Matching Of Asymptotic Expansions Of Solutions Of Boundary Value Problems

Author : A. M. Ilʹin
Publisher : American Mathematical Soc.
Page : 281 pages
File Size : 50,7 Mb
Release : 1992
Category : Mathematics
ISBN : 0821845616

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Matching of Asymptotic Expansions of Solutions of Boundary Value Problems by A. M. Ilʹin Pdf

This book deals with the solution of singularly perturbed boundary value problems for differential equations. It presents, for the first time, a detailed and systematic treatment of the version of the matching method developed by the author and his colleagues. A broad class of problems is considered from a unified point of view, and the procedure for constructing asymptotic expansions is discussed in detail. The book covers formal constructions of asymptotic expansions and provides rigorous justifications of these asymptotics. One highlight is a complete asymptotic analysis of Burger's equation with small diffusion in the neighborhood of the gradient catastrophe point. The book is suitable as a text for graduate study in asymptotic methods in calculus and singularly perturbed equations.

Author : P.A. Lagerstrom
Publisher : Springer Science & Business Media
Page : 263 pages
File Size : 41,9 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475719901

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Matched Asymptotic Expansions by P.A. Lagerstrom Pdf

Content and Aims of this Book Earlier drafts of the manuscript of this book (James A. Boa was then coau thor) contained discussions of many methods and examples of singular perturba tion problems. The ambitious plans of covering a large number of topics were later abandoned in favor of the present goal: a thorough discussion of selected ideas and techniques used in the method of matched asymptotic expansions. Thus many problems and methods are not covered here: the method of av eraging and the related method of multiple scales are mentioned mainly to give reasons why they are not discussed further. Examples which required too sophis ticated and involved calculations, or advanced knowledge of a special field, are not treated; for instance, to the author's regret some very interesting applications to fluid mechanics had to be omitted for this reason. Artificial mathematical examples introduced to show some exotic or unexpected behavior are omitted, except when they are analytically simple and are needed to illustrate mathematical phenomena important for realistic problems. Problems of numerical analysis are not discussed.

Author : M.V. Fedoryuk
Publisher : Springer Science & Business Media
Page : 248 pages
File Size : 54,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642584237

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Partial Differential Equations V by M.V. Fedoryuk Pdf

In this paper we shall discuss the construction of formal short-wave asymp totic solutions of problems of mathematical physics. The topic is very broad. It can somewhat conveniently be divided into three parts: 1. Finding the short-wave asymptotics of a rather narrow class of problems, which admit a solution in an explicit form, via formulas that represent this solution. 2. Finding formal asymptotic solutions of equations that describe wave processes by basing them on some ansatz or other. We explain what 2 means. Giving an ansatz is knowing how to give a formula for the desired asymptotic solution in the form of a series or some expression containing a series, where the analytic nature of the terms of these series is indicated up to functions and coefficients that are undetermined at the first stage of consideration. The second stage is to determine these functions and coefficients using a direct substitution of the ansatz in the equation, the boundary conditions and the initial conditions. Sometimes it is necessary to use different ansiitze in different domains, and in the overlapping parts of these domains the formal asymptotic solutions must be asymptotically equivalent (the method of matched asymptotic expansions). The basis for success in the search for formal asymptotic solutions is a suitable choice of ansiitze. The study of the asymptotics of explicit solutions of special model problems allows us to "surmise" what the correct ansiitze are for the general solution.

Author : F. Verhulst
Publisher : Springer
Page : 249 pages
File Size : 55,6 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540353324

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Asymptotic Analysis by F. Verhulst Pdf

Author : Augustin Fruchard,Reinhard Schafke
Publisher : Springer
Page : 161 pages
File Size : 53,7 Mb
Release : 2012-12-15
Category : Mathematics
ISBN : 9783642340352

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Composite Asymptotic Expansions by Augustin Fruchard,Reinhard Schafke Pdf

The purpose of these lecture notes is to develop a theory of asymptotic expansions for functions involving two variables, while at the same time using functions involving one variable and functions of the quotient of these two variables. Such composite asymptotic expansions (CAsEs) are particularly well-suited to describing solutions of singularly perturbed ordinary differential equations near turning points. CAsEs imply inner and outer expansions near turning points. Thus our approach is closely related to the method of matched asymptotic expansions. CAsEs offer two unique advantages, however. First, they provide uniform expansions near a turning point and away from it. Second, a Gevrey version of CAsEs is available and detailed in the lecture notes. Three problems are presented in which CAsEs are useful. The first application concerns canard solutions near a multiple turning point. The second application concerns so-called non-smooth or angular canard solutions. Finally an Ackerberg-O’Malley resonance problem is solved.

Author : Wolfgang Richard Wasow
Publisher : Unknown
Page : 384 pages
File Size : 47,7 Mb
Release : 1965
Category : Asymptotic expansions
ISBN : UOM:39015015701777

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Asymptotic Expansions for Ordinary Differential Equations by Wolfgang Richard Wasow Pdf

Author : Jens M. Melenk
Publisher : Springer
Page : 326 pages
File Size : 47,6 Mb
Release : 2004-10-20
Category : Mathematics
ISBN : 9783540457817

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hp-Finite Element Methods for Singular Perturbations by Jens M. Melenk Pdf

Many partial differential equations arising in practice are parameter-dependent problems that are of singularly perturbed type. Prominent examples include plate and shell models for small thickness in solid mechanics, convection-diffusion problems in fluid mechanics, and equations arising in semi-conductor device modelling. Common features of these problems are layers and, in the case of non-smooth geometries, corner singularities. Mesh design principles for the efficient approximation of both features by the hp-version of the finite element method (hp-FEM) are proposed in this volume. For a class of singularly perturbed problems on polygonal domains, robust exponential convergence of the hp-FEM based on these mesh design principles is established rigorously.

