This book gives a clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. These methods allow one to analyze physics and engineering problems that may not be solvable in closed form and for which brute-force numerical methods may not converge to useful solutions. The presentation is aimed at teaching the insights that are most useful in approaching new problems; it avoids special methods and tricks that work only for particular problems, such as the traditional transcendental functions.
Intended for graduate students and advanced undergraduates, the book assumes only a limited familiarity with differential equations and complex variables.
The presentation begins with a review of differential and difference equations; develops local asymptotic methods for differential and difference equations; explains perturbation and summation theory; and concludes with a an exposition of global asymptotic methods, including boundary-layer theory, WKB theory, and multiple-scale analysis. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach the reader how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions; over 600 problems, of varying levels of difficulty; and an appendix summarizing the properties of special functions.
Author(s): Carl M. Bender, Steven A. Orszag
Publisher: Springer
Year: 1999
Language: English
Pages: 606
City: New York, NY
PART I: FUNDAMENTALS 1
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 59
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 Linear Equations 66
3.3 Local Series Expansions About Regular Singular Points of Homogeneous 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-> oo: The Stirling Series 218
5.5 Local Behavior Near an Irregular Singular Point at Infinity: 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 III: PERTURBATION METHODS 317
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 Padé Summation 383
8.4 Continued Fractions and Padé Approximants 395
8.5 Convergence of Padé Approximants 400
8.6 Padé Sequences for Stieltjes Functions 405
Problems for Chapter 8 410
PART IV: GLOBAL ANALYSIS 417
9 Boundary Layer Theory 419
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 Asymptotic Approximations: WKB Solution of Inhomogeneous 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
