Perturbation Methods

by Nayfeh, Ali H.

• ISBN13: 9780471399179
• ISBN10: 0471399175
• Edition: 1st
• Format: Paperback
• Copyright: 2000-07-26
• Publisher: Wiley-VCH

Summary

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

Author Biography

Ali H. Nayfeh received his BS in engineering science and his MS and PhD in aeronautics and astronautics from Stanford University. He holds honorary doctorates from Marine Technical University, Russia, Technical University of Munich, Germany, and Politechnika Szczecinska, Poland. He is currently University Distinguished Professor of Engineering at Virginia Tech. He is the Editor of the Wiley Series in Nonlinear Science and Editor in Chief of Nonlinear Dynamics and the Journal of Vibration and Control.

Table of Contents

Introduction	1	(22)
Parameter Perturbations	1	(3)
An Algebraic Equation	2	(1)
The van der Pol Oscillator	3	(1)
Coordinate Perturbations	4	(3)
The Bessel Equation of Zeroth Order	5	(1)
A Simple Example	6	(1)
Order Symbols and Gauge Functions	7	(2)
Asymptotic Expansions and Sequences	9	(6)
Asymptotic Series	9	(3)
Asymptotic Expansions	12	(2)
Uniqueness of Asymptotic Expansions	14	(1)
Convergent versus Asymptotic Series	15	(1)
Nonuniform Expansions	16	(2)
Elementary Operations on Asymptotic Expansions	18	(5)
Exercises	19	(4)
Straightforward Expansions and Sources of Nonuniformity	23	(33)
Infinite Domains	24	(7)
The Duffing Equation	24	(1)
A Model for Weak Nonlinear Instability	25	(1)
Supersonic Flow Past a Thin Airfoil	26	(2)
Small Reynolds Number Flow Past a Sphere	28	(3)
A Small Parameter Multiplying the Highest Derivative	31	(6)
A Second-Order Example	31	(2)
High Reynolds Number Flow Past a Body	33	(1)
Relaxation Oscillations	34	(1)
Unsymmetrical Bending of Prestressed Annular Plates	35	(2)
Type Change of a Partial Differential Equation	37	(5)
A Simple Example	38	(1)
Long Waves on Liquids Flowing down Incline Planes	38	(4)
The Presence of Singularities	42	(7)
Shift in Singularity	42	(1)
The Earth-Moon-Spaceship Problem	43	(2)
Thermoelastic Surface Waves	45	(3)
Turning Point Problems	48	(1)
The Role of Coordinate Systems	49	(7)
Exercises	52	(4)
The Method of Strained Coordinates	56	(54)
The Method of Strained Parameters	58	(19)
The Lindstedt-Poincare Method	58	(2)
Transition Curves for the Mathieu Equation	60	(2)
Characteristic Exponents for the Mathieu Equation (Whittaker's Method)	62	(2)
The Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies	64	(2)
Characteristic Exponents for the Triangular Points in the Elliptic Restricted Problem of Three Bodies	66	(2)
A Simple Linear Eigenvalue Problem	68	(3)
A Quasi-Linear Eigenvalue Problem	71	(5)
The Quasi-Linear Klein-Gordon Equation	76	(1)
Lighthill's Technique	77	(17)
A First-Order Differential Equation	79	(3)
The One-Dimensional Earth-Moon-Spaceship Problem	82	(1)
A Solid Cylinder Expanding Uniformly in Still Air	83	(3)
Supersonic Flow Past a Thin Airfoil	86	(3)
Expansions by Using Exact Characteristics-Nonlinear Elastic Waves	89	(5)
Temple's Technique	94	(1)
Renormalization Technique	95	(3)
The Duffing Equation	95	(1)
A Model for Weak Nonlinear Instability	96	(1)
Supersonic Flow Past a Thin Airfoil	97	(1)
Shift in Singularity	98	(1)
Limitations of the Method of Strained Coordinates	98	(12)
A Model for Weak Nonlinear Instability	99	(1)
A Small Parameter Multiplying the Highest Derivative	100	(2)
The Earth-Moon-Spaceship Problem	102	(1)
Exercises	103	(7)
The Methods of Matched and Composite Asymptotic Expansions	110	(49)
The Method of Matched Asymptotic Expansions	111	(33)
Introduction---Prandtl's Technique	111	(3)
Higher Approximations and Refined Matching Procedures	114	(8)
A Second-Order Equation with Variable Coefficients	122	(3)
Reynolds' Equation for a Slider Bearing	125	(3)
Unsymmetrical Bending of Prestressed Annular Plates	128	(5)
Thermoelastic Surface Waves	133	(4)
The Earth-Moon-Spaceship Problem	137	(2)
Small Reynolds Number Flow Past a Sphere	139	(5)
The Method of Composite Expansions	144	(15)
A Second-Order Equation with Constant Coefficients	145	(3)
A Second-Order Equation with Variable Coefficients	148	(2)
An Initial Value Problem for the Heat Equation	150	(3)
Limitations of the Method of Composite Expansions	153	(1)
Exercises	154	(5)
Variation of Parameters and Methods of Averaging	159	(69)
Variation of Parameters	159	(5)
