Algebraic Analysis Of Singular Perturbation Theory

Author : Takahiro Kawai,Yoshitsugu Takei
Publisher : American Mathematical Soc.
Page : 148 pages
File Size : 45,7 Mb
Release : 2005
Category : Mathematics
ISBN : 0821835475

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Algebraic Analysis of Singular Perturbation Theory by Takahiro Kawai,Yoshitsugu Takei Pdf

The topic of this book is the study of singular perturbations of ordinary differential equations, i.e., perturbations that represent solutions as asymptotic series rather than as analytic functions in a perturbation parameter. The main method used is the so-called WKB (Wentzel-Kramers-Brillouin) method, originally invented for the study of quantum-mechanical systems. The authors describe in detail the WKB method and its applications to the study of monodromy problems for Fuchsian differential equations and to the analysis of Painleve functions. This volume is suitable for graduate students and researchers interested in differential equations and special functions.

Author : Lindsay A. Skinner
Publisher : Springer Science & Business Media
Page : 85 pages
File Size : 42,8 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.

Author : E.M. de Jager,J.F. Furu
Publisher : Elsevier
Page : 339 pages
File Size : 55,8 Mb
Release : 1996-11-08
Category : Mathematics
ISBN : 0080542751

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The Theory of Singular Perturbations by E.M. de Jager,J.F. Furu Pdf

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 : James A. Murdock
Publisher : SIAM
Page : 358 pages
File Size : 41,9 Mb
Release : 1999-01-01
Category : Mathematics
ISBN : 1611971098

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Perturbations by James A. Murdock Pdf

Perturbations: Theory and Methods gives a thorough introduction to both regular and singular perturbation methods for algebraic and differential equations. Unlike most introductory books on the subject, this one distinguishes between formal and rigorous asymptotic validity, which are commonly confused in books that treat perturbation theory as a bag of heuristic tricks with no foundation. The meaning of "uniformity" is carefully explained in a variety of contexts. All standard methods, such as rescaling, multiple scales, averaging, matching, and the WKB method are covered, and the asymptotic validity (in the rigorous sense) of each method is carefully proved. First published in 1991, this book is still useful today because it is an introduction. It combines perturbation results with those known through other methods. Sometimes a geometrical result (such as the existence of a periodic solution) is rigorously deduced from a perturbation result, and at other times a knowledge of the geometry of the solutions is used to aid in the selection of an effective perturbation method. Dr. Murdock's approach differs from other introductory texts because he attempts to present perturbation theory as a natural part of a larger whole, the mathematical theory of differential equations. He explores the meaning of the results and their connections to other ways of studying the same problems.

Author : Michael P. Mortell,Robert E. O'Malley,Alexei Pokrovskii,Vladimir Sobolev
Publisher : SIAM
Page : 356 pages
File Size : 47,8 Mb
Release : 2005-06-01
Category : Mathematics
ISBN : 9780898715972

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Singular Perturbations and Hysteresis by Michael P. Mortell,Robert E. O'Malley,Alexei Pokrovskii,Vladimir Sobolev Pdf

This book unifies many important recent developments in the analysis of singular perturbation and hysteresis phenomena in an accessible and comprehensive fashion. In April 2002 at University College Cork in Ireland, the editors conducted a workshop to provide a forum for experts to share their interests and knowledge. For this book, the editors have compiled research from those practitioners in areas such as reacting systems, semiconductor lasers, shock phenomena in economic modeling, and fluid mechanics, all with an emphasis on hysteresis and singular perturbations. A basic introduction to hysteresis and singular perturbation theory is included, with simple examples from both physics and mathematics. Later chapters address: applications of hysteresis to economics; various aspects of the asymptotic theory of singularly perturbed systems; typical problems of the asymptotic theory of contrast structures; and the geometrical approach to an investigation of models with singular perturbations and hysteresis.

Author : Desineni S. Naidu,Ayalasomayajula K. Rao
Publisher : Lecture Notes in Mathematics
Page : 212 pages
File Size : 44,8 Mb
Release : 1985-11
Category : Language Arts & Disciplines
ISBN : STANFORD:36105032259793

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Singular Perturbation Analysis of Discrete Control Systems by Desineni S. Naidu,Ayalasomayajula K. Rao Pdf

Author : James G. Simmonds,James E. Mann
Publisher : Courier Corporation
Page : 162 pages
File Size : 47,8 Mb
Release : 2013-07-04
Category : Mathematics
ISBN : 9780486315584

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A First Look at Perturbation Theory by James G. Simmonds,James E. Mann Pdf

Undergraduates in engineering and the physical sciences receive a thorough introduction to perturbation theory in this useful and accessible text. Students discover methods for obtaining an approximate solution of a mathematical problem by exploiting the presence of a small, dimensionless parameter — the smaller the parameter, the more accurate the approximate solution. Knowledge of perturbation theory offers a twofold benefit: approximate solutions often reveal the exact solution's essential dependence on specified parameters; also, some problems resistant to numerical solutions may yield to perturbation methods. In fact, numerical and perturbation methods can be combined in a complementary way. The text opens with a well-defined treatment of finding the roots of polynomials whose coefficients contain a small parameter. Proceeding to differential equations, the authors explain many techniques for handling perturbations that reorder the equations or involve an unbounded independent variable. Two disparate practical problems that can be solved efficiently with perturbation methods conclude the volume. Written in an informal style that moves from specific examples to general principles, this elementary text emphasizes the "why" along with the "how"; prerequisites include a knowledge of one-variable calculus and ordinary differential equations. This newly revised second edition features an additional appendix concerning the approximate evaluation of integrals.

