differential equations, linear, non-linear; without perturbation or transformations, but by applying physical principle of action-and-opposite-reaction, giving ‘approximately exact’ series solutions. Historically, Sir Isaac Newton and Gottfried Wihelm Leibniz independently discovered calculus in the middle of the 17th century. In recognition to this remarkable discovery, J.von Neumann remarked, “...the calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more equivocally than anything else the inception of modern mathematical analysis which is logical development, still constitute the greatest technical advance in exact thinking.” This XXI century has thus started to ‘think-exactly’ for advancement in science & technology by growing application of fractional calculus, and this century has started speaking the language which nature understands the best.

## Basic Theory of Fractional Differential Equations

- Author : Yong Zhou,JinRong Wang,Lu Zhang
- Publisher :Unknown
- Release Date :2016-10-20
- Total pages :380
- ISBN : 9789813148185

**Summary :** This invaluable monograph is devoted to a rapidly developing area on the research of qualitative theory of fractional ordinary and partial differential equations. It provides the readers the necessary background material required to go further into the subject and explore the rich research literature. The tools used include many classical and modern nonlinear analysis methods such as fixed point theory, measure of noncompactness method, topological degree method, the technique of Picard operators, critical point theory and semigroup theory. Based on the research work carried out by the authors and other experts during the past seven years, the contents are very recent and comprehensive. In this edition, two new topics have been added, that is, fractional impulsive differential equations, and fractional partial differential equations including fractional Navier–Stokes equations and fractional diffusion equations. Contents:Preliminaries:IntroductionSome Notations, Concepts and LemmasFractional CalculusSome Results from Nonlinear AnalysisSemigroupsFractional Functional Differential Equations:IntroductionNeutral Equations with Bounded Delayp-Type Neutral EquationsNeutral Equations with Infinite DelayIterative Functional Differential EquationsNotes and RemarksFractional Ordinary Differential Equations in Banach Spaces:IntroductionCauchy Problems via Measure of Noncompactness MethodCauchy Problems via Topological Degree MethodCauchy Problems via Picard Operators TechniqueNotes and RemarksFractional Abstract Evolution Equations:IntroductionEvolution Equations with Riemann–Liouville DerivativeEvolution Equations with Caputo DerivativeNonlocal Problems for Evolution EquationsAbstract Cauchy Problems with Almost Sectorial OperatorsNotes and RemarksFractional Impulsive Differential Equations:IntroductionImpulsive Initial Value ProblemsImpulsive Boundary Value ProblemsImpulsive Langevin EquationsImpulsive Evolution EquationsNotes and RemarksFractional Boundary Value Problems:IntroductionSolution for BVP with Left and Right Fractional IntegralsMultiple Solutions for BVP with ParametersInfinite Solutions for BVP with Left and Right Fractional IntegralsSolutions for BVP with Left and Right Fractional DerivativesNotes and RemarksFractional Partial Differential Equations:IntroductionFractional Navier–Stokes EquationsFractional Euler–Lagrange EquationsFractional Diffusion EquationsFractional Schrödinger EquationsNotes and Remarks Readership: Researchers and graduate or PhD students dealing with fractional calculus and applied analysis, differential equations and related areas of research.

## Impulsive Differential Inclusions

- Author : John R. Graef,Johnny Henderson,Abdelghani Ouahab
- Publisher :Unknown
- Release Date :2013-07-31
- Total pages :410
- ISBN : 9783110295313

**Summary :** Differential equations with impulses arise as models of many evolving processes that are subject to abrupt changes, such as shocks, harvesting, and natural disasters. These phenomena involve short-term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. In models involving such perturbations, it is natural to assume these perturbations act instantaneously or in the form of impulses. As a consequence, impulsive differential equations have been developed in modeling impulsive problems in physics, population dynamics, ecology, biotechnology, industrial robotics, pharmacokinetics, optimal control, and so forth. There are also many different studies in biology and medicine for which impulsive differential equations provide good models. During the last 10 years, the authors have been responsible for extensive contributions to the literature on impulsive differential inclusions via fixed point methods. This book is motivated by that research as the authors endeavor to bring under one cover much of those results along with results by other researchers either affecting or affected by the authors' work. The questions of existence and stability of solutions for different classes of initial value problems for impulsive differential equations and inclusions with fixed and variable moments are considered in detail. Attention is also given to boundary value problems. In addition, since differential equations can be viewed as special cases of differential inclusions, significant attention is also given to relative questions concerning differential equations. This monograph addresses a variety of side issues that arise from its simpler beginnings as well.

