Introduction to Asymptotic Methods: 1st Edition (Hardback) book cover

Introduction to Asymptotic Methods

1st Edition

By David Y. Gao, Vadim A. Krysko

Chapman and Hall/CRC

272 pages | 40 B/W Illus.

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pub: 2006-05-03
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Description

Among the theoretical methods for solving many problems of applied mathematics, physics, and technology, asymptotic methods often provide results that lead to obtaining more effective algorithms of numerical evaluation. Presenting the mathematical methods of perturbation theory, Introduction to Asymptotic Methods reviews the most important methods of singular perturbations within the scope of application of differential equations. The authors take a challenging and original approach based on the integrated mathematical-analytical treatment of various objects taken from interdisciplinary fields of mechanics, physics, and applied mathematics. This new hybrid approach will lead to results that cannot be obtained by standard theories in the field.

Emphasizing fundamental elements of the mathematical modeling process, the book provides comprehensive coverage of asymptotic approaches, regular and singular perturbations, one-dimensional non-stationary non-linear waves, Padé approximations, oscillators with negative Duffing type stiffness, and differential equations with discontinuous nonlinearities. The book also offers a method of construction for canonical variables transformation in parametric form along with a number of examples and applications. The book is applications oriented and features results and literature citations that have not been seen in the Western Scientific Community. The authors emphasize the dynamics of the development of perturbation methods and present the development of ideas associated with this wide field of research.

Reviews

“The main aim of this book is to introduce the reader to the mathematical methods of perturbation theory and to review the most important methods of singular perturbations within the scope of application of differential equations. … The authors take a challenging and original approach based on the integrated mathematical-analytical treatment of various objects taken from interdisciplinary fields of mechanics, physics, and applied mathematics. This new hybrid approach leads to results that cannot be obtained by standard theories in the field…. The book contains 9 chapters, and index, and a rich list of references. …”

— Vasile Dragan (Bucharest), in Mathematical Reviews, Issue 2007f

“Providing a state-of-the-art review of asymptotic applications, this book is useful as an introduction into the field of asymptotic methods for beginners as well as a reference for specialists. The choice of topics reflects the authors’ research experience in industrial applications. The authors pay special attention to examples and to the discussion of results rather than to burying the ideas of formalism, notation, and technical details.”

–– Igor Andrianov, in Zentralblatt Math, Vol. 1110 (2007/12)

Table of Contents

Introduction

Elements of Mathematical Modeling

Structure of a Mathematical Model

Examples of Reducing Problems to a Dimensionless Form

Mathematical Model Adequacy and Properties. Regular and Singular Perturbations

Expansion of Functions and Mathematical Methods

Expansions of Elementary Functions into Power Series

Mathematical Methods of Perturbations

Exercises

Regular and Singular Perturbations

Introduction. Asymptotic Approximations with Respect to Parameter

Non-Uniformities of a Classical Perturbation Approach

Method of “Elongated” Parameters

Method of Deformed Variables

Method of Scaling and Full Approximation

Multiple Scale Methods

Variations of Arbitrary Constants

Averaging Methods

Matching Asymptotic Decompositions

On the Sources of Non-Uniformities

On the Influence of Initial Conditions

Analysis of Strongly Nonlinear Dynamical Problems

A Few Perturbation Parameters

Exercises

Wave-Impact Processes

Definition of a Cylinder-Piston Wave

One-Dimensional Non-Stationary Non-Linear Waves

PadÉ Approximations

Determination and Characteristics of Padé Approximations

Application of Padé Approximations

Exercises

Averaging of Ribbed Plates

Averaging in the Theory of Ribbed Plates

Kantorovich-Vlasov –Type Methods

Transverse Vibrations of Rectangular Plates

Deflections of Rectangular Plates

Chaos Foresight

The Analysed System

Melnikov-Gruendler’s Approach

Melnikov-Gruendler Function

Numerical Results

Continuous Approximation of Discontinuous Systems

An Illustrative Example

Higher Dimensional Systems

Nonlinear Dynamics of a Swinging Oscillator

Parametrical Form of Canonical Transformations

Function Derivative

Invariant Normalization of Hamiltonians

Algorithm of Invariant Normalization for Asymptotical Determination of the Poincaré Series

Examples of Asymptotical Solutions

Swinging Oscillator

Normal Form

Normal Form Integral

References

Bibliography

About the Series

Modern Mechanics and Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
SCI041000
SCIENCE / Mechanics / General