Perturbation Methods (Pure and Applied Mathematics)

✍ Scribed by Ali H. Nayfeh

• Publisher: Wiley-VCH
• Year: 1973
• Tongue: English
• Leaves: 442
• Edition: 1
• Category: Library

✦ Synopsis

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.
Currently available in the Series:

T. W. Anderson
The Statistical Analysis of Time Series
T. S. Arthanari & Yadolah Dodge
Mathematical Programming in Statistics
Emil Artin
Geometric Algebra
Norman T. J. Bailey
The Elements of Stochastic Processes
with Applications to the Natural Sciences
Robert G. Bartle
The Elements of Integration and
Lebesgue Measure
George E. P. Box & Norman R. Draper
Evolutionary Operation: A Statistical Method for Process Improvement
George E. P. Box & George C. Tiao
Bayesian Inference in Statistical Analysis
R. W. Carter
Finite Groups of Lie Type: Conjugacy Classes
and Complex Characters
R. W. Carter
Simple Groups of Lie Type
William G. Cochran & Gertrude M. Cox
Experimental Designs, Second Edition
Richard Courant
Differential and Integral Calculus, Volume I
RIchard Courant
Differential and Integral Calculus, Volume II
Richard Courant & D. Hilbert
Methods of Mathematical Physics, Volume I
Richard Courant & D. Hilbert
Methods of Mathematical Physics, Volume II

D. R. Cox
Planning of Experiments
Harold S. M. Coxeter
Introduction to Geometry, Second Edition
Charles W. Curtis & Irving Reiner
Representation Theory of Finite Groups and
Associative Algebras
Charles W. Curtis & Irving Reiner
Methods of Representation Theory
with Applications to Finite Groups
and Orders, Volume I
Charles W. Curtis & Irving Reiner
Methods of Representation Theory
with Applications to Finite Groups
and Orders, Volume II
Cuthbert Daniel
Fitting Equations to Data: Computer Analysis of
Multifactor Data, Second Edition
Bruno de Finetti
Theory of Probability, Volume I
Bruno de Finetti
Theory of Probability, Volume 2
W. Edwards Deming
Sample Design in Business Research

