Advanced Mathematical Methods for Scientists and Engineers I

Asymptotic Methods and Perturbation Theory

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Author: Carl M. Bender,Steven A. Orszag

Publisher: Springer Science & Business Media

ISBN: 1475730691

Category: Mathematics

Page: 593

View: 4992

A clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. Aimed at teaching the most useful insights in approaching new problems, the text avoids special methods and tricks that only work for particular problems. Intended for graduates and advanced undergraduates, it assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations, then develops local asymptotic methods for such equations, and explains perturbation and summation theory before concluding with an exposition of global asymptotic methods. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach readers how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions, over 600 problems of varying levels of difficulty, and an appendix summarizing the properties of special functions.

Advanced Mathematical Methods for Scientists and Engineers I

Asymptotic Methods and Perturbation Theory

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Author: Carl M. Bender,Steven A. Orszag

Publisher: Springer Science & Business Media

ISBN: 9780387989310

Category: Mathematics

Page: 593

View: 5324

This book gives a clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. These methods allow one to analyze physics and engineering problems that may not be solvable in closed form. The presentation provides insights that will be useful in approaching new problems.

Introduction to Perturbation Methods

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Author: Mark H. Holmes

Publisher: Springer Science & Business Media

ISBN: 9780387942032

Category: Mathematics

Page: 356

View: 442

This introductory graduate text is based on a graduate course the author has taught repeatedly over the last ten years to students in applied mathematics, engineering sciences, and physics. Each chapter begins with an introductory development involving ordinary differential equations, and goes on to cover such traditional topics as boundary layers and multiple scales. However, it also contains material arising from current research interest, including homogenisation, slender body theory, symbolic computing, and discrete equations. Many of the excellent exercises are derived from problems of up-to-date research and are drawn from a wide range of application areas.

Perturbations

Theory and Methods

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Author: James A. Murdock

Publisher: SIAM

ISBN: 9781611971095

Category: Perturbation

Page: 509

View: 7134

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.

Perturbation Methods for Engineers and Scientists

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Author: Alan W. Bush

Publisher: CRC Press

ISBN: 9780849386145

Category: Mathematics

Page: 320

View: 5706

The subject of perturbation expansions is a powerful analytical technique which can be applied to problems which are too complex to have an exact solution, for example, calculating the drag of an aircraft in flight. These techniques can be used in place of complicated numerical solutions.

Perturbation Methods in Applied Mathematics

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Author: Jirair Kevorkian,J.D. Cole

Publisher: Springer Science & Business Media

ISBN: 1475742134

Category: Mathematics

Page: 560

View: 2555

This book is a revised and updated version, including a substantial portion of new material, of J. D. Cole's text Perturbation Methods in Applied Mathe matics, Ginn-Blaisdell, 1968. We present the material at a level which 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. The applied mathematician, attempting to understand or solve a physical problem, very often uses a perturbation procedure. In doing this, he usually draws on a backlog of experience gained from the solution of similar examples rather than on some general theory of perturbations. The aim of this book is to survey these perturbation methods, especially in connection with differ ential equations, in order to illustrate certain general features common to many examples. The basic ideas, however, are also applicable to integral equations, integrodifferential equations, and even to_difference equations. In essence, a perturbation procedure consists of constructing the solution for a problem involving a small parameter B, either in the differential equation or the boundary conditions or both, when the solution for the limiting case B = 0 is known. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of B.

A First Look at Perturbation Theory

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Author: James G. Simmonds,James E. Mann

Publisher: Courier Corporation

ISBN: 0486315584

Category: Mathematics

Page: 160

View: 1107

This introductory text explains methods for obtaining approximate solutions to mathematical problems by exploiting the presence of small, dimensionless parameters. For engineering and physical science undergraduates.

Applied Asymptotic Analysis

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Author: Peter David Miller

Publisher: American Mathematical Soc.

ISBN: 0821840789

Category: Mathematics

Page: 467

View: 6049

"The book is intended for a beginning graduate course on asymptotic analysis in applied mathematics and is aimed at students of pure and applied mathematics as well as science and engineering. The basic prerequisite is a background in differential equations, linear algebra, advanced calculus, and complex variables at the level of introductory undergraduate courses on these subjects."--BOOK JACKET.

Perturbation Techniques in Mathematics, Engineering and Physics

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Author: Richard Ernest Bellman

Publisher: Courier Corporation

ISBN: 9780486432588

Category: Science

Page: 118

View: 5416

Graduate students receive a stimulating introduction to analytical approximation techniques for solving differential equations in this text, which introduces scientifically significant problems and indicates useful solutions. 1966 edition.

