Introduction to perturbation methods

✍ Scribed by Mark H Holmes

• Publisher: Springer
• Year: 2013
• Tongue: English
• Leaves: 451
• Series: Texts in applied mathematics, 20
• Edition: 2ed.
• Category: Library

✦ Synopsis

This introductory graduate text is based on a graduate course the author has taught repeatedly over the last twenty or so years to students in applied mathematics, engineering sciences, and physics. Each chapter begins with an introductory development involving ordinary differential equations, and goes on to cover more advanced topics such as systems and partial differential equations. Moreover, it also contains material arising from current research interest, including homogenisation, slender body theory, symbolic computing, and discrete equations. Many of the excellent exercises are derived from problems of up-to-date research and are drawn from a wide range of application areas. For this new edition every section has been updated throughout, many only in minor ways, while others have been completely rewritten. New material has also been added. This includes approximations for weakly coupled oscillators, analysis of problems that involve transcendentally small terms, an expanded discussion of Kummer functions, and metastability. Two appendices have been added, one on solving difference equations and another on delay equations. Additional exercises have been included throughout. Review of first edition:"Those familiar with earlier expositions of singular perturbations for ordinary and partial differential equations will find many traditional gems freshly presented, as well as many new topics. Much of the excitement lies in the examples and the more than 250 exercises, which are guaranteed to provoke and challenge readers and learners with various backgrounds and levels of expertise."(SIAM Review, 1996 ) Read more... Introduction to Asymptotic Approximations -- Matched Asymptotic Expansions -- Multiple Scales -- The WKB and Related Methods -- The Method of Homogenization -- Introduction to Bifurcation and Stability. Preface -- Preface to Second Edition -- Introduction to Asymptotic Approximations -- Matched Asymptotic Expansions -- Multiple Scales -- The WKB and Related Methods -- The Method of Homogenization- Introduction to Bifurcation and Stability -- References -- Index

