《advanced mathematical methods for scientists and engineers P593》

PART IFUNDAMENTALS3

1Ordinary Differential Equations3

1.1 Ordinary Differential Equations3

1.2 Initial-Value and Boundary-Value Problems5

1.3 Theory of Homogeneous Linear Equations7

1.4 Solutions of Homogeneous Linear Equations11

1.5 Inhomogeneous Linear Equations14

1.6 First-Order Nonlinear Differential Equations20

1.7 Higher-Order Nonlinear Differential Equations24

1.8 Eigenvalue Problems27

1.9 Differential Equations in the Complex Plane29

Problems for Chapter 130

2Difference Equations36

2.1 The Calculus of Differences36

2.2 Elementary Difference Equations37

2.3 Homogeneous Linear Difference Equations40

2.4 Inhomogeneous Linear Difference Equations49

2.5 Nonlinear Difference Equations53

Problems for Chapter 253

PART ⅡLOCAL ANALYSIS61

3Approximate Solution of Linear Differential Equations61

3.1 Classification of Singular Points of Homogeneous Linear Equations62

3.2 Local Behavior Near Ordinary Points of Homogeneous LinearEquations66

3.3 Local Series Expansions About Regular Singular Points ofHomogeneous Linear Equations68

3.4 Local Behavior at Irregular Singular Points of Homogeneous Linear Equations76

3.5 Irregular Singular Point at Infinity88

3.6 Local Analysis of Inhomogeneous Linear Equations103

3.7 Asymptotic Relations107

3.8 Asymptotic Series118

Problems for Chapter 3136

4 Approximate Solution of Nonlinear Differential Equations146

4.1Spontaneous Singularities146

4.2 Approximate Solutions of First-Order Nonlinear Differential Equations148

4.3 Approximate Solutions to Higher-Order Nonlinear Differential Equations152

4.4 Nonlinear Autonomous Systems171

4.5 Higher-Order Nonlinear Autonomous Systems185

Problems for Chapter 4196

5Approximate Solution of Difference Equations205

5.1 Introductory Comments205

5.2 Ordinary and Regular Singular Points of Linear Difference Equations206

5.3 Local Behavior Near an Irregular Singular Point at Infinity: Determination of Controlling Factors214

5.4 Asymptotic Behavior of n! as n→∞:The Stirling Series218

5.5 Local Behavior Near an Irregular Singular Point at Intinity:Full Asymptotic Series227

5.6 Local Behavior of Nonlinear Difference Equations233

Problems for Chapter 5240

6Asymptotic Expansion of Integrals247

6.1 Introduction247

6.2 Elementary Examples249

6.3 Integration by Parts252

6.4 Laplace’s Method and Watson’s Lemma261

6.5 Method of Stationary Phase276

6.6 Method of Steepest Descents280

6.7 Asymptotic Evaluation of Sums302

Problems for Chapter 6306

PART ⅢPERTURBATION METHODS319

7Perturbation Series319

7.1 Perturbation Theory319

7.2 Regular and Singular Perturbation Theory324

7.3 Perturbation Methods for Linear Eigenvalue Problems330

7.4 Asymptotic Matching335

7.5 Mathematical Structure of Perturbative Eigenvalue Problems350

Problems for Chapter 7361

8Summation of Series368

8.1 Improvement of Convergence368

8.2 Summation of Divergent Series379

8.3 Pade Summation383

8.4 Continued Fractions and Pade Approximants395

8.5 Convergence of Pade Approximants400

8.6 Pade Sequences for Stieltjes Functions405

Problems for Chapter 8410

PART ⅣGLOBAL ANALYSIS417

9Boundary Layer Theory417

9.1 Introduction to Boundary-Layer Theory419

9.2 Mathematical Structure of Boundary Layers: Inner, Outer, and Intermediate Limits426

9.3 Higher-Order Boundary Layer Theory431

9.4 Distinguished Limits and Boundary Layers of Thickness ≠ ε435

9.5 Miscellaneous Examples of Linear Boundary-Layer Problems446

9.6 Internal Boundary Layers455

9.7 Nonlinear Boundary-Layer Problems463

Problems for Chapter 9479

10WKB Theory484

10.1 The Exponential Approximation for Dissipative and Dispersive Phenomena484

10.2 Conditions for Validity of the WKB Approximation493

10.3 Patched Asvmptotic Approximations: WKB Solution ofInhomogeneous Linear Equations497

10.4 Matched Asymptotic Approximations: Solution of the One-Turning-Point Problem504

10.5 Two-Turning-Point Problems: Eigenvalue Condition519

10.6 Tunneling524

10.7 Brief Discussion of Higher-Order WKB Approximations534

Problems for Chapter 10539

11Multiple-Scale Analysis544

11.1 Resonance and Secular Behavior544

11.2 Multiple-Scale Analysis549

11.3 Examples of Multiple-Scale Analysis551

11.4 The Mathieu Equation and Stability560

Problems for Chapter 11566

Appendix—Useful Formulas569

References577

Index581