Author : W. Eckhaus,E.M. de Jager
Publisher : Springer
Page : 371 pages
File Size : 54,5 Mb
Release : 2007-01-05
Category : Mathematics
ISBN : 9783540393320

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Theory and Applications of Singular Perturbations by W. Eckhaus,E.M. de Jager Pdf

Author : Donald R. Smith
Publisher : Cambridge University Press
Page : 532 pages
File Size : 49,5 Mb
Release : 1985-08-30
Category : Mathematics
ISBN : 0521300428

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Singular-Perturbation Theory by Donald R. Smith Pdf

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 : T. Aoki,H. Majima,Y. Takei,N. Tose
Publisher : Springer Science & Business Media
Page : 354 pages
File Size : 41,8 Mb
Release : 2009-03-15
Category : Mathematics
ISBN : 9784431732402

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Algebraic Analysis of Differential Equations by T. Aoki,H. Majima,Y. Takei,N. Tose Pdf

This volume contains 23 articles on algebraic analysis of differential equations and related topics, most of which were presented as papers at the conference "Algebraic Analysis of Differential Equations – from Microlocal Analysis to Exponential Asymptotics" at Kyoto University in 2005. This volume is dedicated to Professor Takahiro Kawai, who is one of the creators of microlocal analysis and who introduced the technique of microlocal analysis into exponential asymptotics.

Author : M. Konstantinov,D. Wei Gu,V. Mehrmann,P. Petkov
Publisher : Gulf Professional Publishing
Page : 442 pages
File Size : 55,8 Mb
Release : 2003-05-20
Category : Mathematics
ISBN : 9780080538679

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Perturbation Theory for Matrix Equations by M. Konstantinov,D. Wei Gu,V. Mehrmann,P. Petkov Pdf

The book is devoted to the perturbation analysis of matrix equations. The importance of perturbation analysis is that it gives a way to estimate the influence of measurement and/or parametric errors in mathematical models together with the rounding errors done in the computational process. The perturbation bounds may further be incorporated in accuracy estimates for the solution computed in finite arithmetic. This is necessary for the development of reliable computational methods, algorithms and software from the viewpoint of modern numerical analysis. In this book a general perturbation theory for matrix algebraic equations is presented. Local and non-local perturbation bounds are derived for general types of matrix equations as well as for the most important equations arising in linear algebra and control theory. A large number of examples, tables and figures is included in order to illustrate the perturbation techniques and bounds. Key features: • The first book in this field • Can be used by a variety of specialists • Material is self-contained • Results can be used in the development of reliable computational algorithms • A large number of examples and graphical illustrations are given • Written by prominent specialists in the field

Author : Konstantin E. Avrachenkov,Jerzy A. Filar,Phil G. Howlett
Publisher : SIAM
Page : 384 pages
File Size : 48,9 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9781611973143

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Analytic Perturbation Theory and Its Applications by Konstantin E. Avrachenkov,Jerzy A. Filar,Phil G. Howlett Pdf

Mathematical models are often used to describe complex phenomena such as climate change dynamics, stock market fluctuations, and the Internet. These models typically depend on estimated values of key parameters that determine system behavior. Hence it is important to know what happens when these values are changed. The study of single-parameter deviations provides a natural starting point for this analysis in many special settings in the sciences, engineering, and economics. The difference between the actual and nominal values of the perturbation parameter is small but unknown, and it is important to understand the asymptotic behavior of the system as the perturbation tends to zero. This is particularly true in applications with an apparent discontinuity in the limiting behavior?the so-called singularly perturbed problems. Analytic Perturbation Theory and Its Applications includes a comprehensive treatment of analytic perturbations of matrices, linear operators, and polynomial systems, particularly the singular perturbation of inverses and generalized inverses. It also offers original applications in Markov chains, Markov decision processes, optimization, and applications to Google PageRank? and the Hamiltonian cycle problem as well as input retrieval in linear control systems and a problem section in every chapter to aid in course preparation.

Author : Robert E. Jr. O'Malley
Publisher : Elsevier
Page : 215 pages
File Size : 44,5 Mb
Release : 2012-12-02
Category : Mathematics
ISBN : 9780323162272

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Introduction to Singular Perturbations by Robert E. Jr. O'Malley Pdf

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,Gheorghe Morosanu
Publisher : Birkhäuser
Page : 231 pages
File Size : 54,9 Mb
Release : 2009-09-03
Category : Mathematics
ISBN : 3764391979

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Singularly Perturbed Boundary-Value Problems by Luminita Barbu,Gheorghe Morosanu Pdf

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 : Ferdinand Verhulst
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 55,5 Mb
Release : 2006-06-04
Category : Mathematics
ISBN : 9780387283135

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Methods and Applications of Singular Perturbations by Ferdinand Verhulst Pdf

Contains well-chosen examples and exercises A student-friendly introduction that follows a workbook type approach