Symplectic geometry and the theory of Fourier integral operators are modern manifestations of themes that have occupied a central position in mathematical thought for the past three hundred years — the relations between the wave and the corpuscular theories of light.Показать полностьюSymplectic geometry and the theory of Fourier integral operators are modern manifestations of themes that have occupied a central position in mathematical thought for the past three hundred years — the relations between the wave and the corpuscular theories of light. The purpose of this book is to develop these themes, and present some of the recent advances, using the language of differential geometry as a unifying influence. Chapters included in this book are: Chapter I, Introduction. The method of stationary phase; Appendix I, Morse’s lemma and some generalizations; Chapter II, Differential operators and asymptotic solutions; Chapter III, Geometrical optics; Chapter IV, Symplectic geometry; Chapter V, Geometric quantization; Chapter VI, Geometric aspects of distribution; Appendix to Chapter VI, The Plancherel formula for the complex semisimple Lie groups; Chapter VII, Compound Asymptotics; Appendix II, Various functorial constructions; Index.

In this book the author presents the theory and techniques underlying perturbation methods in a manner that will make the book widely appealing to readers in a broad range of disciplines.Показать полностьюIn this book the author presents the theory and techniques underlying perturbation methods in a manner that will make the book widely appealing to readers in a broad range of disciplines. Methods of algebraic equations, asymptotic expansions, integrals, PDEs, strained coordinates, and multiple scales are illustrated by copious use of examples drawn from many areas of mathematics and physics. The philosophy adopted is that there is no single or best method for such problems, but that one may exploit the small parameter given some experience and understanding of similar perturbation problems. The author does not look to perturbation methods to give quantitative answers but rather uses them to give a physical understanding of the subtle balances in a complex problem.

Drawing from an exhaustive variety of mathematical subjects, including real and complex analysis, fluid mechanics and asymptotics, this book demonstrates how mathematics can be intelligently applied within the specific context to a wide range of industrial uses. The volume is directed to undergraduate and graduate students.

Many branches of science and engineering involve applications of mathematical analysis. An important part of applied analysis is asymptotic approximation which is, therefore, an active area of research with new methods and publications being found constantly.Показать полностьюMany branches of science and engineering involve applications of mathematical analysis. An important part of applied analysis is asymptotic approximation which is, therefore, an active area of research with new methods and publications being found constantly. This book gives an introduction to the subject sufficient for scientists and engineers to grasp the fundamental techniques, both those which have been known for some time and those which have been discovered more recently. The asymptotic approximation of both integrals and differential equations is discussed and the discussion includes hyperasymptotics as well as uniform asymptotics. There are many numerical examples to illustrate the relation between theory and practice. Exercises in the chapters enable the book to be used as a text for an introductory course.

This book is a revised and updated version, including a substantial portion of new material, of the authors' widely acclaimed earlier text, Perturbation Methods in Applied Mathematics (AMS 34).Показать полностьюThis book is a revised and updated version, including a substantial portion of new material, of the authors' widely acclaimed earlier text, Perturbation Methods in Applied Mathematics (AMS 34). A new chapter dealing with regular expansions has been added, the discussion of layer-type singular perturbations has been revised, and the coverage of multiple scale and averaging methods has been significantly expanded to reflect recent advances and viewpoints. The result is a comprehensive account of the various perturbation techniques currently used in the sciences and engineering that is suitable both as graduate text and as a reference work on the subject.

Solving nonlinear problems is inherently difficult, and the stronger the nonlinearity, the more intractable solutions become. Analytic approximations often break down as nonlinearity becomes strong, and even perturbation approximations are valid only for problems with weak nonlinearity.This book introduces a powerful new analytic method for nonlinear problems-homotopy analysis-that remains valid even with strong nonlinearity.Показать полностьюSolving nonlinear problems is inherently difficult, and the stronger the nonlinearity, the more intractable solutions become. Analytic approximations often break down as nonlinearity becomes strong, and even perturbation approximations are valid only for problems with weak nonlinearity.This book introduces a powerful new analytic method for nonlinear problems-homotopy analysis-that remains valid even with strong nonlinearity. In Part I, the author starts with a very simple example, then presents the basic ideas, detailed procedures, and the advantages (and limitations) of homotopy analysis. Part II illustrates the application of homotopy analysis to many interesting nonlinear problems. These range from simple bifurcations of a nonlinear boundary-value problem to the Thomas-Fermi atom model, Volterra’s population model, Von Karman swirling viscous flow, and nonlinear progressive waves in deep water.Although the homotopy analysis method has been verified in a number of prestigious journals, it has yet to be fully detailed in book form. Written by a pioneer in its development, Beyond Pertubation: Introduction to the Homotopy Analysis Method is your first opportunity to explore the details of this valuable new approach, add it to your analytic toolbox, and perhaps make contributions to some of the questions that remain open.