Asymptotic Expansions of Integrals

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Author: Norman Bleistein,Richard A. Handelsman

Publisher: Courier Corporation

ISBN: 0486650820

Category: Mathematics

Page: 425

View: 8276

Excellent introductory text, written by two experts, presents a coherent and systematic view of principles and methods. Topics include integration by parts, Watson's lemma, LaPlace's method, stationary phase, and steepest descents. Additional subjects include the Mellin transform method and less elementary aspects of the method of steepest descents. 1975 edition.

Methods of Mathematical Physics

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Author: Harold Jeffreys,Bertha Jeffreys

Publisher: Cambridge University Press

ISBN: 9780521664028

Category: Mathematics

Page: 718

View: 4160

This well-known text and reference contains an account of those parts of mathematics that are most frequently needed in physics. As a working rule, it includes methods which have applications in at least two branches of physics. The authors have aimed at a high standard of rigour and have not accepted the often-quoted opinion that 'any argument is good enough if it is intended to be used by scientists'. At the same time, they have not attempted to achieve greater generality than is required for the physical applications: this often leads to considerable simplification of the mathematics. Particular attention is also paid to the conditions under which theorems hold. Examples of the practical use of the methods developed are given in the text: these are taken from a wide range of physics, including dynamics, hydrodynamics, elasticity, electromagnetism, heat conduction, wave motion and quantum theory. Exercises accompany each chapter.

Asymptotic Expansions

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Author: E. T. Copson,Edward Thomas Copson

Publisher: Cambridge University Press

ISBN: 9780521604826

Category: Mathematics

Page: 120

View: 770

Asymptotic representation of a function os of great importance in many branches of pure and applied mathematics.

Introduction to Perturbation Techniques

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Author: Ali H. Nayfeh

Publisher: John Wiley & Sons

ISBN: 3527618457

Category: Science

Page: 533

View: 724

Similarities, differences, advantages and limitations of perturbation techniques are pointed out concisely. The techniques are described by means of examples that consist mainly of algebraic and ordinary differential equations. Each chapter contains a number of exercises.

Asymptotic Analysis of Differential Equations

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Author: R. B. White

Publisher: World Scientific

ISBN: 1848166087

Category: Mathematics

Page: 405

View: 5019

"This is a useful volume in which a wide selection of asymptotic techniques is clearly presented in a form suitable for both applied mathematicians and Physicists who require an introduction to asymptotic techniques." --Book Jacket.

Numerical Partial Differential Equations

Conservation Laws and Elliptic Equations

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Author: J.W. Thomas

Publisher: Springer Science & Business Media

ISBN: N.A

Category: Mathematics

Page: 556

View: 5854

Of the many different approaches to solving partial differential equations numerically, this book studies difference methods. Written for the beginning graduate student in applied mathematics and engineering, this text offers a means of coming out of a course with a large number of methods that provide both theoretical knowledge and numerical experience. The reader will learn that numerical experimentation is a part of the subject of numerical solution of partial differential equations, and will be shown some uses and taught some techniques of numerical experimentation. Prerequisites suggested for using this book in a course might include at least one semester of partial differential equations and some programming capability. The author stresses the use of technology throughout the text, allowing the student to utilize it as much as possible. The use of graphics for both illustration and analysis is emphasized, and algebraic manipulators are used when convenient. This is the second volume of a two-part book.

Ordinary Differential Equations

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Author: George F. Carrier,Carl E. Pearson

Publisher: SIAM

ISBN: 9781611971293

Category: Differential equations

Page: 220

View: 5142

Offers an alternative to the "rote" approach of presenting standard categories of differential equations accompanied by routine problem sets. The exercises presented amplify and provide perspective for the material, often giving readers opportunity for ingenuity. Little or no previous acquaintance with the subject is required to learn usage of techniques for constructing solutions of differential equations in this reprint volume.

Singular Perturbation in the Physical Sciences

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Author: John C. Neu

Publisher: American Mathematical Soc.

ISBN: 1470425556

Category: Asymptotic expansions

Page: 336

View: 8795

This book is the testimony of a physical scientist whose language is singular perturbation analysis. Classical mathematical notions, such as matched asymptotic expansions, projections of large dynamical systems onto small center manifolds, and modulation theory of oscillations based either on multiple scales or on averaging/transformation theory, are included. The narratives of these topics are carried by physical examples: Let's say that the moment when we "see" how a mathematical pattern fits a physical problem is like "hitting the ball." Yes, we want to hit the ball. But a powerful stroke includes the follow-through. One intention of this book is to discern in the structure and/or solutions of the equations their geometric and physical content. Through analysis, we come to sense directly the shape and feel of phenomena. The book is structured into a main text of fundamental ideas and a subtext of problems with detailed solutions. Roughly speaking, the former is the initial contact between mathematics and phenomena, and the latter emphasizes geometric and physical insight. It will be useful for mathematicians and physicists learning singular perturbation analysis of ODE and PDE boundary value problems as well as the full range of related examples and problems. Prerequisites are basic skills in analysis and a good junior/senior level undergraduate course of mathematical physics.