Homogenization is a method for modelling processes in complex structures. These processes are far too complex for analytic and numerical methods and are best described by PDEs with rapidly oscillating coefficients - a technique that has become increasingly important in the last three decades due to its multiple applications in the areas of optimization, radiophysics, filtration theory, rheology, elasticity theory, and other domains of mechanics, physics, and technology. The present monograph is a comprehensive study of homogenization problems, describing various physical processes in micro-inhomogeneous media. A variety of techniques are used - specifically functional analysis, the spectral theory for differential operators, the Laplace transform, and, most importantly, a new variational PDE method for studying the asymptotic behavior of solutions of stationary boundary value problems. Along with complete proofs of all main results, numerous examples of typical structures of micro-inhomogeneous media with their corresponding homogenized models are provided. Graduate students, applied mathematicians, physicists, engineers, and specialists in mechanics will benefit from this monograp

This substantially revised second edition teaches the bifurcation of asymptotic solutions to evolution problems governed by nonlinear differential equations. Written not just for mathematicians, it appeals to the widest audience of learners, including engineers, biologists, chemists, physicists and economists. For this reason, it uses only well-known methods of classical analysis at foundation level, while the applications and examples are specially chosen to be as varied as possible.

This book is the first systematic presentation of the theory of dynamical systems under the influence of randomness. It includes products of random mappings as well as random and stochastic differential equations. The basic mulitplicative ergodic theorem is presented and provides a random substitute for linear algebra. On its basis random invariant manifolds are constructed, systems are simplified by smooth random coordinate transformations (random normal forms), and qualitative changes in families of random systems (random bifurcation theory) are studied. Numerous instructive examples are treated analytically or numerically. The main intention, however, is to present a reliable and rather complete source of reference which lays the foundation for future work and applications.

This monograph describes global propagation of regular nonlinear hyperbolic waves described by first-order quasilinear hyperbolic systems in one dimension. The exposition is clear, concise, and unfolds systematically beginning with introductory material and leading to the original research of the authors. Topics are motivated with a number of physical examples from the areas of elastic materials, one-dimensional gas dynamics, and waves. Aimed at researchers and graduate students in partial differential equations and related topics, this book will stimulate further research and help readers further understand important aspects and recent progress of regular nonlinear hyperbolic waves.

Infinite interval problems abound in nature and yet until now there has been no book dealing with such problems. The main reason for this seems to be that until the 1970's for the infinite interval problem all the theoretical results available required rather technical hypotheses and were applicable only to narrowly defined classes of problems. Thus scientists mainly offer d and used special devices to construct the numerical solution assuming tacitly the existence of a solution. In recent years a mixture of classical analysis and modern fixed point theory has been employed to study the existence of solutions to infinite interval problems. This has resulted in widely applicable results. This monograph is a cumulation mainly of the authors' research over a period of more than ten years and offers easily verifiable existence criteria for differential, difference and integral equations over the infinite interval. An important feature of this monograph is that we illustrate almost all the results with examples. The plan of this monograph is as follows. In Chapter 1 we present the existence theory for second order boundary value problems on infinite intervals. We begin with several examples which model real world phenom ena. A brief history of the infinite interval problem is also included. We then present general existence results for several different types of boundary value problems. Here we note that for the infinite interval problem only two major approaches are available in the literature."

Bifurcation theory and catastrophe theory are two well-known areas within the field of dynamical systems. Both are studies of smooth systems, focusing on properties that seem to be manifestly non-smooth. Bifurcation theory is concerned with the sudden changes that occur in a system when one or more parameters are varied. Examples of such are familiar to students of differential equations, from phase portraits. Understanding the bifurcations of the differential equations that describe real physical systems provides important information about the behavior of the systems. Catastrophe theory became quite famous during the 1970's, mostly because of the sensation caused by the usually less than rigorous applications of its principal ideas to "hot topics," such as the characterization of personalities and the difference between a "genius" and a "maniac." Catastrophe theory is accurately described as singularity theory and its (genuine) applications. The authors of this book, previously published as Volume 5 of the Encyclopaedia, have given a masterly exposition of these two theories, with penetrating insight.