✦ Table of Contents

Cover......Page 1
Introduction to Perturbation
Methods......Page 4
Preface......Page 8
Preface to the Second Edition......Page 12
Contents......Page 14
1.1 Introduction......Page 20
1.2 Taylor's Theorem and l'Hospital's Rule......Page 22
1.3 Order Symbols......Page 23
1.4 Asymptotic Approximations......Page 26
1.4.1 Asymptotic Expansions......Page 28
1.4.2 Accuracy Versus Convergence of an Asymptotic Series......Page 32
1.4.3 Manipulating Asymptotic Expansions......Page 34
1.5 Asymptotic Solution of Algebraic and Transcendental Equations......Page 41
1.6 Introduction to the Asymptotic Solution of Differential Equations......Page 52
1.7 Uniformity......Page 66
1.8 Symbolic Computing......Page 73
2.1 Introduction......Page 76
2.2 Introductory Example......Page 77
2.2.1 Step 1: Outer Solution......Page 78
2.2.2 Step 2: Boundary Layer......Page 79
2.2.3 Step 3: Matching......Page 81
2.2.5 Matching Revisited......Page 82
2.2.6 Second Term......Page 86
2.2.7 Discussion......Page 88
2.3.2 Steps 2 and 3: Boundary Layers and Matching......Page 93
2.3.3 Step 4: Composite Expansion......Page 95
2.4 Transcendentally Small Terms......Page 105
2.5.2 Step 1.5: Locating the Layer......Page 112
2.5.3 Steps 2 and 3: Interior Layer and Matching......Page 114
2.5.4 Step 3.5: Missing Equation......Page 115
2.5.5 Step 4: Composite Expansion......Page 116
2.5.6 Kummer Functions......Page 117
2.6 Corner Layers......Page 125
2.6.2 Step 2: Corner Layer......Page 126
2.6.4 Step 4: Composite Expansion......Page 128
2.7.1 Elliptic Problem......Page 133
2.7.2 Outer Expansion......Page 135
2.7.3 Boundary-Layer Expansion......Page 137
2.7.4 Composite Expansion......Page 139
2.7.5 Parabolic Boundary Layer......Page 140
2.7.6 Parabolic Problem......Page 141
2.7.7 Outer Expansion......Page 142
2.7.8 Inner Expansion......Page 143
2.8 Difference Equations......Page 150
2.8.2 Boundary-Layer Approximation......Page 151
2.8.3 Numerical Solution of Differential Equations......Page 154
3.1 Introduction......Page 157
3.2.1 Regular Expansion......Page 158
3.2.2 Multiple-Scale Expansion......Page 159
3.2.3 Labor-Saving Observations......Page 162
3.2.4 Discussion......Page 163
3.3 Introductory Example (continued)......Page 170
3.3.1 Three Time Scales......Page 171
3.3.4 Uniqueness and Minimum Error......Page 173
3.4 Forced Motion Near Resonance......Page 176
3.5 Weakly Coupled Oscillators......Page 186
3.6 Slowly Varying Coefficients......Page 194
3.7 Boundary Layers......Page 201
3.8 Introduction to Partial Differential Equations......Page 202
3.9 Linear Wave Propagation......Page 207
3.10.1 Nonlinear Wave Equation......Page 212
3.10.2 Wave–Wave Interactions......Page 215
3.10.3 Nonlinear Diffusion......Page 217
3.10.3.1 Example: Fisher's Equation......Page 221
3.11.1 Weakly Nonlinear Difference Equation......Page 227
3.11.2 Chain of Oscillators......Page 230
3.11.2.1 Example: Exact Solution......Page 231
3.11.2.2 Example: Plane Wave Solution......Page 232
4.1 Introduction......Page 240
4.2 Introductory Example......Page 241
4.2.1 Second Term of Expansion......Page 244
4.2.2 General Discussion......Page 246
4.3.1 The Case Where q'(xt)>0......Page 253
4.3.1.1 Solution in Transition Layer......Page 254
4.3.1.3 Matching for x > x t......Page 256
4.3.1.5 Summary......Page 257
4.3.2 The Case Where q'(xt)<0......Page 259
4.3.3 Multiple Turning Points......Page 260
4.3.4 Uniform Approximation......Page 261
4.4 Wave Propagation and Energy Methods......Page 267
4.4.1 Energy Methods......Page 269
4.5 Wave Propagation and Slender-Body Approximations......Page 273
4.5.1 Solution in Transition Region......Page 276
4.5.2 Matching......Page 277
4.6 Ray Methods......Page 281
4.6.1 WKB Expansion......Page 283
4.6.2 Surfaces and Wave Fronts......Page 284
4.6.3 Solution of Eikonal Equation......Page 285
4.6.4 Solution of Transport Equation......Page 286
4.6.5 Ray Equations......Page 287
4.6.6 Summary......Page 288
4.7 Parabolic Approximations......Page 298
4.7.1 Heuristic Derivation......Page 299
4.7.2 Multiple-Scale Derivation......Page 300
4.8 Discrete WKB Method......Page 303
4.8.1 Turning Points......Page 306
5.2 Introductory Example......Page 314
5.2.2 Summary......Page 322
5.3.1 Implications of Periodicity......Page 326
5.3.2 Homogenization Procedure......Page 328
5.4 Porous Flow......Page 333
5.4.1 Reduction Using Homogenization......Page 334
5.4.2 Averaging......Page 336
5.4.3 Homogenized Problem......Page 337
6.1 Introduction......Page 342
6.2 Introductory Example......Page 343
6.3 Analysis of a Bifurcation Point......Page 344
6.3.1 Lyapunov–Schmidt Method......Page 346
6.3.2 Linearized Stability......Page 348
6.3.3 Example: Delay Equation......Page 351
6.4 Quasi-Steady States and Relaxation......Page 358
6.4.2 Initial Layer Expansion......Page 360
6.4.3 Corner-Layer Expansion......Page 361
6.4.4 Interior-Layer Expansion......Page 362
6.5 Bifurcation of Periodic Solutions......Page 368
6.6.1 Linearized Stability Analysis......Page 374
6.6.2 Limit Cycles......Page 379
6.7 Weakly Coupled Nonlinear Oscillators......Page 386
6.8 An Example Involving a Nonlinear Partial Differential Equation......Page 393
6.8.1 Steady State Solutions......Page 394
6.8.2 Linearized Stability Analysis......Page 396
6.8.3 Stability of Zero Solution......Page 397
6.8.4 Stability of the Branches that Bifurcatefrom the Zero Solution......Page 398
6.9 Metastability......Page 403
A.2 Two Variables......Page 410
A.4 Useful Examples for x Near Zero......Page 411
A.6 Trig Functions......Page 412
A.8 Hyperbolic Functions......Page 413
B.1.2 General Solution......Page 414
B.1.4 Asymptotic Approximations......Page 415
B.2.1 Differential Equation......Page 416
B.2.2 General Solution......Page 417
B.2.5 Special Cases......Page 418
B.2.7 Asymptotic Approximations......Page 419
B.2.8 Related Special Functions......Page 420
B.3.3 Asymptotic Approximations......Page 421
C.2 Watson's Lemma......Page 423
C.3 Laplace's Approximation......Page 424
C.4 Stationary Phase Approximation......Page 425
Appendix D Second-Order Difference Equations
......Page 427
D.1 Initial-Value Problems......Page 428
D.2 Boundary-Value Problems......Page 429
E.1 Differential Delay Equations......Page 431
E.2 Integrodifferential Delay Equations......Page 432
E.2.1 Basis Function Approach......Page 433
E.2.2 Differential Equation Approach......Page 434
References......Page 436
Index......Page 448