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Applied Asymptotic Analysis

Publisher: 
American Mathematical Society
Number of Pages: 
467
Price: 
69.00
ISBN: 
0821840789
Date Received: 
Monday, July 3, 2006
Reviewable: 
No
Include In BLL Rating: 
No
Reviewer Email Address: 
Peter D. Miller
Series: 
Graduate Studies in Mathematics 75
Publication Date: 
2006
Format: 
Hardcover
Category: 
Textbook

Preface xiii

Part 1. Fundamentals

Chapter 0. Themes of Asymptotic Analysis 3

§0.1. Theme: Asymptotics, Convergent and Divergent

Asymptotic Series 3

§0.2. Theme: Other Parameters and Nonuniformity 8

0.2.1. First example. Oscillations 8

0.2.2. Second example. Boundary layers 10

§0.3. Theme: Differential Equations 12

§0.4. Theme: Universal Partial Differential Equations and

Canonical Physical Models 13

Chapter 1. The Nature of Asymptotic Approximations 15

§1.1. Asymptotic Approximations and Errors 15

1.1.1. Order relations among functions 15

1.1.2. Statements following from the order relations 20

1.1.3. Absolute and relative errors 23

§1.2. Convergent versus Asymptotic Series: Concepts 24

1.2.1. Convergent power series 24

1.2.2. Introduction to asymptotic series 26

§1.3. Asymptotic Sequences and Series: General Definitions 28

§1.4. How to “Sum” an Asymptotic Series 32

§1.5. Asymptotic Root Finding 36

1.5.1. A regular perturbation problem 38

vii

viii Contents

1.5.2. A singular perturbation problem. Rescaling and the

principle of dominant balance 40

§1.6. Notes and References 43

Part 2. Asymptotic Analysis of Exponential Integrals

Chapter 2. Fundamental Techniques for Integrals 47

§2.1. Review of Basic Methods 47

§2.2. Exponential Integrals and Watson’s Lemma 52

§2.3. Elementary Generalizations of Watson’s Lemma 56

Chapter 3. Laplace’s Method for Asymptotic Expansions of

Integrals 61

§3.1. Introduction 61

§3.2. Nonlocal Contributions 62

§3.3. Contributions from Endpoints 64

§3.4. Contributions from Interior Maxima 67

§3.5. Summary of Generic Leading-order Behavior 70

§3.6. Application: Weakly Diffusive Regularization of Shock

Waves 73

3.6.1. The method of characteristics 75

3.6.2. Regularization of shocks by diffusion. Burgers’ equation 78

3.6.3. The Cole-Hopf transformation and the solution of the

initial-value problem for Burgers’ equation 80

3.6.4. Analysis of the solution in the limit of vanishing

diffusion 82

§3.7. Multidimensional Integrals 87

§3.8. Notes and References 93

Chapter 4. The Method of Steepest Descents for Asymptotic

Expansions of Integrals 95

§4.1. Introduction 95

§4.2. Contour Deformation 97

§4.3. Paths of Steepest Descent 98

§4.4. Saddle Points 103

§4.5. Parametrization-independent Local Contributions 107

§4.6. Application: Long-time Asymptotic Behavior of Diffusion

Processes 108

4.6.1. A derivation of the diffusion equation 109

Contents ix

4.6.2. Solution of the diffusion equation and the corresponding

initial-value problem 110

4.6.3. Long-time asymptotics via the method of steepest

descents 112

§4.7. Application: Asymptotic Behavior of Special Functions,

Airy Functions and the Stokes Phenomenon 116

4.7.1. Integral representations for Airy functions 116

4.7.2. Preliminary transformations necessary for asymptotic

analysis of Ai(x) for large x 117

4.7.3. Determination of the path. Dependence of the path on κ 119

4.7.4. Asymptotic behavior of Ai(x) for large x. The Stokes

phenomenon 122

§4.8. The Effect of Branch Points 125

4.8.1. Application: Asymptotics of transform integrals 135

4.8.2. Application: Selection of particular solutions of linear

differential equations admitting integral representations 142

§4.9. Notes and References 147

Chapter 5. The Method of Stationary Phase for Asymptotic

Analysis of Oscillatory Integrals 149

§5.1. Introduction 149

§5.2. Nonlocal Contributions 151

§5.3. Contributions from Interior Stationary Phase Points 156

5.3.1. Putting the exponent in normal form by a change of

variables 156

5.3.2. Analysis of J1(λ) by the method of steepest descents 158

5.3.3. Analysis of J2(λ) using integration by parts 160

5.3.4. The asymptotic contribution of a stationary phase point 161

§5.4. Summary of Generic Leading-order Behavior 162

§5.5. Application: Long-time Behavior of Linear Dispersive

Waves 164

5.5.1. Partial differential equations for linear dispersive waves 164

5.5.2. Analysis of the solution formula. Longtime

asymptotics using the method of stationary

phase 167

5.5.3. Structure of the wave field for large time. Modulated

wavetrains and group velocity 169

§5.6. Application: Semiclassical Dynamics of Free Particles in

Quantum Mechanics 171