A self-contained and systematic development of an aspect of analysis which deals with the theory of fundamental solutions for differential operators, and their applications to boundary value problems of mathematical physics, applied mathematics, and engineering, with the related computational aspects.

Whether costs are to be reduced, profits to be maximized, or scarce resources to be used wisely, optimization methods are available to guide decision making. In online optimization the main issue is incomplete data, and the scientific challenge: How well can an online algorithm perform? Can one guarantee solution quality, even without knowing all data in advance? In real-time optimization there is an additional requirement, decisions have to be computed very fast in relation to the time frame of the instance we consider. Online and real-time optimization problems occur in all branches of optimization. These areas have developed their own techniques but they are addressing the same issues: quality, stability, and robustness of the solutions. To fertilize this emerging topic of optimization theory and to foster cooperation between the different branches of optimization, the Deutsche Forschungsgemeinschaft (DFG) has supported a Priority Programme "Online Optimization of Large Systems".

This volume is a collection of papers devoted to the 70th birthday of Professor Vladimir Rabinovich. The opening article (by Stefan Samko) includes a short biography of Vladimir Rabinovich, along with some personal recollections and bibliography of his work. It is followed by twenty research and survey papers in various branches of analysis (pseudodifferential operators and partial differential equations, Toeplitz, Hankel, and convolution type operators, variable Lebesgue spaces, etc.) close to Professor Rabinovich's research interests. Many of them are written by participants of the International workshop Analysis, Operator Theory, and Mathematical Physics (Ixtapa, Mexico, January 23 27, 2012) having a long history of scientific collaboration with Vladimir Rabinovich, and are partially based on the talks presented there.The volume will be of great interest to researchers and graduate students in differential equations, operator theory, functional and harmonic analysis, and mathematical physics. "

(NOTES)This text focuses on the topics which are an essential part of the engineering mathematics course:ordinary differential equations, vector calculus, linear algebra and partial differential equations. Advantages over competing texts: 1. The text has a large number of examples and problems - a typical section having 25 quality problems directly related to the text. 2. The authors use a practical engineering approach based upon solving equations. All ideas and definitions are introduced from this basic viewpoint, which allows engineers in their second year to understand concepts that would otherwise be impossibly abstract. Partial differential equations are introduced in an engineering and science context based upon modelling of physical problems. A strength of the manuscript is the vast number of applications to real-world problems, each treated completely and in sufficient depth to be self-contained. 3. Numerical analysis is introduced in the manuscript at a completely elementary calculus level. In fact, numerics are advertised as just an extension of the calculus and used generally as enrichment, to help communicate the role of mathematics in engineering applications. 4.The authors have used and updated the book as a course text over a 10 year period. 5. Modern outline, as contrasted to the outdated outline by Kreysig and Wylie. 6. This is now a one year course. The text is shorter and more readable than the current reference type manuals published all at around 1300-1500 pages.

This is the first book to systematically state the fundamental theory of integrability and its development of ordinary differential equations with emphasis on the Darboux theory of integrability and local integrability together with their applications. It summarizes the classical results of Darboux integrability and its modern development together with their related Darboux polynomials and their applications in the reduction of Liouville and elementary integrabilty and in the center-focus problem, the weakened Hilbert 16th problem on algebraic limit cycles and the global dynamical analysis of some realistic models in fields such as physics, mechanics and biology. Although it can be used as a textbook for graduate students in dynamical systems, it is intended as supplementary reading for graduate students from mathematics, physics, mechanics and engineering in courses related to the qualitative theory, bifurcation theory and the theory of integrability of dynamical systems.