Time-Dependent Solutions of the Schrodinger Equation	160	(2)
A Nonlinear Stability Example	162	(2)
The Method of Averaging	164	(7)
Van der Pol's Technique	164	(1)
The Krylov--Bogoliubov Technique	165	(3)
The Generalized Method of Averaging	168	(3)
Struble's Technique	171	(3)
The Krylov--Bogoliubov--Mitropolski Technique	174	(5)
The Duffing Equation	175	(1)
The van der Pol Oscillator	176	(2)
The Klein-Gordon Equation	178	(1)
The Method of Averaging by Using Canonical Variables	179	(10)
The Duffing Equation	182	(1)
The Mathieu Equation	183	(2)
A Swinging Spring	185	(4)
Von Zeipel's Procedure	189	(11)
The Duffing Equation	192	(2)
The Mathieu Equation	194	(6)
Averaging by Using the Lie Series and Transforms	200	(16)
The Lie Series and Transforms	201	(1)
Generalized Algorithms	202	(4)
Simplified General Algorithms	206	(2)
A Procedure Outline	208	(4)
Algorithms for Canonical Systems	212	(4)
Averaging by Using Lagrangians	216	(12)
A Model for Dispersive Waves	217	(2)
A Model for Wave-Wave Interaction	219	(2)
The Nonlinear Klein-Gordon Equation	221	(2)
Exercises	223	(5)
The Method of Multiple Scales	228	(80)
Description of the Method	228	(15)
Many-Variable Version (The Derivative-Expansion Procedure)	236	(4)
The Two-Variable Expansion Procedure	240	(1)
Generalized Method-Nonlinear Scales	241	(2)
Applications of the Derivative-Expansion Method	243	(27)
The Duffing Equation	243	(2)
The van der Pol Oscillator	245	(3)
Forced Oscillations of the van der Pol Equation	248	(5)
Parametric Resonances-The Mathieu Equation	253	(4)
The van der Pol Oscillator with Delayed Amplitude Limiting	257	(2)
The Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies	259	(3)
A Swinging Spring	262	(2)
A Model for Weak Nonlinear Instability	264	(2)
A Model for Wave-Wave Interaction	266	(3)
Limitations of the Derivative-Expansion Method	269	(1)
The Two-Variable Expansion Procedure	270	(6)
The Duffing Equation	270	(2)
The van der Pol Oscillator	272	(3)
The Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies	275	(1)
Limitations of This Technique	275	(1)
Generalized Method	276	(32)
A Second-Order Equation with Variable Coefficients	276	(4)
A General Second-Order Equation with Variable Coefficients	280	(2)
A Linear Oscillator with a Slowly Varying Restoring Force	282	(2)
An Example with a Turning Point	284	(2)
The Duffing Equation with Slowly Varying Coefficients	286	(5)
Reentry Dynamics	291	(4)
The Earth-Moon-Spaceship Problem	295	(3)
A Model for Dispersive Waves	298	(3)
The Nonlinear Klein-Gordon Equation	301	(1)
Advantages and Limitations of the Generalized Method	302	(1)
Exercises	303	(5)
Asymptotic Solutions of Linear Equations	308	(79)
Second-Order Differential Equations	309	(16)
Expansions Near an Irregular Singularity	309	(3)
An Expansion of the Zeroth-Order Bessel Function for Large Argument	312	(2)
Liouville's Problem	314	(1)
Higher Approximations for Equations Containing a Large Parameter	315	(2)
A Small Parameter Multiplying the Highest Derivative	317	(1)
Homogeneous Problems with Slowly Varying Coefficients	318	(2)
Reentry Missile Dynamics	320	(1)
Inhomogeneous Problems with Slowly Varying Coefficients	321	(3)
Successive Liouville-Green (WKB) Approximations	324	(1)
Systems of First-Order Ordinary Equations	325	(10)
Expansions Near an Irregular Singular Point	326	(1)
Asymptotic Partitioning of Systems of Equations	327	(4)
Subnormal Solutions	331	(1)
Systems Containing a Parameter	332	(1)
Homogeneous Systems with Slowly Varying Coefficients	333	(2)
Turning Point Problems	335	(25)
The Method of Matched Asymptotic Expansions	336	(3)
The Langer Transformation	339	(3)
Problems with Two Turning Points	342	(3)
Higher-Order Turning Point Problems	345	(1)
Higher Approximations	346	(6)
An Inhomogeneous Problem with a Simple Turning Point-First Approximation	352	(2)
An Inhomogeneous Problem with a Simple Turning Point-Higher Approximations	354	(2)
An Inhomogeneous Problem with a Second-Order Turning Point	356	(2)
Turning Point Problems about Singularities	358	(2)
Turning Point Problems of Higher Order	360	(1)
Wave Equations	360	(27)
The Born or Neumann Expansion and the Feynman Diagrams	361	(6)
Renormalization Techniques	367	(6)
Rytov's Method	373	(1)
A Geometrical Optics Approximation	374	(3)
A Uniform Expansion at a Caustic	377	(3)
The Method of Smoothing	380	(2)
Exercises	382	(5)
References and Author Index	387	(30)
Subject Index	417