Perturbation methods.

Format:

Book

Author/Creator:

Nayfeh, Ali Hasan, 1933-

Series:

Chun cui shu xue yu ying yong shu xue zhuan zhu

Series in pure and applied mathematics

Language:

English

Subjects (All):

Perturbation (Mathematics).

Asymptotic expansions.

Differential equations--Numerical solutions.

Differential equations.

Physical Description:

xii, 425 pages ; 24 cm.

Place of Publication:

New York : Wiley, [1973]

Summary:

The Method of Perturbations (asymptotic expansions) is an approximations technique for solving complicated problems in mathematics, engineering and physics involving nonlinear equations, variable coefficients and nonlinear boundary conditions. The purpose of this book is to present in a unified way an account of some of these techniques, pointing out their similarities, differences, and advantages, as well as their limitations.

Contents:

1.1. Parameter Perturbations 1

1.1.1. An Algebraic Equation 2

1.1.2. The van der Pol Oscillator 3

1.2. Coordinate Perturbations 4

1.2.1. The Bessel Equation of Zeroth Order 5

1.2.2. A Simple Example 6

1.3. Order Symbols and Gauge Functions 7

1.4. Asymptotic Expansions and Sequences 9

1.4.1. Asymptotic Series 9

1.4.2. Asymptotic Expansions 12

1.4.3. Uniqueness of Asymptotic Expansions 14

1.5. Convergent versus Asymptotic Series 15

1.6. Nonuniform Expansions 16

1.7. Elementary Operations on Asymptotic Expansions 18

2. Straightforward Expansions and Sources of Nonuniformity 23

2.1. Infinite Domains 24

2.1.1. The Duffing Equation 24

2.1.2. A Model for Weak Nonlinear Instability 25

2.1.3. Supersonic Flow Past a Thin Airfoil 26

2.1.4. Small Reynolds Number Flow Past a Sphere 28

2.2. A Small Parameter Multiplying the Highest Derivative 31

2.2.1. A Second-Order Example 31

2.2.2. High Reynolds Number Flow Past a Body 33

2.2.3. Relaxation Oscillations 34

2.2.4. Unsymmetrical Bending of Prestressed Annular Plates 35

2.3. Type Change of a Partial Differential Equation 37

2.3.1. A Simple Example 38

2.3.2. Long Waves on Liquids Flowing down Incline Planes 38

2.4. The Presence of Singularities 42

2.4.1. Shift in Singularity 42

2.4.2. The Earth-Moon-Spaceship Problem 43

2.4.3. Thermoelastic Surface Waves 45

2.4.4. Turning Point Problems 48

2.5. The Role of Coordinate Systems 49

3. The Method of Strained Coordinates 56

3.1. The Method of Strained Parameters 58

3.1.1. The Lindstedt-Poincare Method 58

3.1.2. Transition Curves for the Mathieu Equation 60

3.1.3. Characteristic Exponents for the Mathieu Equation (Whittaker's Method) 62

3.1.4. The Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies 64

3.1.5. Characteristic Exponents for the Triangular Points in the Elliptic Restricted Problem of Three Bodies 66

3.1.6. A Simple Linear Eigenvalue Problem 68

3.1.7. A Quasi-Linear Eigenvalue Problem 71

3.1.8. The Quasi-Linear Klein-Gordon Equation 76

3.2. Lighthill's Technique 77

3.2.1. A First-Order Differential Equation 79

3.2.2. The One-Dimensional Earth-Moon-Spaceship Problem 82

3.2.3. A Solid Cylinder Expanding Uniformly in Still Air 83

3.2.4. Supersonic Flow Past a Thin Airfoil 86

3.2.5. Expansions by Using Exact Characteristics

Nonlinear Elastic Waves 89

3.3. Temple's Technique 94

3.4. Renormalization Technique 95

3.4.1. The Duffing Equation 95

3.4.2. A Model for Weak Nonlinear Instability 96

3.4.3. Supersonic Flow Past a Thin Airfoil 97

3.4.4. Shift in Singularity 98

3.5. Limitations of the Method of Strained Coordinates 98

3.5.1. A Model for Weak Nonlinear Instability 99

3.5.2. A Small Parameter Multiplying the Highest Derivative 100

3.5.3. The Earth-Moon-Spaceship Problem 102

4. The Methods of Matched and Composite Asymptotic Expansions 110

4.1. The Method of Matched Asymptotic Expansions 111

4.1.1. Introduction

Prandtl's Technique 111

4.1.2. Higher Approximations and Refined Matching Procedures 114

4.1.3. A Second-Order Equation with Variable Coefficients 122

4.1.4. Reynolds' Equation for a Slider Bearing 125

4.1.5. Unsymmetrical Bending of Prestressed Annular Plates 128

4.1.6. Thermoelastic Surface Waves 133

4.1.7. The Earth-Moon-Spaceship Problem 137

4.1.8. Small Reynolds Number Flow Past a Sphere 139

4.2. The Method of Composite Expansions 144

4.2.1. A Second-Order Equation with Constant Coefficients 145

4.2.2. A Second-Order Equation with Variable Coefficients 148

4.2.3. An Initial Value Problem for the Heat Equation 150

4.2.4. Limitations of the Method of Composite Expansions 153

5. Variation of Parameters and Methods of Averaging 159

5.1. Variation of Parameters 159