Boundary element methods are very important for solving boundary value problems in PDEs. Many boundary value problems of partial differential equations can be reduced into boundary integral equations by the natural boundary reduction. In this book the natural boundary integral method, suggested and developed by Feng and Yu, is introduced systematically. It is quite different from popular boundary element methods and has many distinctive advantages. The variational principle is conserved after the natural boundary reduction, and some useful properties are also preserved faithfully. Moreover, it can be applied directly and naturally in the coupling method and the domain decomposition method of finite and boundary elements. Most of the material in this book has only appeared in the author's previous papers. Compared with its Chinese edition (Science Press, Beijing, 1993), many new research results such as the domain decomposition methods based on the natural boundary reduction are added.

Two-and three-level difference schemes for discretisation in time, in conjunction with finite difference or finite element approximations with respect to the space variables, are often used to solve numerically non stationary problems of mathematical physics. In the theoretical analysis of difference schemes our basic attention is paid to the problem of sta bility of a difference solution (or well posedness of a difference scheme) with respect to small perturbations of the initial conditions and the right hand side. The theory of stability of difference schemes develops in various di rections. The most important results on this subject can be found in the book by A.A. Samarskii and A.V. Goolin [Samarskii and Goolin, 1973]. The survey papers of V. Thomee [Thomee, 1969, Thomee, 1990], A.V. Goolin and A.A. Samarskii [Goolin and Samarskii, 1976], E. Tad more [Tadmor, 1987] should also be mentioned here. The stability theory is a basis for the analysis of the convergence of an approximative solu tion to the exact solution, provided that the mesh width tends to zero. In this case the required estimate for the truncation error follows from consideration of the corresponding problem for it and from a priori es timates of stability with respect to the initial data and the right hand side. Putting it briefly, this means the known result that consistency and stability imply convergence.

Topics of this volume are close to scientific interests of Professor Maz'ya and use, directly or indirectly, the fundamental influential Maz'ya's works penetrating, in a sense, the theory of PDEs.

In particular, recent advantages in the study of semilinear elliptic equations, stationary Navier-Stokes equations, the Stokes system in convex polyhedra, periodic scattering problems, problems with perturbed boundary at a conic point, singular perturbations arising in elliptic shells and other important problems in mathematical physics are presented.

This book surveys recent developments in numerical techniques for global atmospheric models. It is based upon a collection of lectures prepared by leading experts in the field. The chapters reveal the multitude of steps that determine the global atmospheric model design. They encompass the choice of the equation set, computational grids on the sphere, horizontal and vertical discretizations, time integration methods, filtering and diffusion mechanisms, conservation properties, tracer transport, and considerations for designing models for massively parallel computers. A reader interested in applied numerical methods but also the many facets of atmospheric modeling should find this book of particular relevance.

The book collects many techniques that are helpul in obtaining regularity results for solutions of nonlinear systems of partial differential equations. They are then applied in various cases to provide useful examples and relevant results, particularly in fields like fluid mechanics, solid mechanics, semiconductor theory, or game theory.In general, these techniques are scattered in the journal literature and developed in the strict context of a given model. In the book, they are presented independently of specific models, so that the main ideas are explained, while remaining applicable to various situations. Such a presentation will facilitate application and implementation by researchers, as well as teaching to students.

In recent years there has been a resurgence of interest in the study of delay differential equations motivated largely by new applications in physics, biology, ecology, and physiology. The aim of this monograph is to present a reasonably self-contained account of the advances in the oscillation theory of this class of equations. Throughout, the main topics of study are shown in action, with applications to such diverse problems as insect population estimations, logistic equations in ecology, the survival of red blood cells in animals, integro-differential equations, and the motion of the tips of growing plants. The authors begin by reviewing the basic theory of delay differential equations, including the fundamental results of existence and uniqueness of solutions and the theory of the Laplace and z-transforms. Little prior knowledge of the subject is required other than a firm grounding in the main techniques of differential equation theory. As a result, this book provides an invaluable reference to the recent work both for mathematicians and for all those whose research includes the study of this fascinating class of differential equations.

