Book Description
This book is an introductory graduate text dealing with many of the perturbation methods currently used by applied mathematicians, scientists, and engineers. The author has based his book on a graduate course he has taught several times over the last ten years to students in applied mathematics, engineering sciences, and physics. The only prerequisite for the course is a background in differential equations. Each chapter begins with an introductory development involving ordinary differential equations. The book covers traditional topics, such as boundary layers and multiple scales. However, it also contains material arising from current research interest. This includes homogenization, slender body theory, symbolic computing, and discrete equations. One of the more important features of this book is contained in the exercises. Many are derived from problems of up- to-date research and are from a wide range of application areas.
Customer Reviews:
nice introduction.......2006-05-30
I found this book explains the ideas very well and also provides examples that help to understand the methods. I also liked all the exercies at the end of each chapter. This is one book that anyone who studies applied mathematics should have on their bookshelf.
Best available introduction to perturbation methods.......2004-05-20
I have read several books on perturbation methods, and this book absolutely the best. Perturbation methods are used in problems where the equations (algebraic, ODEs, or PDEs) involve a small parameter, or that evolve on multiple time scales that differ by orders of magnitude. The basic idea is to build an approximate solution to the equation based on a series expansion involving the small parameter. Usually two or more series expansions are built and then combined to yield a solution to the equation that incorporates dynamics on both the slow and fast time scales (for example).
The first two chapters introduce the basic concepts of asymptotic expansions. Several informative, though relatively simple, examples are developed in detail to illustrate the techniques. The third chapter introduces techniques for solving problems involving multiple scales. Chapters four and five discuss the WKB method and homogenization techniques. The final chapter shows how perturbation methods can be used in conjunction with techniques from dynamical systems analysis, especially stability analysis and bifurcation theory.
The contents of the book are extremely well-chosen and pertinent. Many of the problems and techniques are based on realistic physical systems, and represent the types of problems an applied mathematician is likely to encounter in practice. The book's prose is clear and informative. Only occasionally did I feel that the author had explained something poorly or left out an important piece of information.
There are of course a couple of problems with the book. First, most of the exercises are significantly more difficult than the examples, and no hints or solutions are given. This can be a draw-back if you are using the book for self-study. There are a number of typos and errors in the text, though the author's website has a list of errata. Finally, solving perturbation problems in practice often involves subtle tricks, and unfortunately, this book does not discuss or give examples of many of those tricks.
Overall, this is definitely the best introductory book on perturbation methods. You should definitely consider this book if you work in applied math or with differential equations.
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Beyond Perturbation: Introduction to the Homotopy Analysis Method (Modern Mathematics and Mechanics)
Shijun Liao
Manufacturer: Chapman & Hall/CRC
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ASIN: 158488407X |
Book Description
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. 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 Kármán 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.
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Foundation of Quantum Chromodynamics: an Introduction to Perturbative Methods in Gauge Theories
Taizo Muta
Manufacturer: World Scientific Publishing Company
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Quantum Field Theory
ASIN: 9810226748 |
Book Description
This volume develops the techniques of perturbative QCD in great pedagogical detail starting with field theory. Aside from extensive treatments of the renormalization group technique, the operator product expansion formalism and their applications to short-distance reactions, this book provides a comprehensive introduction to gauge theories. Examples and exercises are provided to amplify the discussions on important topics. This is an ideal textbook on the subject of quantum chromodynamics and is essential for researchers and graduate students in high energy physics, nuclear physics and mathematical physics.
Product Description
Techniques for treating regular and singular perturbations are illustrated by application to problems of fluid motion. In particular, the method of matched asymptotic expansions is applied to the aerodynamics of airfoils and wings, and to viscous flow at high and low Reynolds numbers. Other topics include the methods of strained coordinates and of multiple scales, and the improvement of series.
Customer Reviews:
An excellent introduction to a powerful analysis tool........2000-06-29
Dr. Van Dyke presents a good discussion of the role of perturbation methods to fluid mechanics. Topics include an overview of perturbation methods (matched asympotic expansions, successive approximations, strained coordinates, etc.) as well a broad range of applications ranging from low speed viscous (Stokes) flows to hypersonic flows. I found this book far more approachable than other perturbation texts (e.g., Kevorkian & Cole). And, like his renowned "Album of Fluid Motion," the price has been kept low. It should be on the shelf of any serious fluid dynamicist.