5.6.1. Derivation of the dispersion relation for “matter waves” 171

x Contents

5.6.2. The Schr¨odinger equation for a free particle.

Interpretation of the Schr¨odinger wave function 173

5.6.3. The semiclassical limit. Heuristic reasoning 174

5.6.4. Rigorous semiclassical asymptotics using the method of

stationary phase 177

§5.7. Multidimensional Integrals 181

§5.8. Notes and References 193

Part 3. Asymptotic Analysis of Differential Equations

Chapter 6. Asymptotic Behavior of Solutions of Linear Secondorder

Differential Equations in the Complex

Plane 197

§6.1. Qualitative Theory of Solutions 198

6.1.1. Reduction to canonical form 198

6.1.2. Solutions viewed as analytic functions of the complex

variable z 200

6.1.3. Reduction of order 213

§6.2. Asymptotic Behavior near Ordinary and Regular Singular

Points 214

6.2.1. Series solutions at ordinary points 215

6.2.2. Series solutions at regular singular points. The method

of Frobenius 216

§6.3. Asymptotic Behavior near Irregular Singular Points 223

6.3.1. Formal asymptotic series 223

6.3.2. Existence of true solutions described by the formal

asymptotic series. The Stokes phenomenon 229

6.3.3. Another approach to the existence of true solutions and

the Stokes phenomenon. Borel summation 246

§6.4. Notes and References 251

Chapter 7. Introduction to Asymptotics of Solutions of Ordinary

Differential Equations with Respect to Parameters 253

§7.1. Regular Perturbation Problems 254

7.1.1. Formal power series expansions 255

7.1.2. Solving for yn(x). Variation of parameters 256

7.1.3. Justification of the formal expansion 260

§7.2. Singular Asymptotics 263

7.2.1. The WKB method 263

7.2.2. The special case of an asymptotic power series for

f(x; λ) 268

7.2.3. Turning points 277

Contents xi

7.2.4. Problems with more than one turning point. The Bohr-

Sommerfeld quantization rule 300

7.2.5. Uniform asymptotics near turning points. Langer

transformations 304

§7.3. Notes and References 310

Chapter 8. Asymptotics of Linear Boundary-value Problems 311

§8.1. Asymptotic Existence of Solutions 312

8.1.1. Case I: a(x) = 0 on [α, β] and is positive but

sufficiently small 314

8.1.2. Case II: b(x) a(x)/2 0 on [α, β] and is positive 314

§8.2. An Exactly Solvable Boundary-value Problem:

Phenomenology of Boundary Layers 315

§8.3. Outer Asymptotics 318

§8.4. Rescaling and Inner Asymptotics for Boundary Layers and

Internal Layers 321

§8.5. Matching of Asymptotic Expansions, Intermediate

Variables, and Uniformly Valid Asymptotics 325

§8.6. Examples 328

§8.7. Proving the Validity of Uniform Approximations 342

§8.8. The Method of Multiple Scales 350

§8.9. Notes and References 353

Chapter 9. Asymptotics of Oscillatory Phenomena 355

§9.1. Perturbation Theory in Linear Algebra and Eigenvalue

Problems 356

9.1.1. Nondegenerate theory 357

9.1.2. Degenerate theory 362

9.1.3. More on solvability conditions. Inner products and

adjoints 365

§9.2. Periodic Boundary Conditions and Mathieu’s Equation 368

9.2.1. Floquet theory 368

9.2.2. Periodic and antiperiodic solutions. Formal asymptotics 371

9.2.3. Justification of the expansions 377

§9.3. Weakly Nonlinear Oscillations 382

9.3.1. Periodic solutions near equilibrium 383

9.3.2. A perturbative approach to weak cubic nonlinearity.

Secular terms 384

9.3.3. Removal of secular terms. Strained coordinates and the

Poincar´e-Lindstedt method 388

xii Contents

9.3.4. The method of multiple scales 391

9.3.5. Justification of the expansions 397

§9.4. Notes and References 400

Chapter 10. Weakly Nonlinear Waves 401

§10.1. Derivation of Universal Partial Differential Equations

Using the Method of Multiple Scales 401

10.1.1. Modulated wavetrains with dispersion and nonlinear

effects. The cubic nonlinear Schr¨odinger equation 402

10.1.2. Spontaneous excitation of a mean flow 410

10.1.3. Multiple wave resonances 417

10.1.4. Long wave asymptotics. The Boussinesq equation and

the Korteweg-de Vries equation 423

§10.2. Waves in Molecular Chains 425

10.2.1. The Fermi-Pasta-Ulam model 426

10.2.2. Derivation of the cubic nonlinear Schr¨odinger equation 427

10.2.3. Derivation of the Boussinesq and Kortewegde

Vries equations 432

§10.3. Water Waves 433

10.3.1. Derivation of the cubic nonlinear Schr¨odinger equation 436

10.3.2. Derivation of the Korteweg-de Vries equation 444

§10.4. Notes and References 447

Appendix: Fundamental Inequalities 451

Triangle Inequalities 451

Minkowski Inequalities 452

H¨older Inequalities 452

Bibliography 453

Index of Names 455

Subject Index 457

Publish Book: 
Modify Date: 
Sunday, December 31, 2006

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