Drawing examplesfrom mathematics, physics, chemistry, biology, engineering, economics, medicine, politics, and sports, this book illustrates how nonlinear dynamics plays a vital role in our world. Examples cover a wide range from the spread and possible control of communicable diseases, to the lack of predictability in long-range weather forecasting, to competition between political groups and nations.

After an introductorychapter that explores what it means to be nonlinear, the book covers the mathematical conceptssuch as limit cycles, fractals, chaos, bifurcations, and solitons, that will be applied throughout the book. Numerous computer simulations and exercises allow students to explore topics in greater depth using the Maple computer algebra system. The mathematical level of the text assumes prior exposure to ordinary differential equations and familiarity with the wave and diffusion equations.No prior knowledge of Maple is assumed.

The book may be used at the undergraduate or graduate level to prepare science and engineering students for problems in the "real world," or for self-study by practicing scientists and engineers."

In this book the author presents the dynamical systems in infinite dimension, especially those generated by dissipative partial differential equations. This book attempts a systematic study of infinite dimensional dynamical systems generated by dissipative evolution partial differential equations arising in mechanics and physics and in other areas of sciences and technology. This second edition has been updated and extended.

This volume collects selected papers presented at the Ninth International Workshop on Meshfree Methods held in Bonn, Germany in September 2017. They address various aspects of this very active research field and cover topics from applied mathematics, physics and engineering. The numerical treatment of partial differential equations with meshfree discretization techniques has been a very active research area in recent years. While the fundamental theory of meshfree methods has been developed and considerable advances of the various methods have been made, many challenges in the mathematical analysis and practical implementation of meshfree methods remain. This symposium aims to promote collaboration among engineers, mathematicians, and computer scientists and industrial researchers to address the development, mathematical analysis, and application of meshfree and particle methods especially to multiscale phenomena. It continues the 2-year-cycled Workshops on Meshfree Methods for Partial Differential Equations.

Haim Brezis has made significant contributions in the fields of partial differential equations and functional analysis, and this volume collects contributions by his former students and collaborators in honor of his 60th anniversary at a conference in Gaeta. It presents new developments in the theory of partial differential equations with emphasis on elliptic and parabolic problems.

The fundamental contributions of Professor Maz'ya to the theory of function spaces and especially Sobolev spaces are well known and often play a key role in the study of different aspects of the theory, which is demonstrated, in particular, by presented new results and reviews from world-recognized specialists. Sobolev type spaces, extensions, capacities, Sobolev inequalities, pseudo-Poincare inequalities, optimal Hardy-Sobolev-Maz'ya inequalities, Maz'ya's isocapacitary inequalities in a measure-metric space setting and many other actual topics are discussed.

Topics in Fractional Differential Equationsis 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. "

In the last decades, functional methods played an increasing role in the qualita tive theory of partial differential equations. The spectral methods and theory of C 0 semigroups of linear operators as well as Leray-Schauder degree theory, ?xed point theorems, and theory of maximal monotone nonlinear operators are now essential functional tools for the treatment of linear and nonlinear boundary value problems associated with partial differential equations. An important step was the extension in the early seventies of the nonlinear dy namics of accretive (dissipative) type of the Hille-Yosida theory of C semigroups 0 of linear continuous operators. The main achievement was that the Cauchy problem associated with nonlinear m accretive operators in Banach spaces is well posed and the corresponding dynamic is expressed by the Peano exponential formula from ?nite dimensional theory. This fundamental result is the corner stone of the whole existence theory of nonlinear in?nite dynamics of dissipative type and its contri bution to the development of the modern theory of nonlinear partial differential equations cannot be underestimated.

This book teaches basic methods of partial differential equations and introduces related important ideas associated with the analysis of numerical methods for those partial differential equations. Coverage details such topics as separation of variables, Fourier analysis, maximum principles, and energy estimates. The book introduces numerical methods in parallel to the classical theory and also includes many engaging exercises.