Author : A. Georgescu
Publisher : CRC Press
Page : 282 pages
File Size : 45,6 Mb
Release : 1995-05-15
Category : Mathematics
ISBN : 0412558602

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Asymptotic Treatment of Differential Equations by A. Georgescu Pdf

The main definitions and results of asymptotic analysis and the theory of regular and singular perturbations are summarized in this book. They are applied to the asymptotic study of several mathematical models from mechanics, fluid dynamics, statistical mechanics, meteorology and elasticity. Due to the generality of presentation this applications-oriented book is suitable for the solving of differential equations from any other field of interest.

Author : Dmitrii Korikov,Boris Plamenevskii,Oleg Sarafanov
Publisher : Springer Nature
Page : 404 pages
File Size : 46,8 Mb
Release : 2021-04-01
Category : Mathematics
ISBN : 9783030653729

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Asymptotic Theory of Dynamic Boundary Value Problems in Irregular Domains by Dmitrii Korikov,Boris Plamenevskii,Oleg Sarafanov Pdf

This book considers dynamic boundary value problems in domains with singularities of two types. The first type consists of "edges" of various dimensions on the boundary; in particular, polygons, cones, lenses, polyhedra are domains of this type. Singularities of the second type are "singularly perturbed edges" such as smoothed corners and edges and small holes. A domain with singularities of such type depends on a small parameter, whereas the boundary of the limit domain (as the parameter tends to zero) has usual edges, i.e. singularities of the first type. In the transition from the limit domain to the perturbed one, the boundary near a conical point or an edge becomes smooth, isolated singular points become small cavities, and so on. In an "irregular" domain with such singularities, problems of elastodynamics, electrodynamics and some other dynamic problems are discussed. The purpose is to describe the asymptotics of solutions near singularities of the boundary. The presented results and methods have a wide range of applications in mathematical physics and engineering. The book is addressed to specialists in mathematical physics, partial differential equations, and asymptotic methods.

Author : J.A. Leach,D.J. Needham
Publisher : Springer Science & Business Media
Page : 290 pages
File Size : 41,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9780857293961

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Matched Asymptotic Expansions in Reaction-Diffusion Theory by J.A. Leach,D.J. Needham Pdf

This volume contains a wealth of results and methodologies applicable to a wide range of problems arising in reaction-diffusion theory. It can be viewed both as a handbook, and as a detailed description of the methodology. The authors present new methods based on matched asymptotic expansions.

Author : N.N. Uraltseva (Mathematikerin, Russland)
Publisher : American Mathematical Soc.
Page : 252 pages
File Size : 53,9 Mb
Release : 2024-05-20
Category : Mathematics
ISBN : 0821896040

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Proceedings of the St. Petersburg Mathematical Society by N.N. Uraltseva (Mathematikerin, Russland) Pdf

This collection presents new results in algebra, functional analysis, and mathematical physics. In particular, evolution and spectral problems related to small motions of viscoelastic fluid are considered. Specific areas covered in the book include functional equations and functional operator equations from the point of view of the $C*$-algebraic approach, the existence of an isomorphism between certain ideals regarded as Galois modules, spectral problems in singularly perturbed domains, scattering theory, the existence of bounded solutions to the equation $\operatorname{div} u = f$ in a plane domain, and a compactification of a locally compact group. Also given is an historic overview of the mathematical seminars held at St. Petersburg State University. The results, ideas, and methods given in the book will be of interest to a broad range of specialists.

Author : Nina Nikolaevna Ural_t_seva
Publisher : American Mathematical Soc.
Page : 288 pages
File Size : 50,9 Mb
Release : 1999-07-01
Category : Electronic
ISBN : 0821896024

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Proceedings of the St. Petersburg Mathematical Society Volume V by Nina Nikolaevna Ural_t_seva Pdf

This volume contains 10 papers with new results on problems in mathematical physics, differential equations, and probability. Included also is an article on the dramatic history of mathematics in Leningrad in the 1930s.

Author : Ari Laptev
Publisher : Springer Science & Business Media
Page : 404 pages
File Size : 51,8 Mb
Release : 2009-12-05
Category : Mathematics
ISBN : 9781441913432

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Around the Research of Vladimir Maz'ya II by Ari Laptev Pdf

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.

Author : Anonim
Publisher : Unknown
Page : 198 pages
File Size : 46,5 Mb
Release : 1973
Category : Electronic
ISBN : NASA:31769000494701

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Application of Matched Asymptotic Expansions to Lunar and Interplanetary Trajectories. Volume 2: Derivations of Second-order Asymptotic Boundary Value Solutions by Anonim Pdf

Author : Lindsay A. Skinner
Publisher : Springer Science & Business Media
Page : 85 pages
File Size : 40,5 Mb
Release : 2011-05-11
Category : Mathematics
ISBN : 9781441999580

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Singular Perturbation Theory by Lindsay A. Skinner Pdf

This book is a rigorous presentation of the method of matched asymptotic expansions, the primary tool for attacking singular perturbation problems. A knowledge of conventional asymptotic analysis is assumed. The first chapter introduces the theory and is followed by four chapters of applications to ordinary differential equation problems of increasing complexity. Exercises are included as well as several Maple programs for computing the terms of the various asymptotic expansions that arise in solving the problems.