## Nonlocal Diffusion and Applications

- Author : Claudia Bucur,Enrico Valdinoci
- Publisher :Unknown
- Release Date :2016-04-08
- Total pages :155
- ISBN : 9783319287393

**Summary :** Working in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation, some applications to water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrödinger equations are given. Furthermore, an example of an s-harmonic function, its harmonic extension and some insight into a fractional version of a classical conjecture due to De Giorgi are presented. Although the aim is primarily to gather some introductory material concerning applications of the fractional Laplacian, some of the proofs and results are new. The work is entirely self-contained, and readers who wish to pursue related subjects of interest are invited to consult the rich bibliography for guidance.

## Theory and Applications of Fractional Differential Equations

- Author : A.A. Kilbas,H. M. Srivastava,J.J. Trujillo
- Publisher :Unknown
- Release Date :2006-03-02
- Total pages :523
- ISBN : 0444518320

**Summary :** This work aims to present, in a systematic manner, results including the existence and uniqueness of solutions for the Cauchy Type and Cauchy problems involving nonlinear ordinary fractional differential equations.

## One-Parameter Semigroups for Linear Evolution Equations

- Author : Klaus-Jochen Engel,Rainer Nagel
- Publisher :Unknown
- Release Date :2006-04-06
- Total pages :589
- ISBN : 9780387226422

**Summary :** This book explores the theory of strongly continuous one-parameter semigroups of linear operators. A special feature of the text is an unusually wide range of applications such as to ordinary and partial differential operators, to delay and Volterra equations, and to control theory. Also, the book places an emphasis on philosophical motivation and the historical background.

## Implicit Fractional Differential and Integral Equations

- Author : Saïd Abbas,Mouffak Benchohra,John R. Graef,Johnny Henderson
- Publisher :Unknown
- Release Date :2018-02-05
- Total pages :359
- ISBN : 9783110553819

**Summary :** This book deals with the existence and stability of solutions to initial and boundary value problems for functional differential and integral equations and inclusions involving the Riemann-Liouville, Caputo, and Hadamard fractional derivatives and integrals. A wide variety of topics is covered in a mathematically rigorous manner making this work a valuable source of information for graduate students and researchers working with problems in fractional calculus. ContentsPreliminary Background Nonlinear Implicit Fractional Differential Equations Impulsive Nonlinear Implicit Fractional Differential Equations Boundary Value Problems for Nonlinear Implicit Fractional Differential Equations Boundary Value Problems for Impulsive NIFDE Integrable Solutions for Implicit Fractional Differential Equations Partial Hadamard Fractional Integral Equations and Inclusions Stability Results for Partial Hadamard Fractional Integral Equations and Inclusions Hadamard–Stieltjes Fractional Integral Equations Ulam Stabilities for Random Hadamard Fractional Integral Equations

## Harmonic Analysis Method for Nonlinear Evolution Equations, I

- Author : Baoxiang Wang,Zhaohui Huo,Zihua Guo,Chengchun Hao
- Publisher :Unknown
- Release Date :2011
- Total pages :283
- ISBN : 9789814360746

**Summary :** This monograph provides a comprehensive overview on a class of nonlinear evolution equations, such as nonlinear SchrAdinger equations, nonlinear KleinOCoGordon equations, KdV equations as well as NavierOCoStokes equations and Boltzmann equations. The global wellposedness to the Cauchy problem for those equations is systematically studied by using the harmonic analysis methods. This book is self-contained and may also be used as an advanced textbook by graduate students in analysis and PDE subjects and even ambitious undergraduate students.

