Existence Families, Functional Calculi and Evolution Equations
Existence Families, Functional Calculi and Evolution Equations

Author: Ralph DeLaubenfels

Publisher: Springer

Published: 2006-11-15

Total Pages: 254

ISBN-13: 3540483225

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This book presents an operator-theoretic approach to ill-posed evolution equations. It presents the basic theory, and the more surprising examples, of generalizations of strongly continuous semigroups known as 'existent families' and 'regularized semigroups'. These families of operators may be used either to produce all initial data for which a solution in the original space exists, or to construct a maximal subspace on which the problem is well-posed. Regularized semigroups are also used to construct functional, or operational, calculi for unbounded operators. The book takes an intuitive and constructive approach by emphasizing the interaction between functional calculus constructions and evolution equations. One thinks of a semigroup generated by A as etA and thinks of a regularized semigroup generated by A as etA g(A), producing solutions of the abstract Cauchy problem for initial data in the image of g(A). Material that is scattered throughout numerous papers is brought together and presented in a fresh, organized way, together with a great deal of new material.

From Divergent Power Series to Analytic Functions
From Divergent Power Series to Analytic Functions

Author: Werner Balser

Publisher: Springer

Published: 1994-08-29

Total Pages: 124

ISBN-13: 9783540582687

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Multisummability is a method which, for certain formal power series with radius of convergence equal to zero, produces an analytic function having the formal series as its asymptotic expansion. This book presents the theory of multisummabi- lity, and as an application, contains a proof of the fact that all formal power series solutions of non-linear meromorphic ODE are multisummable. It will be of use to graduate students and researchers in mathematics and theoretical physics, and especially to those who encounter formal power series to (physical) equations with rapidly, but regularly, growing coefficients.

Fractional Evolution Equations and Inclusions
Fractional Evolution Equations and Inclusions

Author: Yong Zhou

Publisher: Academic Press

Published: 2016-02-05

Total Pages: 296

ISBN-13: 0128047755

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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

Basic Theory Of Fractional Differential Equations (Second Edition)
Basic Theory Of Fractional Differential Equations (Second Edition)

Author: Yong Zhou

Publisher: World Scientific

Published: 2016-10-20

Total Pages: 380

ISBN-13: 9813148187

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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.

Basic Theory Of Fractional Differential Equations (Third Edition)
Basic Theory Of Fractional Differential Equations (Third Edition)

Author: Yong Zhou

Publisher: World Scientific

Published: 2023-10-06

Total Pages: 516

ISBN-13: 9811271704

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This accessible monograph is devoted to a rapidly developing area on the research of qualitative theory of fractional ordinary differential equations and evolution equations. It is self-contained and unified in presentation, and 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, Picard operators technique, critical point theory and semigroups theory. This book is based on the research work done so far by the author and other experts, and contains comprehensive up-to-date materials on the topic.In this third edition, four new topics have been added: Hilfer fractional evolution equations and infinite interval problems, oscillations and nonoscillations, fractional Hamiltonian systems, fractional Rayleigh-Stokes equations, and wave equations. The bibliography has also been updated and expanded.This book is useful to researchers, graduate or PhD students dealing with fractional calculus and applied analysis, differential equations, and related areas of research.

Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations
Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations

Author: Marko Kostić

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2019-05-06

Total Pages: 508

ISBN-13: 3110641259

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This book discusses almost periodic and almost automorphic solutions to abstract integro-differential Volterra equations that are degenerate in time, and in particular equations whose solutions are governed by (degenerate) solution operator families with removable singularities at zero. It particularly covers abstract fractional equations and inclusions with multivalued linear operators as well as abstract fractional semilinear Cauchy problems.

The Cauchy Problem for Higher Order Abstract Differential Equations
The Cauchy Problem for Higher Order Abstract Differential Equations

Author: Ti-Jun Xiao

Publisher: Springer

Published: 2013-12-11

Total Pages: 314

ISBN-13: 3540494790

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The main purpose of this book is to present the basic theory and some recent de velopments concerning the Cauchy problem for higher order abstract differential equations u(n)(t) + ~ AiU(i)(t) = 0, t ~ 0, { U(k)(O) = Uk, 0 ~ k ~ n-l. where AQ, Ab . . . , A - are linear operators in a topological vector space E. n 1 Many problems in nature can be modeled as (ACP ). For example, many n initial value or initial-boundary value problems for partial differential equations, stemmed from mechanics, physics, engineering, control theory, etc. , can be trans lated into this form by regarding the partial differential operators in the space variables as operators Ai (0 ~ i ~ n - 1) in some function space E and letting the boundary conditions (if any) be absorbed into the definition of the space E or of the domain of Ai (this idea of treating initial value or initial-boundary value problems was discovered independently by E. Hille and K. Yosida in the forties). The theory of (ACP ) is closely connected with many other branches of n mathematics. Therefore, the study of (ACPn) is important for both theoretical investigations and practical applications. Over the past half a century, (ACP ) has been studied extensively.

Recent Developments in Evolution Equations
Recent Developments in Evolution Equations

Author: G F Roach

Publisher: CRC Press

Published: 1995-04-28

Total Pages: 268

ISBN-13: 9780582246690

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This book presents the majority of talks given at an International Converence held recently at the University of Strathclyde in Glasgow. The works presented focus on the analysis of mathematical models of systems evolving with time. The main topics are semigroups and related subjects connected with applications to partial differential equations of evolution type. Topics of particular interest include spectral and asymptotic properties of semigroups, B evolution scattering theory, and coagulation fragmentation phenomena.

Evolution Semigroups in Dynamical Systems and Differential Equations
Evolution Semigroups in Dynamical Systems and Differential Equations

Author: Carmen Chicone

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 375

ISBN-13: 0821811851

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The main theme of the book is the spectral theory for evolution operators and evolution semigroups, a subject tracing its origins to the classical results of J. Mather on hyperbolic dynamical systems and J. Howland on nonautonomous Cauchy problems. The authors use a wide range of methods and offer a unique presentation. The authors give a unifying approach for a study of infinite-dimensional nonautonomous problems, which is based on the consistent use of evolution semigroups. This unifying idea connects various questions in stability of semigroups, infinite-dimensional hyperbolic linear skew-product flows, translation Banach algebras, transfer operators, stability radii in control theory, Lyapunov exponents, magneto-dynamics and hydro-dynamics. Thus the book is much broader in scope than existing books on asymptotic behavior of semigroups. Included is a solid collection of examples from different areas of analysis, PDEs, and dynamical systems. This is the first monograph where the spectral theory of infinite dimensional linear skew-product flows is described together with its connection to the multiplicative ergodic theorem; the same technique is used to study evolution semigroups, kinematic dynamos, and Ruelle operators; the theory of stability radii, an important concept in control theory, is also presented. Examples are included and non-traditional applications are provided.

The Functional Calculus for Sectorial Operators
The Functional Calculus for Sectorial Operators

Author: Markus Haase

Publisher: Springer Science & Business Media

Published: 2006-08-18

Total Pages: 399

ISBN-13: 3764376988

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This book contains a systematic and partly axiomatic treatment of the holomorphic functional calculus for unbounded sectorial operators. The account is generic so that it can be used to construct and interrelate holomorphic functional calculi for other types of unbounded operators. Particularly, an elegant unified approach to holomorphic semigroups is obtained. The last chapter describes applications to PDE, evolution equations and approximation theory as well as the connection with harmonic analysis.