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Theory of Orbits treats celestial mechanics as well as stellar dynamics from the common point of view of orbit theory, making use of concepts and techniques from modern geometric mechanics. It starts with elementary Newtonian mechanics and ends with the dynamics of chaotic motion. The two volumes are meant for students in astronomy and physics alike. Prerequisite is a physicist's knowledge of calculus and differential geometry.
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Perturbative Quantum Electrodynamics and Axiomatic Field Theory (Texts and Monographs in Physics)
Othmar Steinmann
Manufacturer: Springer
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ASIN: 3540670246 |
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This book demonstrates that fundamental concepts and methods from phenomenological particle physics can be derived rigorously from well-defined general assumptions in a mathematically clean way. Starting with the Wightman formulation of relativistic quantum field theory, the perturbative formulation of quantum electrodynamics is derived avoiding the usual formalism based on the canonical commutation relations. A scattering formalism based on the local-observables approach is developed, directly yielding expressions for the observable inclusive cross-sections without having to introduce the S-matrix. Neither ultraviolet nor infrared regularizations are required in this approach. Although primarily intended for researchers working in this field, anyone with a basic working knowledge of relativistic quantum field theory can benefit from this book.
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Perturbation Methods and Semilinear Elliptic Problems on R^n (Progress in Mathematics)
Antonio Ambrosetti , and
Andrea Malchiodi
Manufacturer: Birkhäuser Basel
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Nonlinear Analysis and Semilinear Elliptic Problems (Cambridge Studies in Advanced Mathematics)
ASIN: 3764373210 |
Book Description
The aim of this monograph is to discuss several elliptic problems on R
n with two main features: they are variational and perturbative in nature, and standard tools of nonlinear analysis based on compactness arguments cannot be used in general. For these problems, a more specific approach that takes advantage of such a perturbative setting seems to be the most appropriate. The first part of the book is devoted to these abstract tools, which provide a unified frame for several applications, often considered different in nature.
Such applications are discussed in the second part, and include semilinear elliptic
problems on R
n, bifurcation from the essential spectrum, the prescribed scalar
curvature problem, nonlinear Schrödinger equations, and singularly perturbed
elliptic problems in domains. These topics are presented in a systematic and
unified way.
Book Description
The Wiley Classics Library consists of selected books that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists.
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William G. Cochran & Gertrude M. Cox
Experimental Designs, Second Edition
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Differential and Integral Calculus, Volume I
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Methods of Mathematical Physics, Volume I
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D. R. Cox
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Harold S. M. Coxeter
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Charles W. Curtis & Irving Reiner
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Charles W. Curtis & Irving Reiner
Methods of Representation Theory
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Methods of Representation Theory
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W. Edwards Deming
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Customer Reviews:
Good and delineating theory for perturbation method.......2006-03-27
understandable for engineers with good reasoning and a little mathematical knowledge
Book Description
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. 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.
Customer Reviews:
Starting Perturbations.......2005-01-30
This is a nice place to start when learning basic perturbation methods. There are numerous examples and the material is clearly presented. However, there are weak points. One is brevity, as pointed out by the first reviewer. The coverage of the subject of perturbation methods is minimal and some of the topics that are covered are done so with little depth. Another drawback is the emphasis, when discussing applications, on fluids and to a lesser extent mechanics. This makes it somewhat old fashioned. The greatest drawback is the exercises. It is impossible to learn mathematics, and perturbation methods in particular, without doing lots of problems. This text, unfortunately, falls far short in the number and range of problems it provides the reader.
Great Intro to Perturbation Methods.......2000-09-14
This is a wonderful first introduction to perturbation methods. Extremley readable, my only criticism of this text is its brevity. In some places, it lacks detail and doesn't go very deeply into several areas. However, it is the type of book that one can easily pick up, read and learn something usefull. The author's do lot's of examples and once this text is mastered, the student is certaintly ready to read Kevorkian and Cole's classic text on the subject.
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