This user-friendly work presents perturbation theory for the solution of algebraic and differential equations, especially ordinary differential equations.Показать полностьюThis user-friendly work presents perturbation theory for the solution of algebraic and differential equations, especially ordinary differential equations. It covers all of the methods commonly used in both regular and singular perturbations: Taylor series, Lindstedt series, strained parameters, multiple scales, averaging, matching and WKB methods. Its unique feature is the attention given to the interpretation of results. Considerable attention is given to the mathematical validity of the methods presented and to the estimation of error in a perturbation approximation. Makes accessible to all users of perturbations theory many of the subtler points often known only by specialists.

Many vibration problems in engineering are nonlinear in nature. The usual linear analysis may be inadequate for many applications. An essential difference in the study of nonlinear systems is that general solutions cannot be obtained by superposition, as in the case of linear systems.Показать полностьюMany vibration problems in engineering are nonlinear in nature. The usual linear analysis may be inadequate for many applications. An essential difference in the study of nonlinear systems is that general solutions cannot be obtained by superposition, as in the case of linear systems. Moreover, the nonlinearity brings many new phenomena, which do not occur in linear systems. To study nonlinear systems, one has to learn new mathematical techniques, which have been developed in many branches of applied mathematics, physics, and engineering. In the past several years, a number of powerful computer software packages have been developed that allow one to perform complicated symbolic manipulations. In this book, we use Mathematica’s symbolic programming techniques to implement various perturbation methods for studying the dynamics of weakly nonlinear systems. Instead of being burdened by the tedious algebra required to obtain the solutions, Mathematica enables us to focus our attention on understanding the techniques and the physics, thereby free our time for creative thinking.

A classic reference, intended for graduate students, mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.

Through this book the reader will become acquainted with certain fundamental techniques for obtaining asymptotic solutions to boundary value problems. It is intended for engineers and applied mathematicians, students of these disciplines, and others interested in the topic. It does not suppose many mathematical prerequisites.

This text summarizes parts of the exercises of the tutorial on ‘Asymptotic Analysis’ held in the winter term 1993/94 at the University of Kaiserslautern.Показать полностьюThis text summarizes parts of the exercises of the tutorial on ‘Asymptotic Analysis’ held in the winter term 1993/94 at the University of Kaiserslautern. The lecture was designed and held by Prof. Dr. H. Neunzert. The main aspect of the lecture and the tutorial was to investigate the basic techniques used in Asymptotic Analysis. This first part covers the following topics 1) Simple Operators in IR 2) Asymptotic Expansions 3) Asymptotic Expansions of Integrals 4) Perturbation Methods in Partial Differential Equations 5) Singular Perturbation: Matching In order to understand the exercises of the sections 4)--6) the reader should be fa­miliar with basic analytical solution techniques for ordinary and partial differential equations. Each section is provided with a small review on the main topics which were discussed in the lecture. A list of references is given at the beginning of a section.

We study the behaviour of truncated Rayleigh-Schrodinger series for the low-lying eigenvalues of the one-dimensional, time-independent Schrodinger equation, in the semiclassical limit h->0.Показать полностьюWe study the behaviour of truncated Rayleigh-Schrodinger series for the low-lying eigenvalues of the one-dimensional, time-independent Schrodinger equation, in the semiclassical limit h->0. Under certain hypotheses on the potential V(x), we prove that for any given small h>0 there is an optimal truncation of the series for the approximate eigenvalues, such that the difference between an approximate and exact eigenvalue is smaller than exp (-C/h) for some positive constant C. We also prove the analogous results concerning the eigenfunctions.

A new method is proposed for the solution of multiple-scale perturbation problems. The straightforward perturbation solution, with secular terms, is used to construct a uniformly-valid solution by enforcing an invariance property implicit in the expansion procedure.Показать полностьюA new method is proposed for the solution of multiple-scale perturbation problems. The straightforward perturbation solution, with secular terms, is used to construct a uniformly-valid solution by enforcing an invariance property implicit in the expansion procedure. It is suggested that this method is more systematic than many methods currently in use and requires fewer a priori assumptions about the nature of the solution. A proof of the asymptotic validity of the method is given and some examples are presented.