✦ Table of Contents

Perturbation Methods
Copyright
Contents
1. Introduction
1.1. Parameter Perturbations
1.1.1. An Algebraic Equation
1.1.2. The van der Pol Oscillator
1.2. Coordinate Perturbations
1.2.1. The Bessel Equation of Zeroth Order
1.2.2. A Simple Example
1.3. Order Symbols and Gauge Functions
1.4. Asymptotic Expansions and Sequences
1.4.1 . Asymptotic Series
1.4.2. Asymptotic Expansions
1.4.3. Uniqueness of Asymptotic Expansions
1.5. Convergent versus Asymptotic Series
1.6. Nonuniform Expansions
1.7. Elementary Operations on Asymptotic Expansions
Exercises
2. Straightforward Expansions and Sources of Nonuniformity
2.1. Infinite Domains
2.1.1.The Duffing Equation
2.1.2. A Model for Weak Nonlinear Instability
2.1.3. Supersonic Flow Past a Thin Airfoil
2.1.4. Small Reynolds Number FIow Past a Sphere
2.2. A Small Parameter Multiplying the Highest Derivative
2.2.1. A Second-Order Example
2.2.2. High Reynolds Number Flow Past a Body
2.2.3. Relaxation Oscillations
2.2.4. Unsymmetrical Bending of Prestressed Annular Plates
2.3. Type Change of a Partial Differential Equation
2.3.1. A Simple Example
2.3.2. Long Waves on Liquids Flowing down Incline Planes
2.4. The Presence of Singularities
2.4.1. Shift in Singularity
2.4.2. The Earth-Moon-Spaceship Problem
2.4.3. Thermoelastic Surface Waves
2.4.4. Turning Point Problems
2.5. The Role of Coordinate Systems
Exercises
3. The Method of Strained Coordinates
3.1. The Method of Strained Parameters
3.1.1. The Lindstedt-Poincaré Method
3.1.2. Transition Curves for the Mathieu quation
3.1.3. Characteristic Exponents for the Mathieu Equation (Whittaker’s Method)
3.1.4. The Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies
3.1.5. Characteristic Exponents for the Triangular Points in the Elliptic Restricted Problem of Three Bodies
3.1.6. A Simple Linear Eigenvalue Problem
3.1.7. A Quasi-Linear Eigenvalue Problem
3.1.8. The Quasi-Linear Klein-Gordon Equation
3.2. Lighthill’s Technique
3.2.1. A First-Order Differential Equation
3.2.2. The One-Dimensional Earth-Moon-Spaceship Problem
3.2.3. A Solid Cylinder Expanding Uniformly in Still Air
3.2.4. Supersonic Flow Past a Thin Airfoil
3.2.5. Expansions by Using Exact Characteristics - Nonlinear Elastic Waves
3.3. Temple’s Technique
3.4. Renormalization Technique
3.4.1. The Duffing Equation
3.4.2. A Model for Weak Nonlinear Instability
3.4.3. Supersonic Flow Past a Thin Airfoil
3.4.4. Shift in Singularity
3.5. Limitations of the Method of Strained Coordinates
3.5.1. A Model for Weak Nonlinear Instability
3.5.2. A Small Parameter Multiplying the Highest Derivative
3.5.3. The Earth-Moon-Spaceship Problem
Exercises
4. The Methods of Matched and Composite Asymptotic Expansions
4.1. The Method of Matched Asymptotic Expansions
4.1.1 . Introduction - Prandtl's Technique
4.1.2. Higher Approximations and Refined Matching Procedures
4.1.3. A Second-Order Equation with Variable Coefficients
4.1.4. Reynolds’ Equation for a Slider Bearing
4.1.5. Unsymmetrical Bending of Prestressed Annular Plates
4.1.6. Thermoelastic Surface Waves
4.1.7. The Earth-Moon-Spaceship Problem
4.1.8. Small Reynolds Number Flow Past a Sphere
4.2. The Method of Composite Expansions
4.2. 1. A Second-Order Equation with Constant Coefficients
4.2.2. A Second-Order Equation with Variable Coefficients
4.2.3. An Initial Value Problem for the Heat Equation
4.2.4. Limitations of the Method of Composite Expansions
Exercises
5. Variation of Parameters and Methods of Averaging
5.1. Variation of Parameters
5.1.1. Time- Dependent Solutions of the Schrödinger Equation
5.1.2. A Nonlinear Stability Example
5.2. The Method of Averaging
5.2.1. Van der Pol’s Technique
5.2.2. The Krylov-Bogoliubov Technique
5.2.3. The Generalized Method of Averaging
5.3. Struble’s Technique
5.4. The Krylov–Bogoliubov–Mitropolski Technique
5.4.1. The Duffing Equation
5.4.2. The van der Pol Oscillator
5.4.3. The Klein-Gordon Equation
5.5. The Method of Averaging by Using Canonical Variables
5.5.1. The Duffing Equation
5.5.2. The Mathieu Equation
5.5.3. A Swinging Spring
5.6. Von Zeipel’s Procedure
5.6.1 . The Dufing Equation
5.6.2. The Mathieu Equation
5.7. Averaging by Using the Lie Series and Transforms
5.7. 1. The Lie Series and Transforms
5.7.2. Generalized Algorithms
5.7.3. Simplified General Algorithms
5.7.4. A Procedure Outline
5.7.5. Algorithms for Canonical Systems
5.8. Averaging by Using Lagrangians
5.8.1. A Model for Dispersive Waves
5.8.2. A Model for Wave–Wave Interaction
5.8.3. The Nonlinear Klein–Gordon Equation
Exercises
6. The Method of Multiple Scales
6.1. Description of the Method
6.1.1. Many-Variable Version (The Derivatiue-Expansion Procedure)
6.1.2. The Two-Variable Expansion Procedure
6.1.3. Generalized Method—Nonlinear Scales
6.2. Applications of the Derivative-Expansion Method
6.2.1. The Duffing Equation
6.2.2. The van der Pol Oscillator
6.2.3. Forced Oscillations of the van der Pol Equation
6.2.4. Parametric Resonances—The Mathieu Equation
6.2.5. The van der Pol Oscillator with Delayed Amplitude Limiting
6.2.6. The Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies
6.2.7. A Swinging Spring
6.2.8. A Model for Weak Nonlinear Instability
6.2.9. A Model for Wave-Wave Interaction
6.2.10. Limitations of the Derivative-Expansion Method
6.3. The Two-Variable Expansion Procedure
6.3.1. The Duffng Equation
6.3.2. The van der Pol Oscillator
6.3.3. The Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies
6.3.4. Limitations of This Technique
6.4. Generalized Method
6.4.1. A Second-Order Equation with Variable Coefficients
6.4.2. A General Second-Order Equation with Variable Coefficients
6.4.3. A Linear Oscillator with a Slowly Varying Restoring Force
6.4.4. An Example with a Turning Point
6.4.5. The Duffing Equation with Slowly Varying Coeficients
6.4.6. Reentry Dynamics
6.4.7. The Earth-Moon-Spaceship Problem
6.4.8. A Model for Dispersive Waves
6.4.9. The Nonlinear Klein-Gordon Equation
6.4.10. Advantages and Limitations of the Generalized Method
Exercises
7. Asymptotic Solutions of Linear Equations
7.1. Second-Order Differential Equations
7. 1. 1. Expansions Near an Irregular Singularity
7.1.2. An Expansion of the Zeroth-Order Bessel Function for Large Argument
7.1.3. Liouville’s Problem
7.1.4. Higher Approximations for Equations Containing a Large Parameter
7.1 .5. A Small Parameter Multiplying the Highest Derivative
7.1.6. Homogeneous Problems with Slowly Varying Coefficients
7.1.7. Reentry Missile Dynamics
7.1.8. Inhomogeneous Problems with Slowly Varying Coefficients
7.1.9. Successive Liouville-Green (WKB) Approximations
7.2. Systems of First-Order Ordinary Equations
7.2.1. Expansions Near an Irregular Singular Point
7.2.2. Asymptotic Partitioning of System of Equations
7.2.3. Subnormal Solutions
7.2.4. Systems Containing a Parameter
7.2.5. Homogeneous Systems with Slowly Varying Cofficients
7.3. Turning Point Problems
7.3.1. The Method of Matched Asymptotic Expansions
7.3.2. The Langer Transformation
7.3.3. Problems with Two Turning Points
7.3.4. Higher-Order Turning Point Problems
7.3.5. Higher Approximations
7.3.6. An Inhomogeneous Problem with a Simple Turning Point—First Approximation
7.3.7. An Inhomogeneous Problem with a Simple Turning Point—Higher Approximations
7.3.8. An Inhomogeneous Problem with a Second-Order Turning Point
7.3.9. Turning Point Problems about Singularities
7.3.10. Turning Point Problems of Higher Order
7.4. Wave Equations
7.4.1. The Born or Neumann Expansion and The Feynman Diagrams
7.4.2. Renormalization Techniques
7.4.3. Rytov’s Method
7.4.4. A Geometrical Optics Approximation
7.4.5. A Uniform Expansion at a Caustic
7.4.6. The Method of Smoothing
Exercises
References and Author Index
Subject Index