Singular Perturbation Theory

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Geometric Singular Perturbation Theory Beyond the Standard Form

Martin Wechselberger
Geometric Singular Perturbation Theory Beyond the Standard Form Book PDF

Author : Martin Wechselberger
Publisher : Springer Nature
Page : 143 pages
File Size : 51,6 Mb
Release : 2020-02-21
Category : Mathematics
ISBN : 9783030363994

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Geometric Singular Perturbation Theory Beyond the Standard Form by Martin Wechselberger Pdf

This volume provides a comprehensive review of multiple-scale dynamical systems. Mathematical models of such multiple-scale systems are considered singular perturbation problems, and this volume focuses on the geometric approach known as Geometric Singular Perturbation Theory (GSPT). It is the first of its kind that introduces the GSPT in a coordinate-independent manner. This is motivated by specific examples of biochemical reaction networks, electronic circuit and mechanic oscillator models and advection-reaction-diffusion models, all with an inherent non-uniform scale splitting, which identifies these examples as singular perturbation problems beyond the standard form. The contents cover a general framework for this GSPT beyond the standard form including canard theory, concrete applications, and instructive qualitative models. It contains many illustrations and key pointers to the existing literature. The target audience are senior undergraduates, graduate students and researchers interested in using the GSPT toolbox in nonlinear science, either from a theoretical or an application point of view. Martin Wechselberger is Professor at the School of Mathematics & Statistics, University of Sydney, Australia. He received the J.D. Crawford Prize in 2017 by the Society for Industrial and Applied Mathematics (SIAM) for achievements in the field of dynamical systems with multiple time-scales.

Singular Perturbation Theory

R.S. Johnson
Singular Perturbation Theory Book PDF

Author : R.S. Johnson
Publisher : Springer Science & Business Media
Page : 292 pages
File Size : 46,5 Mb
Release : 2006-01-16
Category : Technology & Engineering
ISBN : 9780387232171

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Singular Perturbation Theory by R.S. Johnson Pdf

The importance of mathematics in the study of problems arising from the real world, and the increasing success with which it has been used to model situations ranging from the purely deterministic to the stochastic, is well established. The purpose of the set of volumes to which the present one belongs is to make available authoritative, up to date, and self-contained accounts of some of the most important and useful of these analytical approaches and techniques. Each volume provides a detailed introduction to a specific subject area of current importance that is summarized below, and then goes beyond this by reviewing recent contributions, and so serving as a valuable reference source. The progress in applicable mathematics has been brought about by the extension and development of many important analytical approaches and techniques, in areas both old and new, frequently aided by the use of computers without which the solution of realistic problems would otherwise have been impossible.

Algebraic Analysis of Singular Perturbation Theory

Takahiro Kawai,Yoshitsugu Takei
Algebraic Analysis of Singular Perturbation Theory Book PDF

Author : Takahiro Kawai,Yoshitsugu Takei
Publisher : American Mathematical Soc.
Page : 148 pages
File Size : 40,8 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.

Methods and Applications of Singular Perturbations

Ferdinand Verhulst
Methods and Applications of Singular Perturbations Book PDF

Author : Ferdinand Verhulst
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 40,8 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

Singular Perturbation Theory

Lindsay A. Skinner
Singular Perturbation Theory Book PDF

Author : Lindsay A. Skinner
Publisher : Springer Science & Business Media
Page : 85 pages
File Size : 42,7 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.

Singular Perturbation Methods for Ordinary Differential Equations

Robert E., Jr. O'Malley
Singular Perturbation Methods for Ordinary Differential Equations Book PDF

Author : Robert E., Jr. O'Malley
Publisher : Springer Science & Business Media
Page : 234 pages
File Size : 40,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461209775

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Singular Perturbation Methods for Ordinary Differential Equations by Robert E., Jr. O'Malley Pdf

This book results from various lectures given in recent years. Early drafts were used for several single semester courses on singular perturbation meth ods given at Rensselaer, and a more complete version was used for a one year course at the Technische Universitat Wien. Some portions have been used for short lecture series at Universidad Central de Venezuela, West Vir ginia University, the University of Southern California, the University of California at Davis, East China Normal University, the University of Texas at Arlington, Universita di Padova, and the University of New Hampshire, among other places. As a result, I've obtained lots of valuable feedback from students and listeners, for which I am grateful. This writing continues a pattern. Earlier lectures at Bell Laboratories, at the University of Edin burgh and New York University, and at the Australian National University led to my earlier works (1968, 1974, and 1978). All seem to have been useful for the study of singular perturbations, and I hope the same will be true of this monograph. I've personally learned much from reading and analyzing the works of others, so I would especially encourage readers to treat this book as an introduction to a diverse and exciting literature. The topic coverage selected is personal and reflects my current opin ions. An attempt has been made to encourage a consistent method of ap proaching problems, largely through correcting outer limits in regions of rapid change. Formal proofs of correctness are not emphasized.