5.1.1. Time-Dependent Solutions of the Schrodinger Equation 160

5.1.2. A Nonlinear Stability Example 162

5.2. The Method of Averaging 164

5.2.1. Van der Pol's Technique 164

5.2.2. The Krylov-Bogoliubov Technique 165

5.2.3. The Generalized Method of Averaging 168

5.3. Struble's Technique 171

5.4. The Krylov-Bogoliubov-Mitropolski Technique 174

5.4.1. The Duffiing Equation 175

5.4.2. The van der Pol Oscillator 176

5.4.3. The Klein-Gordon Equation 178

5.5. The Method of Averaging by Using Canonical Variables 179

5.5.1. The Duffing Equation 182

5.5.2. The Mathieu Equation 183

5.5.3. A Swinging Spring 185

5.6. Von Zeipel's Procedure 189

5.6.1. The Duffing Equation 192

5.6.2. The Mathieu Equation 194

5.7. Averaging by Using the Lie Series and Transforms 200

5.7.1. The Lie Series and Transforms 201

5.7.2. Generalized Algorithms 202

5.7.3. Simplified General Algorithms 206

5.7.5. Algorithms for Canonical Systems 212

5.8. Averaging by Using Lagrangians 216

5.8.1. A Model for Dispersive Waves 217

5.8.2. A Model for Wave-Wave Interaction 219

5.8.3. The Nonlinear Klein-Gordon Equation 221

6. The Method of Multiple Scales 228

6.1.1. Many-Variable Version (The Derivative-Expansion Procedure) 236

6.1.2. The Two-Variable Expansion Procedure 240

6.1.3. Generalized Method

Nonlinear Scales 241

6.2. Applications of the Derivative-Expansion Method 243

6.2.1. The Duffing Equation 243

6.2.2. The van der Pol Oscillator 245

6.2.3. Forced Oscillations of the van der Pol Equation 248

6.2.4. Parametric Resonances

The Mathieu Equation 253

6.2.5. The van der Pol Oscillator with Delayed Amplitude Limiting 257

6.2.6. The Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies 259

6.2.7. A Swinging Spring 262

6.2.8. A Model for Weak Nonlinear Instability 264

6.2.9. A Model for Wave-Wave Interaction 266

6.2.10. Limitations of the Derivative-Expansion Method 269

6.3. The Two-Variable Expansion Procedure 270

6.3.1. The Duffing Equation 270

6.3.2. The van der Pol Oscillator 272

6.3.3. The Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies 275

6.3.4. Limitations of This Technique 275

6.4. Generalized Method 276

6.4.1. A Second-Order Equation with Variable Coefficients 276

6.4.2. A General Second-Order Equation with Variable Coefficients 280

6.4.3. A Linear Oscillator with a Slowly Varying Restoring Force 282

6.4.4. An Example with a Turning Point 284

6.4.5. The Duffing Equation with Slowly Varying Coefficients 286

6.4.6. Reentry Dynamics 291

6.4.7. The Earth-Moon-Spaceship Problem 295

6.4.8. A Model for Dispersive Waves 298

6.4.9. The Nonlinear Klein-Gordon Equation 301

6.4.10. Advantages and Limitations of the Generalized Method 302

7. Asymptotic Solutions of Linear Equations 308

7.1. Second-Order Differential Equations 309

7.1.1. Expansions Near an Irregular Singularity 309

7.1.2. An Expansion of the Zeroth-Order Bessel Function for Large Argument 312

7.1.3. Liouville's Problem 314

7.1.4. Higher Approximations for Equations Containing a Large Parameter 315

7.1.5. A Small Parameter Multiplying the Highest Derivative 317

7.1.6. Homogeneous Problems with Slowly Varying Coefficients 318

7.1.7. Reentry Missile Dynamics 320

7.1.8. Inhomogeneous Problems with Slowly Varying Coefficients 321

7.1.9. Successive Liouville-Green (WKB) Approximations 324

7.2. Systems of First-Order Ordinary Equations 325

7.2.1. Expansions Near an Irregular Singular Point 326

7.2.2. Asymptotic Partitioning of Systems of Equations 327

7.2.3. Subnormal Solutions 331

7.2.4. Systems Containing a Parameter 332

7.2.5. Homogeneous Systems with Slowly Varying Coefficients 333

7.3. Turning Point Problems 335

7.3.1. The Method of Matched Asymptotic Expansions 336

7.3.2. The Langer Transformation 339

7.3.3. Problems with Two Turning Points 342

7.3.4. Higher-Order Turning Point Problems 345

7.3.5. Higher Approximations 346

7.3.6. An Inhomogeneous Problem with a Simple Turning Point

First Approximation 352

7.3.7. An Inhomogeneous Problem with a Simple Turning Point

Higher Approximations 354

7.3.8. An Inhomogeneous Problem with a Second-Order Turning Point 356

7.3.9. Turning Point Problems about Singularities 358

7.3.10. Turning Point Problems of Higher Order 360

7.4. Wave Equations 360

7.4.1. The Born or Neumann Expansion and The Feynman Diagrams 361

7.4.2. Renormalization Techniques 367

7.4.3. Rytov's Method 373

7.4.4. A Geometrical Optics Approximation 374

7.4.5. A Uniform Expansion at a Caustic 377

7.4.6. The Method of Smoothing 380.

Notes:

"A Wiley-Interscience publication."

Bibliography: page.

ISBN:

0471630594

OCLC:

415430