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| Author | : Stephen John Kirschvink |
| Publisher | : |
| Total Pages | : 284 |
| Release | : 1987 |
| Genre | : Boundary value problems |
| ISBN | : |
| Author | : Frederick A. Howes |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 83 |
| Release | : 1976 |
| Genre | : Boundary value problems |
| ISBN | : 0821818686 |
The author discusses the singularly perturbed second-order boundary value problem [lowercase Greek]Epsilon [italic]y′′ = [italic]f([italic]t,[italic]y,[italic]y′, [lowercase Greek]Epsilon), by means of several second-order differential inequality theorems. This article not only gives a unified presentation of much of the body of results on this boundary value problem obtained in the last twenty years or so, but contains very considerable improvements involving less demanding conditions (sometimes leading to weaker results) in some cases, more precise results (sometimes under more severe restrictions) in other cases, and a more thorough investigation of the general boundary conditions. Some potential extensions to transition point problems and the like are indicated (but not carried out in detail) in the last section.
| Author | : Luminita Barbu |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 231 |
| Release | : 2007-12-14 |
| Genre | : Mathematics |
| ISBN | : 3764383313 |
This book offers a detailed asymptotic analysis of some important classes of singularly perturbed boundary value problems which are mathematical models for phenomena in biology, chemistry, and engineering. The authors are particularly interested in nonlinear problems, which have gone little-examined so far in literature dedicated to singular perturbations. The treatment presented here combines successful results from functional analysis, singular perturbation theory, partial differential equations, and evolution equations.
| Author | : Hans-Görg Roos |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 364 |
| Release | : 2013-06-29 |
| Genre | : Mathematics |
| ISBN | : 3662032066 |
The analysis of singular perturbed differential equations began early in this century, when approximate solutions were constructed from asymptotic ex pansions. (Preliminary attempts appear in the nineteenth century [vD94].) This technique has flourished since the mid-1960s. Its principal ideas and methods are described in several textbooks. Nevertheless, asymptotic ex pansions may be impossible to construct or may fail to simplify the given problem; then numerical approximations are often the only option. The systematic study of numerical methods for singular perturbation problems started somewhat later - in the 1970s. While the research frontier has been steadily pushed back, the exposition of new developments in the analysis of numerical methods has been neglected. Perhaps the only example of a textbook that concentrates on this analysis is [DMS80], which collects various results for ordinary differential equations, but many methods and techniques that are relevant today (especially for partial differential equa tions) were developed after 1980.Thus contemporary researchers must comb the literature to acquaint themselves with earlier work. Our purposes in writing this introductory book are twofold. First, we aim to present a structured account of recent ideas in the numerical analysis of singularly perturbed differential equations. Second, this important area has many open problems and we hope that our book will stimulate further investigations.Our choice of topics is inevitably personal and reflects our own main interests.
| Author | : Frederick A. Howes |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 116 |
| Release | : 1978 |
| Genre | : Boundary value problems |
| ISBN | : 0821822039 |
For three classes of singularly perturbed boundary value problems we study the existence of solutions which possess boundary, shock and corner layer behavior and we examine how these nonuniformities arise and how they influence one another. The keys to our analysis are the stability properties of solutions of corresponding reduced problems and the geometric properties of solutions of the boundary value problems inside such layers. Several examples of the theory are discussed in detail with a view to illustrating the naturalness of our approach.
| Author | : Vladimir D. Liseikin |
| Publisher | : Walter de Gruyter GmbH & Co KG |
| Total Pages | : 300 |
| Release | : 2018-11-05 |
| Genre | : Mathematics |
| ISBN | : 3110941945 |
The approach of layer-damping coordinate transformations to treat singularly perturbed equations is a relatively new, and fast growing area in the field of applied mathematics. This monograph aims to present a clear, concise, and easily understandable description of the qualitative properties of solutions to singularly perturbed problems as well as of the essential elements, methods and codes of the technology adjusted to numerical solutions of equations with singularities by applying layer-damping coordinate transformations and corresponding layer-resolving grids. The first part of the book deals with an analytical study of estimates of the solutions and their derivatives in layers of singularities as well as suitable techniques for obtaining results. In the second part, a technique for building the coordinate transformations eliminating boundary and interior layers, is presented. Numerical algorithms based on the technique which is developed for generating layer-damping coordinate transformations and their corresponding layer-resolving meshes are presented in the final part of this volume. This book will be of value and interest to researchers in computational and applied mathematics.
| Author | : E.M. de Jager |
| Publisher | : Elsevier |
| Total Pages | : 353 |
| Release | : 1996-11-08 |
| Genre | : Mathematics |
| ISBN | : 0080542751 |
The subject of this textbook is the mathematical theory of singular perturbations, which despite its respectable history is still in a state of vigorous development. Singular perturbations of cumulative and of boundary layer type are presented. Attention has been given to composite expansions of solutions of initial and boundary value problems for ordinary and partial differential equations, linear as well as quasilinear; also turning points are discussed. The main emphasis lies on several methods of approximation for solutions of singularly perturbed differential equations and on the mathematical justification of these methods. The latter implies a priori estimates of solutions of differential equations; this involves the application of Gronwall's lemma, maximum principles, energy integrals, fixed point theorems and Gåding's theorem for general elliptic equations. These features make the book of value to mathematicians and researchers in the engineering sciences, interested in the mathematical justification of formal approximations of solutions of practical perturbation problems. The text is selfcontained and each chapter is concluded with some exercises.
| Author | : Donald R. Smith |
| Publisher | : Cambridge University Press |
| Total Pages | : 532 |
| Release | : 1985-08-30 |
| Genre | : Mathematics |
| ISBN | : 9780521300421 |
Introduction to singular perturbation problems. Since the nature of the nonuniformity can vary from case to case, the author considers and solves a variety of problems, mostly for ordinary differential equations.
| Author | : Hans-Görg Roos |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 599 |
| Release | : 2008-09-17 |
| Genre | : Mathematics |
| ISBN | : 3540344675 |
This new edition incorporates new developments in numerical methods for singularly perturbed differential equations, focusing on linear convection-diffusion equations and on nonlinear flow problems that appear in computational fluid dynamics.
| Author | : K. W. Chang |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 191 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 146121114X |
Our purpose in writing this monograph is twofold. On the one hand, we want to collect in one place many of the recent results on the exist ence and asymptotic behavior of solutions of certain classes of singularly perturbed nonlinear boundary value problems. On the other, we hope to raise along the way a number of questions for further study, mostly ques tions we ourselves are unable to answer. The presentation involves a study of both scalar and vector boundary value problems for ordinary dif ferential equations, by means of the consistent use of differential in equality techniques. Our results for scalar boundary value problems obeying some type of maximum principle are fairly complete; however, we have been unable to treat, under any circumstances, problems involving "resonant" behavior. The linear theory for such problems is incredibly complicated already, and at the present time there appears to be little hope for any kind of general nonlinear theory. Our results for vector boundary value problems, even those admitting higher dimensional maximum principles in the form of invariant regions, are also far from complete. We offer them with some trepidation, in the hope that they may stimulate further work in this challenging and important area of differential equa tions. The research summarized here has been made possible by the support over the years of the National Science Foundation and the National Science and Engineering Research Council.