Multiple Scale and Singular Perturbation Methods

J.K. Kevorkian,J.D. Cole
Multiple Scale and Singular Perturbation Methods Book PDF

Author : J.K. Kevorkian,J.D. Cole
Publisher : Springer Science & Business Media
Page : 642 pages
File Size : 43,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461239680

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Multiple Scale and Singular Perturbation Methods by J.K. Kevorkian,J.D. Cole Pdf

This book is a revised and updated version, including a substantial portion of new material, of our text Perturbation Methods in Applied Mathematics (Springer Verlag, 1981). We present the material at a level that assumes some familiarity with the basics of ordinary and partial differential equations. Some of the more advanced ideas are reviewed as needed; therefore this book can serve as a text in either an advanced undergraduate course or a graduate-level course on the subject. Perturbation methods, first used by astronomers to predict the effects of small disturbances on the nominal motions of celestial bodies, have now become widely used analytical tools in virtually all branches of science. A problem lends itself to perturbation analysis if it is "close" to a simpler problem that can be solved exactly. Typically, this closeness is measured by the occurrence of a small dimensionless parameter, E, in the governing system (consisting of differential equations and boundary conditions) so that for E = 0 the resulting system is exactly solvable. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of E. In a regular perturbation problem, a straightforward procedure leads to a system of differential equations and boundary conditions for each term in the asymptotic expansion. This system can be solved recursively, and the accuracy of the result improves as E gets smaller, for all values of the independent variables throughout the domain of interest. We discuss regular perturbation problems in the first chapter.

Singular-Perturbation Theory

Donald R. Smith
Singular Perturbation Theory Book PDF

Author : Donald R. Smith
Publisher : Cambridge University Press
Page : 532 pages
File Size : 44,7 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.

Singular Perturbation Methods in Control

Petar Kokotovic,Hassan K. Khalil,John O'Reilly
Singular Perturbation Methods in Control Book PDF

Author : Petar Kokotovic,Hassan K. Khalil,John O'Reilly
Publisher : SIAM
Page : 386 pages
File Size : 47,8 Mb
Release : 1999-01-01
Category : Mathematics
ISBN : 161197111X

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Singular Perturbation Methods in Control by Petar Kokotovic,Hassan K. Khalil,John O'Reilly Pdf

Singular perturbations and time-scale techniques were introduced to control engineering in the late 1960s and have since become common tools for the modeling, analysis, and design of control systems. In this SIAM Classics edition of the 1986 book, the original text is reprinted in its entirety (along with a new preface), providing once again the theoretical foundation for representative control applications. This book continues to be essential in many ways. It lays down the foundation of singular perturbation theory for linear and nonlinear systems, it presents the methodology in a pedagogical way that is not available anywhere else, and it illustrates the theory with many solved examples, including various physical examples and applications. So while new developments may go beyond the topics covered in this book, they are still based on the methodology described here, which continues to be their common starting point.

Nonlinear Singular Perturbation Phenomena

K. W. Chang,F. A. Howes
Nonlinear Singular Perturbation Phenomena Book PDF

Author : K. W. Chang,F. A. Howes
Publisher : Springer Science & Business Media
Page : 191 pages
File Size : 49,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461211143

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Nonlinear Singular Perturbation Phenomena by K. W. Chang,F. A. Howes Pdf

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.

Singular Perturbation Methodology in Control Systems

Desineni S. Naidu
Singular Perturbation Methodology in Control Systems Book PDF

Author : Desineni S. Naidu
Publisher : IET
Page : 314 pages
File Size : 41,7 Mb
Release : 1988
Category : Technology & Engineering
ISBN : 086341107X

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Singular Perturbation Methodology in Control Systems by Desineni S. Naidu Pdf

This book presents the twin topics of singular perturbation methods and time scale analysis to problems in systems and control. The heart of the book is the singularly perturbed optimal control systems, which are notorious for demanding excessive computational costs. The book addresses both continuous control systems (described by differential equations) and discrete control systems (characterised by difference equations).

hp-Finite Element Methods for Singular Perturbations

Jens M. Melenk
hp Finite Element Methods for Singular Perturbations Book PDF

Author : Jens M. Melenk
Publisher : Springer
Page : 326 pages
File Size : 41,7 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.

Perturbations

James A. Murdock
Perturbations Book PDF

Author : James A. Murdock
Publisher : SIAM
Page : 358 pages
File Size : 49,7 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.

A First Look at Perturbation Theory

James G. Simmonds,James E. Mann
A First Look at Perturbation Theory Book PDF

Author : James G. Simmonds,James E. Mann
Publisher : Courier Corporation
Page : 164 pages
File Size : 41,5 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.

Fitted Numerical Methods For Singular Perturbation Problems: Error Estimates In The Maximum Norm For Linear Problems In One And Two Dimensions (Revised Edition)

Miller John J H,O'riordan Eugene,Shishkin G I
Fitted Numerical Methods For Singular Perturbation Problems Error Estimates In The Maximum Norm For Linear Problems In One And Two Dimensions Revised Edition Book PDF

Author : Miller John J H,O'riordan Eugene,Shishkin G I
Publisher : World Scientific
Page : 192 pages
File Size : 49,9 Mb
Release : 2012-02-29
Category : Mathematics
ISBN : 9789814452779

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Fitted Numerical Methods For Singular Perturbation Problems: Error Estimates In The Maximum Norm For Linear Problems In One And Two Dimensions (Revised Edition) by Miller John J H,O'riordan Eugene,Shishkin G I Pdf

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.