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PART I FUNDAMENTALS 3

1 Ordinary Differential Equations 3

1.1 Ordinary Differential Equations 3

1.2 Initial-Value and Boundary-Value Problems 5

1.3 Theory of Homogeneous Linear Equations 7

1.4 Solutions of Homogeneous Linear Equations 11

1.5 Inhomogeneous Linear Equations 14

1.6 First-Order Nonlinear Differential Equations 20

1.7 Higher-Order Nonlinear Differential Equations 24

1.8 Eigenvalue Problems 27

1.9 Differential Equations in the Complex Plane 29

Problems for Chapter 1 30

2 Difference Equations 36

2.1 The Calculus of Differences 36

2.2 Elementary Difference Equations 37

2.3 Homogeneous Linear Difference Equations 40

2.4 Inhomogeneous Linear Difference Equations 49

2.5 Nonlinear Difference Equations 53

Problems for Chapter 2 53

PART Ⅱ LOCAL ANALYSIS 61

3 Approximate Solution of Linear Differential Equations 61

3.1 Classification of Singular Points of Homogeneous Linear Equations 62

3.2 Local Behavior Near Ordinary Points of Homogeneous LinearEquations 66

3.3 Local Series Expansions About Regular Singular Points ofHomogeneous Linear Equations 68

3.4 Local Behavior at Irregular Singular Points of Homogeneous Linear Equations 76

3.5 Irregular Singular Point at Infinity 88

3.6 Local Analysis of Inhomogeneous Linear Equations 103

3.7 Asymptotic Relations 107

3.8 Asymptotic Series 118

Problems for Chapter 3 136

4 Approximate Solution of Nonlinear Differential Equations 146

4.1 Spontaneous Singularities 146

4.2 Approximate Solutions of First-Order Nonlinear Differential Equations 148

4.3 Approximate Solutions to Higher-Order Nonlinear Differential Equations 152

4.4 Nonlinear Autonomous Systems 171

4.5 Higher-Order Nonlinear Autonomous Systems 185

Problems for Chapter 4 196

5 Approximate Solution of Difference Equations 205

5.1 Introductory Comments 205

5.2 Ordinary and Regular Singular Points of Linear Difference Equations 206

5.3 Local Behavior Near an Irregular Singular Point at Infinity: Determination of Controlling Factors 214

5.4 Asymptotic Behavior of n! as n→∞:The Stirling Series 218

5.5 Local Behavior Near an Irregular Singular Point at Intinity:Full Asymptotic Series 227

5.6 Local Behavior of Nonlinear Difference Equations 233

Problems for Chapter 5 240

6 Asymptotic Expansion of Integrals 247

6.1 Introduction 247

6.2 Elementary Examples 249

6.3 Integration by Parts 252

6.4 Laplace’s Method and Watson’s Lemma 261

6.5 Method of Stationary Phase 276

6.6 Method of Steepest Descents 280

6.7 Asymptotic Evaluation of Sums 302

Problems for Chapter 6 306

PART ⅢPERTURBATION METHODS 319

7 Perturbation Series 319

7.1 Perturbation Theory 319

7.2 Regular and Singular Perturbation Theory 324

7.3 Perturbation Methods for Linear Eigenvalue Problems 330

7.4 Asymptotic Matching 335

7.5 Mathematical Structure of Perturbative Eigenvalue Problems 350

Problems for Chapter 7 361

8 Summation of Series 368

8.1 Improvement of Convergence 368

8.2 Summation of Divergent Series 379

8.3 Pade Summation 383

8.4 Continued Fractions and Pade Approximants 395

8.5 Convergence of Pade Approximants 400

8.6 Pade Sequences for Stieltjes Functions 405

Problems for Chapter 8 410

PART Ⅳ GLOBAL ANALYSIS 417

9 Boundary Layer Theory 417

9.1 Introduction to Boundary-Layer Theory 419

9.2 Mathematical Structure of Boundary Layers: Inner, Outer, and Intermediate Limits 426

9.3 Higher-Order Boundary Layer Theory 431

9.4 Distinguished Limits and Boundary Layers of Thickness ≠ ε 435

9.5 Miscellaneous Examples of Linear Boundary-Layer Problems 446

9.6 Internal Boundary Layers 455

9.7 Nonlinear Boundary-Layer Problems 463

Problems for Chapter 9 479

10 WKB Theory 484

10.1 The Exponential Approximation for Dissipative and Dispersive Phenomena 484

10.2 Conditions for Validity of the WKB Approximation 493

10.3 Patched Asvmptotic Approximations: WKB Solution ofInhomogeneous Linear Equations 497

10.4 Matched Asymptotic Approximations: Solution of the One-Turning-Point Problem 504

10.5 Two-Turning-Point Problems: Eigenvalue Condition 519

10.6 Tunneling 524

10.7 Brief Discussion of Higher-Order WKB Approximations 534

Problems for Chapter 10 539

11 Multiple-Scale Analysis 544

11.1 Resonance and Secular Behavior 544

11.2 Multiple-Scale Analysis 549

11.3 Examples of Multiple-Scale Analysis 551

11.4 The Mathieu Equation and Stability 560

Problems for Chapter 11 566

Appendix—Useful Formulas 569

References 577

Index 581