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Singular perturbations, one of the central topics in asymptotic analysis, also play a special role in describing physical phenomena such as the propagation of waves in media in the presence of small energy dissipations or dispersions, the appearance of boundary or interior layers in fluid and gas dynamics, as well as in elasticity theory, semi-classical asymptotic approximations in quantum mechanics etc. Elliptic and coercive singular perturbations are of special interest for the asymptotic solution of problems which are characterized by boundary layer phenomena, e.g. the theory of thin buckli

Perturbation (Mathematics) --- Singular perturbations (Mathematics) --- Differential equations --- Perturbation equations --- Perturbation theory --- Approximation theory --- Dynamics --- Functional analysis --- Mathematical physics --- Asymptotic theory

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This book offers the first systematic account of canard cycles, an intriguing phenomenon in the study of ordinary differential equations. The canard cycles are treated in the general context of slow-fast families of two-dimensional vector fields. The central question of controlling the limit cycles is addressed in detail and strong results are presented with complete proofs. In particular, the book provides a detailed study of the structure of the transitions near the critical set of non-isolated singularities. This leads to precise results on the limit cycles and their bifurcations, including the so-called canard phenomenon and canard explosion. The book also provides a solid basis for the use of asymptotic techniques. It gives a clear understanding of notions like inner and outer solutions, describing their relation and precise structure. The first part of the book provides a thorough introduction to slow-fast systems, suitable for graduate students. The second and third parts will be of interest to both pure mathematicians working on theoretical questions such as Hilbert's 16th problem, as well as to a wide range of applied mathematicians looking for a detailed understanding of two-scale models found in electrical circuits, population dynamics, ecological models, cellular (FitzHugh–Nagumo) models, epidemiological models, chemical reactions, mechanical oscillators with friction, climate models, and many other models with tipping points.

Differential equations --- Mathematics --- Classical mechanics. Field theory --- differentiaalvergelijkingen --- dynamica --- Singular perturbations (Mathematics) --- Vector fields. --- Bifurcation theory. --- Pertorbacions singulars (Matemàtica) --- Camps vectorials --- Teoria de la bifurcació

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Le but du livre est de donner aux enseignants et aux étudiants (à partir de Bac+4) en mathématiques appliquées et en mécanique des fluides un outil d'enseignement et d'apprentissage illustré par cinquante problèmes accompagnés de leur correction détaillée. Ce livre présente une nouvelle méthode d'analyse asymptotique pour des problèmes de "couche limite". Celle-ci est appelée MASC "Méthode des Approximations Successives Complémentaires". La première moitié du livre est consacrée, outre la présentation de la MASC, à ordonner les connaissances nécessaires à l'analyse asymptotique et à donner les clés permettant la compréhension de ce qu'est un problème dit de "couche limite" et des méthodes permettant de construire une approximation. La seconde partie est consacrée à l'application de la MASC en mécanique des fluides et à la comparaison avec les méthodes plus traditionnelles issues de la célèbre MDAR, "Méthode des Développements Asymptotiques Raccordés".

Differential equations --- Singular perturbations (Mathematics) --- Boundary layer. --- Asymptotic theory. --- Aerodynamics --- Fluid dynamics --- Perturbation (Mathematics) --- 517.91 Differential equations --- Asymptotic theory --- Mathematics. --- Operator theory. --- Differential Equations. --- Differential equations, partial. --- Approximations and Expansions. --- Operator Theory. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Partial differential equations --- Functional analysis --- Math --- Science --- Approximation theory. --- Differential equations. --- Partial differential equations. --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems

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The construction of solutions of singularly perturbed systems of equations and boundary value problems that are characteristic for the mechanics of thin-walled structures are the main focus of the book. The theoretical results are supplemented by the analysis of problems and exercises. Some of the topics are rarely discussed in the textbooks, for example, the Newton polyhedron, which is a generalization of the Newton polygon for equations with two or more parameters. After introducing the important concept of the index of variation for functions special attention is devoted to eigenvalue problems containing a small parameter. The main part of the book deals with methods of asymptotic solutions of linear singularly perturbed boundary and boundary value problems without or with turning points, respectively. As examples, one-dimensional equilibrium, dynamics and stability problems for rigid bodies and solids are presented in detail. Numerous exercises and examples as well as vast references to the relevant Russian literature not well known for an English speaking reader makes this a indispensable textbook on the topic.

Mathematics. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Mechanics. --- Differential Equations. --- Differential equations, partial. --- Mathématiques --- Mécanique --- Mechanics, Analytic --- Mathematical physics --- Singular perturbations (Mathematics) --- Boundary value problems --- Differential equations --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Asymptotic theory --- Asymptotic theory. --- Classical mechanics --- Newtonian mechanics --- Asymptotic theory in mathematical physics --- 517.91 Differential equations --- Differential equations. --- Partial differential equations. --- Physics --- Dynamics --- Quantum theory --- Classical Mechanics. --- Partial differential equations

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Singular perturbations occur when a small coefficient affects the highest order derivatives in a system of partial differential equations. From the physical point of view singular perturbations generate in the system under consideration thin layers located often but not always at the boundary of the domains that are called boundary layers or internal layers if the layer is located inside the domain. Important physical phenomena occur in boundary layers. The most common boundary layers appear in fluid mechanics, e.g., the flow of air around an airfoil or a whole airplane, or the flow of air around a car. Also in many instances in geophysical fluid mechanics, like the interface of air and earth, or air and ocean. This self-contained monograph is devoted to the study of certain classes of singular perturbation problems mostly related to thermic, fluid mechanics and optics and where mostly elliptic or parabolic equations in a bounded domain are considered. This book is a fairly unique resource regarding the rigorous mathematical treatment of boundary layer problems. The explicit methodology developed in this book extends in many different directions the concept of correctors initially introduced by J. L. Lions, and in particular the lower- and higher-order error estimates of asymptotic expansions are obtained in the setting of functional analysis. The review of differential geometry and treatment of boundary layers in a curved domain is an additional strength of this book. In the context of fluid mechanics, the outstanding open problem of the vanishing viscosity limit of the Navier-Stokes equations is investigated in this book and solved for a number of particular, but physically relevant cases. This book will serve as a unique resource for those studying singular perturbations and boundary layer problems at the advanced graduate level in mathematics or applied mathematics and may be useful for practitioners in other related fields in science and engineering such as aerodynamics, fluid mechanics, geophysical fluid mechanics, acoustics and optics.

Boundary layer. --- Differential equations, Partial. --- Singular perturbations (Mathematics) --- Differential equations --- Perturbation (Mathematics) --- Partial differential equations --- Aerodynamics --- Fluid dynamics --- Asymptotic theory --- Functional analysis. --- Mathematics. --- Functional Analysis. --- Approximations and Expansions. --- Math --- Science --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Approximation theory. --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems