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| Author | : K. W. Chang |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 191 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 146121114X |
Download Nonlinear Singular Perturbation Phenomena Book in PDF, ePub and Kindle
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.
| Author | : Kosaku Yosida |
| Publisher | : Springer |
| Total Pages | : 170 |
| Release | : 1984-07-30 |
| Genre | : Mathematics |
| ISBN | : 9780387960470 |
Download Operational Calculus Book in PDF, ePub and Kindle
In the end of the last century, Oliver Heaviside inaugurated an operational calculus in connection with his researches in electromagnetic theory. In his operational calculus, the operator of differentiation was denoted by the symbol "p". The explanation of this operator p as given by him was difficult to understand and to use, and the range of the valid ity of his calculus remains unclear still now, although it was widely noticed that his calculus gives correct results in general. In the 1930s, Gustav Doetsch and many other mathematicians began to strive for the mathematical foundation of Heaviside's operational calculus by virtue of the Laplace transform -pt e f(t)dt. ( However, the use of such integrals naturally confronts restrictions con cerning the growth behavior of the numerical function f(t) as t ~ ~. At about the midcentury, Jan Mikusinski invented the theory of con volution quotients, based upon the Titchmarsh convolution theorem: If f(t) and get) are continuous functions defined on [O,~) such that the convolution f~ f(t-u)g(u)du =0, then either f(t) =0 or get) =0 must hold. The convolution quotients include the operator of differentiation "s" and related operators. Mikusinski's operational calculus gives a satisfactory basis of Heaviside's operational calculus; it can be applied successfully to linear ordinary differential equations with constant coefficients as well as to the telegraph equation which includes both the wave and heat equa tions with constant coefficients.
| Author | : Ferdinand Verhulst |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 332 |
| Release | : 2006-06-04 |
| Genre | : Mathematics |
| ISBN | : 0387283137 |
Download Methods and Applications of Singular Perturbations Book in PDF, ePub and Kindle
Contains well-chosen examples and exercises A student-friendly introduction that follows a workbook type approach
| Author | : Jane Cronin |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 201 |
| Release | : 1999 |
| Genre | : Mathematics |
| ISBN | : 0821809296 |
Download Analyzing Multiscale Phenomena Using Singular Perturbation Methods Book in PDF, ePub and Kindle
To understand multiscale phenomena, it is essential to employ asymptotic methods to construct approximate solutions and to design effective computational algorithms. This volume consists of articles based on the AMS Short Course in Singular Perturbations held at the annual Joint Mathematics Meetings in Baltimore (MD). Leading experts discussed the following topics which they expand upon in the book: boundary layer theory, matched expansions, multiple scales, geometric theory, computational techniques, and applications in physiology and dynamic metastability. Readers will find that this text offers an up-to-date survey of this important field with numerous references to the current literature, both pure and applied.
| Author | : Gung-Min Gie |
| Publisher | : Springer |
| Total Pages | : 412 |
| Release | : 2018-11-21 |
| Genre | : Mathematics |
| ISBN | : 3030006387 |
Download Singular Perturbations and Boundary Layers Book in PDF, ePub and Kindle
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.
| Author | : Robert E. Jr. O'Malley |
| Publisher | : Elsevier |
| Total Pages | : 215 |
| Release | : 2012-12-02 |
| Genre | : Mathematics |
| ISBN | : 0323162274 |
Download Introduction to Singular Perturbations Book in PDF, ePub and Kindle
Introduction to Singular Perturbations provides an overview of the fundamental techniques for obtaining asymptomatic solutions to boundary value problems. This text explores singular perturbation techniques, which are among the basic tools of several applied scientists. This book is organized into eight chapters, wherein Chapter 1 discusses the method of matched asymptomatic expansions, which has been frequently applied to several physical problems involving singular perturbations. Chapter 2 considers the nonlinear initial value problem to illustrate the regular perturbation method, and Chapter 3 explains how to construct asymptotic solutions for general linear equations. Chapter 4 discusses scalar equations and nonlinear system, whereas Chapters 5 and 6 explain the contrasts for initial value problems where the outer expansion cannot be determined without obtaining the initial values of the boundary layer correction. Chapters 7 and 8 deal with boundary value problem that arises in the study of adiabatic tubular chemical flow reactors with axial diffusion. This monograph is a valuable resource for applied mathematicians, engineers, researchers, students, and readers whose interests span a variety of fields.
| Author | : Luminita Barbu |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 231 |
| Release | : 2007-12-14 |
| Genre | : Mathematics |
| ISBN | : 3764383313 |
Download Singularly Perturbed Boundary-Value Problems Book in PDF, ePub and Kindle
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 | : Donald R. Smith |
| Publisher | : Cambridge University Press |
| Total Pages | : 532 |
| Release | : 1985-08-30 |
| Genre | : Mathematics |
| ISBN | : 9780521300421 |
Download Singular-Perturbation Theory Book in PDF, ePub and Kindle
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 | : 364 |
| Release | : 2013-06-29 |
| Genre | : Mathematics |
| ISBN | : 3662032066 |
Download Numerical Methods for Singularly Perturbed Differential Equations Book in PDF, ePub and Kindle
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 | : John J H Miller |
| Publisher | : World Scientific |
| Total Pages | : 191 |
| Release | : 2012-02-29 |
| Genre | : Mathematics |
| ISBN | : 9814452777 |
Download Fitted Numerical Methods For Singular Perturbation Problems: Error Estimates In The Maximum Norm For Linear Problems In One And Two Dimensions (Revised Edition) Book in PDF, ePub and Kindle
Since the first edition of this book, the literature on fitted mesh methods for singularly perturbed problems has expanded significantly. Over the intervening years, fitted meshes have been shown to be effective for an extensive set of singularly perturbed partial differential equations. In the revised version of this book, the reader will find an introduction to the basic theory associated with fitted numerical methods for singularly perturbed differential equations. Fitted mesh methods focus on the appropriate distribution of the mesh points for singularly perturbed problems. The global errors in the numerical approximations are measured in the pointwise maximum norm. The fitted mesh algorithm is particularly simple to implement in practice, but the theory of why these numerical methods work is far from simple. This book can be used as an introductory text to the theory underpinning fitted mesh methods.
