Introduction to Perturbation Methods

Cover......Page 3
Preface......Page 7
Preface to the Second Edition......Page 11
Contents......Page 13
1.1 Introduction......Page 19
1.2 Taylor's Theorem and l'Hospital's Rule......Page 21
1.3 Order Symbols......Page 22
1.4 Asymptotic Approximations......Page 25
1.4.1 Asymptotic Expansions......Page 27
1.4.2 Accuracy Versus Convergence of an Asymptotic Series......Page 31
1.4.3 Manipulating Asymptotic Expansions......Page 33
1.5 Asymptotic Solution of Algebraic and Transcendental Equations......Page 40
1.6 Introduction to the Asymptotic Solution of Differential Equations......Page 51
1.7 Uniformity......Page 65
1.8 Symbolic Computing......Page 72
2.1 Introduction......Page 75
2.2 Introductory Example......Page 76
2.2.1 Step 1: Outer Solution......Page 77
2.2.2 Step 2: Boundary Layer......Page 78
2.2.3 Step 3: Matching......Page 80
2.2.5 Matching Revisited......Page 81
2.2.6 Second Term......Page 85
2.2.7 Discussion......Page 87
2.3.2 Steps 2 and 3: Boundary Layers and Matching......Page 92
2.3.3 Step 4: Composite Expansion......Page 94
2.4 Transcendentally Small Terms......Page 104
2.5.2 Step 1.5: Locating the Layer......Page 111
2.5.3 Steps 2 and 3: Interior Layer and Matching......Page 113
2.5.4 Step 3.5: Missing Equation......Page 114
2.5.5 Step 4: Composite Expansion......Page 115
2.5.6 Kummer Functions......Page 116
2.6 Corner Layers......Page 124
2.6.2 Step 2: Corner Layer......Page 125
2.6.4 Step 4: Composite Expansion......Page 127
2.7.1 Elliptic Problem......Page 132
2.7.2 Outer Expansion......Page 134
2.7.3 Boundary-Layer Expansion......Page 136
2.7.4 Composite Expansion......Page 138
2.7.5 Parabolic Boundary Layer......Page 139
2.7.6 Parabolic Problem......Page 140
2.7.7 Outer Expansion......Page 141
2.7.8 Inner Expansion......Page 142
2.8 Difference Equations......Page 149
2.8.2 Boundary-Layer Approximation......Page 150
2.8.3 Numerical Solution of Differential Equations......Page 153
3.1 Introduction......Page 156
3.2.1 Regular Expansion......Page 157
3.2.2 Multiple-Scale Expansion......Page 158
3.2.3 Labor-Saving Observations......Page 161
3.2.4 Discussion......Page 162
3.3 Introductory Example (continued)......Page 169
3.3.1 Three Time Scales......Page 170
3.3.4 Uniqueness and Minimum Error......Page 172
3.4 Forced Motion Near Resonance......Page 175
3.5 Weakly Coupled Oscillators......Page 185
3.6 Slowly Varying Coefficients......Page 193
3.7 Boundary Layers......Page 200
3.8 Introduction to Partial Differential Equations......Page 201
3.9 Linear Wave Propagation......Page 206
3.10.1 Nonlinear Wave Equation......Page 211
3.10.2 Wave–Wave Interactions......Page 214
3.10.3 Nonlinear Diffusion......Page 216
3.10.3.1 Example: Fisher's Equation......Page 220
3.11.1 Weakly Nonlinear Difference Equation......Page 226
3.11.2 Chain of Oscillators......Page 229
3.11.2.1 Example: Exact Solution......Page 230
3.11.2.2 Example: Plane Wave Solution......Page 231
4.1 Introduction......Page 239
4.2 Introductory Example......Page 240
4.2.1 Second Term of Expansion......Page 243
4.2.2 General Discussion......Page 245
4.3.1 The Case Where q'(xt)>0......Page 252
4.3.1.1 Solution in Transition Layer......Page 253
4.3.1.3 Matching for x > x  t......Page 255
4.3.1.5 Summary......Page 256
4.3.2 The Case Where q'(xt)<0......Page 258
4.3.3 Multiple Turning Points......Page 259
4.3.4 Uniform Approximation......Page 260
4.4 Wave Propagation and Energy Methods......Page 266
4.4.1 Energy Methods......Page 268
4.5 Wave Propagation and Slender-Body Approximations......Page 272
4.5.1 Solution in Transition Region......Page 275
4.5.2 Matching......Page 276
4.6 Ray Methods......Page 280
4.6.1 WKB Expansion......Page 282
4.6.2 Surfaces and Wave Fronts......Page 283
4.6.3 Solution of Eikonal Equation......Page 284
4.6.4 Solution of Transport Equation......Page 285
4.6.5 Ray Equations......Page 286
4.6.6 Summary......Page 287
4.7 Parabolic Approximations......Page 297
4.7.1 Heuristic Derivation......Page 298
4.7.2 Multiple-Scale Derivation......Page 299
4.8 Discrete WKB Method......Page 302
4.8.1 Turning Points......Page 305
5.2 Introductory Example......Page 313
5.2.2 Summary......Page 321
5.3.1 Implications of Periodicity......Page 325
5.3.2 Homogenization Procedure......Page 327
5.4 Porous Flow......Page 332
5.4.1 Reduction Using Homogenization......Page 333
5.4.2 Averaging......Page 335
5.4.3 Homogenized Problem......Page 336
6.1 Introduction......Page 341
6.2 Introductory Example......Page 342
6.3 Analysis of a Bifurcation Point......Page 343
6.3.1 Lyapunov–Schmidt Method......Page 345
6.3.2 Linearized Stability......Page 347
6.3.3 Example: Delay Equation......Page 350
6.4 Quasi-Steady States and Relaxation......Page 357
6.4.2 Initial Layer Expansion......Page 359
6.4.3 Corner-Layer Expansion......Page 360
6.4.4 Interior-Layer Expansion......Page 361
6.5 Bifurcation of Periodic Solutions......Page 367
6.6.1 Linearized Stability Analysis......Page 373
6.6.2 Limit Cycles......Page 378
6.7 Weakly Coupled Nonlinear Oscillators......Page 385
6.8 An Example Involving a Nonlinear Partial Differential Equation......Page 392
6.8.1 Steady State Solutions......Page 393
6.8.2 Linearized Stability Analysis......Page 395
6.8.3 Stability of Zero Solution......Page 396
6.8.4 Stability of the Branches that Bifurcatefrom the Zero Solution......Page 397
6.9 Metastability......Page 402
A.2 Two Variables......Page 409
A.4 Useful Examples for x Near Zero......Page 410
A.6 Trig Functions......Page 411
A.8 Hyperbolic Functions......Page 412
B.1.2 General Solution......Page 413
B.1.4 Asymptotic Approximations......Page 414
B.2.1 Differential Equation......Page 415
B.2.2 General Solution......Page 416
B.2.5 Special Cases......Page 417
B.2.7 Asymptotic Approximations......Page 418
B.2.8 Related Special Functions......Page 419
B.3.3 Asymptotic Approximations......Page 420
C.2 Watson's Lemma......Page 422
C.3 Laplace's Approximation......Page 423
C.4 Stationary Phase Approximation......Page 424
Appendix D Second-Order Difference Equations
......Page 426
D.1 Initial-Value Problems......Page 427
D.2 Boundary-Value Problems......Page 428
E.1 Differential Delay Equations......Page 430
E.2 Integrodifferential Delay Equations......Page 431
E.2.1 Basis Function Approach......Page 432
E.2.2 Differential Equation Approach......Page 433
References......Page 435
Index......Page 447