Теория особенностей дифференцируемых отображений — бурно развивающаяся область современной математики, являющаяся обобщением исследования функций на максимум и минимум и имеющая многочисленные приложения в математике, естествознании и технике (так называемые теории бифуркаций и катастроф).Показать полностьюТеория особенностей дифференцируемых отображений — бурно развивающаяся область современной математики, являющаяся обобщением исследования функций на максимум и минимум и имеющая многочисленные приложения в математике, естествознании и технике (так называемые теории бифуркаций и катастроф). Монография является продолжением книги «Особенности дифференцируемых отображений. Классификация критических точек, каустик и волновых фронтов» тех же авторов, опубликованной издательством «Наука» в 1982 году. Она посвящена семействам комплексных гиперповерхностей, асимптотике интегралов многомерных методов стационарной фазы и перевала, приложениям методов алгебраической геометрии к исследованию критических точек функций. Для математиков — научных работников, аспирантов, а также для специалистов в области механики, физики, техники и других наук, интересующихся теорий особенностей дифференцируемых отображений.

В зхой небольшой книге излагается метод пограничного слоя — весьма универсальный метод,позволяющий находить: коротковолновую асимпототику решений многих дифракционных задач. Книга рассчитана на специалистов-теоретиков, работающих в области теории дифракции акустических, элекхромагнихных и других волн, и на математиков, интересующихся асимптотическими методами математической физики.

Книга посвящена современным способам построения коротковолновой асимптотики дифракционных задач: лучевому методу, методу параболического уравнения и методу эталонных задач.

Книга содержит современное изложение анализа размерностей и теории подобия с многочисленными примерами разной степени сложности. Излагаемые новые подходы анализа размерностей и гипотезы автомодельности нового типа применяются для исследования ряда задач (в частности, турбулентных течений в приземном слое атмосферы и пристеночном слое турбулентного потока с поперечным сдвигом, динамики турбулентных пятен в жидкости с сильно устойчивой стратификацией и т.д.). Рассчитана на широкий круг читателей — математиков, механиков, специалистов по физике атмосферы и океана.

Книга посвящена приближенным асимптотическим методам решения задач теории нелинейных колебаний, встречающихся во многих областях физики и техники. Второе издание дополнено изложением некоторых методов, весьма широко использующихся сейчас на практике.Показать полностьюКнига посвящена приближенным асимптотическим методам решения задач теории нелинейных колебаний, встречающихся во многих областях физики и техники. Второе издание дополнено изложением некоторых методов, весьма широко использующихся сейчас на практике. Увеличено количество примеров решений типичных задач. Книга рассчитана на широкий круг технических и научных работников, интересующихся колебательными процессами.

Труды американского математика В.Вазова уже известны советскому читателю (Вазов В., Форсайт Д., Разностные методы решения дифференциальных уравнений в частных производных, ИЛ, 1963).Показать полностьюТруды американского математика В.Вазова уже известны советскому читателю (Вазов В., Форсайт Д., Разностные методы решения дифференциальных уравнений в частных производных, ИЛ, 1963). Настоящая его книга посвящена методам асимптотических разложений для обыкновенных дифференциальных уравнений. Эти цетоды могут быть использованы во многих задачах механики, электроники, астрофизики и др. Монография содержит много примеров и задач для самостоятельного решения, а также обширную библиографию. Книга представляет интерес как для математиков, так и для физиков, механиков и инженеров-исследователей. Она может быть использована как учебное пособие для студентов старших курсов университетов и технических вузов.

Книга посвящена актуальным вопросам теории нелинейных уравнений с аналитическими операторами, зависящими от числового или функционального параметра. В ней излагаются методы отыскания всех решений нелинейного уравнения, ответвляющихся от известного решения этого уравнения при изменении параметра.Показать полностьюКнига посвящена актуальным вопросам теории нелинейных уравнений с аналитическими операторами, зависящими от числового или функционального параметра. В ней излагаются методы отыскания всех решений нелинейного уравнения, ответвляющихся от известного решения этого уравнения при изменении параметра. Такие математические задачи возникают при изучении различных вопросов механики и современной техники. Методы, изложенные в книге, тесно связаны с давно известным в механике методом малого параметра, причем в наиболее важных для приложений случаях доказывается и сходимость этого метода. Значительная часть книги посвящена вопросам, не излагавшимся до сих пор в монографиях.