## Functional Analysis and Semi-groups

- Author : Einar Hille,Ralph Saul Phillips
- Publisher :Unknown
- Release Date :1996-02-06
- Total pages :808
- ISBN : 9780821810316

**Summary :** Early in 1952 it became obvious that a new printing would be needed, and new advances in the theory called for extensive revision. It has been completely rewritten, mostly by Phillips, and much has been added while keeping the existing framework. Thus, the algebraic tools play a major role, and are introduced early, leading to a more satisfactory operational calculus and spectral theory. The Laplace-Stieltjes transform methods, used by Hille, have not been replaced but rather supplemented by the new tools. - Foreword.

## Stability and Oscillations in Delay Differential Equations of Population Dynamics

- Author : K. Gopalsamy
- Publisher :Unknown
- Release Date :2013-03-14
- Total pages :502
- ISBN : 9789401579209

**Summary :** This monograph provides a definitive overview of recent advances in the stability and oscillation of autonomous delay differential equations. Topics include linear and nonlinear delay and integrodifferential equations, which have potential applications to both biological and physical dynamic processes. Chapter 1 deals with an analysis of the dynamical characteristics of the delay logistic equation, and a number of techniques and results relating to stability, oscillation and comparison of scalar delay and integrodifferential equations are presented. Chapter 2 provides a tutorial-style introduction to the study of delay-induced Hopf bifurcation to periodicity and the related computations for the analysis of the stability of bifurcating periodic solutions. Chapter 3 is devoted to local analyses of nonlinear model systems and discusses many methods applicable to linear equations and their perturbations. Chapter 4 considers global convergence to equilibrium states of nonlinear systems, and includes oscillations of nonlinear systems about their equilibria. Qualitative analyses of both competitive and cooperative systems with time delays feature in both Chapters 3 and 4. Finally, Chapter 5 deals with recent developments in models of neutral differential equations and their applications to population dynamics. Each chapter concludes with a number of exercises and the overall exposition recommends this volume as a good supplementary text for graduate courses. For mathematicians whose work involves functional differential equations, and whose interest extends beyond the boundaries of linear stability analysis.

## Semilinear Evolution Equations and Their Applications

- Author : Toka Diagana
- Publisher :Unknown
- Release Date :2018-10-23
- Total pages :189
- ISBN : 9783030004491

**Summary :** This book, which is a continuation of Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, presents recent trends and developments upon fractional, first, and second order semilinear difference and differential equations, including degenerate ones. Various stability, uniqueness, and existence results are established using various tools from nonlinear functional analysis and operator theory (such as semigroup methods). Various applications to partial differential equations and the dynamic of populations are amply discussed. This self-contained volume is primarily intended for advanced undergraduate and graduate students, post-graduates and researchers, but may also be of interest to non-mathematicians such as physicists and theoretically oriented engineers. It can also be used as a graduate text on evolution equations and difference equations and their applications to partial differential equations and practical problems arising in population dynamics. For completeness, detailed preliminary background on Banach and Hilbert spaces, operator theory, semigroups of operators, and almost periodic functions and their spectral theory are included as well.

## Advances in Fractional Calculus

- Author : J. Sabatier,O. P. Agrawal,J. A. Tenreiro Machado
- Publisher :Unknown
- Release Date :2007-07-28
- Total pages :552
- ISBN : 9781402060427

**Summary :** 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.

## The Analysis of Fractional Differential Equations

- Author : Kai Diethelm
- Publisher :Unknown
- Release Date :2010-08-18
- Total pages :247
- ISBN : 9783642145742

**Summary :** Fractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations.

## Analytic and Elementary Number Theory

- Author : Krishnaswami Alladi,P.D.T.A. Elliott,A. Granville,G. Tenenbaum
- Publisher :Unknown
- Release Date :2013-12-21
- Total pages :300
- ISBN : 9781475745078

**Summary :** This volume contains a collection of papers in Analytic and Elementary Number Theory in memory of Professor Paul Erdös, one of the greatest mathematicians of this century. Written by many leading researchers, the papers deal with the most recent advances in a wide variety of topics, including arithmetical functions, prime numbers, the Riemann zeta function, probabilistic number theory, properties of integer sequences, modular forms, partitions, and q-series. Audience: Researchers and students of number theory, analysis, combinatorics and modular forms will find this volume to be stimulating.