Introduction to perturbation methods:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
New York, NY [u.a.]
Springer
2013
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Texts in applied mathematics
20 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 436 S. Ill., graph. Darst. |
| ISBN: | 9781461454762 |
Internformat
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| 100 | 1 | |a Holmes, Mark H. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to perturbation methods |c Mark H. Holmes |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York, NY [u.a.] |b Springer |c 2013 | |
| 300 | |a XVII, 436 S. |b Ill., graph. Darst. | ||
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Datensatz im Suchindex
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| adam_text | Titel: Introduction to perturbation methods
Autor: Holmes, Mark H
Jahr: 2013
Contents
Introduction to Asymptotic Approximations.............. 1
1.1 Introduction........................................... 1
1.2 Taylor s Theorem and l Hospital s Rule.................... 3
1.3 Order Symbols......................................... 4
1.4 Asymptotic Approximations............................. 7
1.4.1 Asymptotic Expansions........................... 9
1.4.2 Accuracy Versus Convergence
of an Asymptotic Series........................... 13
1.4.3 Manipulating Asymptotic Expansions............... 15
1.5 Asymptotic Solution of Algebraic and Transcendental
Equations............................................. 22
1.6 Introduction to the Asymptotic Solution of Differential
Equations............................................. 33
1.7 Uniformity ............................................ 47
1.8 Symbolic Computing.................................... 54
Matched Asymptotic Expansions......................... 57
2.1 Introduction........................................... 57
2.2 Introductory Example................................... 58
2.2.1 Step 1: Outer Solution............................ 59
2.2.2 Step 2: Boundary Layer........................... 60
2.2.3 Step 3: Matching................................. 62
2.2.4 Step 4: Composite Expansion...................... 63
2.2.5 Matching Revisited............................... 63
2.2.6 Second Term..................................... 67
2.2.7 Discussion....................................... 69
2.3 Examples Involving Boundary Layers..................... 74
2.3.1 Step 1: Outer Expansion.......................... 74
2.3.2 Steps 2 and 3: Boundary Layers and Matching....... 74
2.3.3 Step 4: Composite Expansion...................... 76
2.4 Transcendentally Small Terms............................ 86
2.5 Interior Layers......................................... 93
2.5.1 Step 1: Outer Expansion.......................... 93
2.5.2 Step 1.5: Locating the Layer....................... 93
2.5.3 Steps 2 and 3: Interior Layer and Matching.......... 95
2.5.4 Step 3.5: Missing Equation........................ 96
2.5.5 Step 4: Composite Expansion...................... 97
2.5.6 Kummer Functions............................... 98
2.6 Corner Layers..........................................106
2.6.1 Step 1: Outer Expansion..........................107
2.6.2 Step 2: Corner Layer..............................107
2.6.3 Step 3: Matching.................................109
2.6.4 Step 4: Composite Expansion......................109
2.7 Partial Differential Equations............................114
2.7.1 Elliptic Problem..................................114
2.7.2 Outer Expansion.................................116
2.7.3 Boundary-Layer Expansion........................118
2.7.4 Composite Expansion.............................120
2.7.5 Parabolic Boundary Layer.........................121
2.7.6 Parabolic Problem................................122
2.7.7 Outer Expansion.................................123
2.7.8 Inner Expansion..................................124
2.8 Difference Equations....................................131
2.8.1 Outer Expansion.................................132
2.8.2 Boundary-Layer Approximation....................132
2.8.3 Numerical Solution of Differential Equations.........135
Multiple Scales...........................................139
3.1 Introduction...........................................139
3.2 Introductory Example...................................140
3.2.1 Regular Expansion...............................140
3.2.2 Multiple-Scale Expansion..........................141
3.2.3 Labor-Saving Observations........................144
3.2.4 Discussion.......................................145
3.3 Introductory Example (continued)........................152
3.3.1 Three Time Scales................................ 153
3.3.2 Two-Term Expansion............................. 155
3.3.3 Some Comparisons............................... 155
3.3.4 Uniqueness and Minimum Error.................... 155
3.4 Forced Motion Near Resonance........................... 158
3.5 Weakly Coupled Oscillators.............................. 168
3.6 Slowly Varying Coefficients.............................. 176
3.7 Boundary Layers....................................... 183
3.8 Introduction to Partial Differential Equations.............. 184
3.9 Linear Wave Propagation................................ 189
3.10 Nonlinear Waves.......................................194
3.10.1 Nonlinear Wave Equation.........................194
3.10.2 Wave-Wave Interactions ..........................197
3.10.3 Nonlinear Diffusion...............................199
3.11 Difference Equations....................................209
3.11.1 Weakly Nonlinear Difference Equation..............209
3.11.2 Chain of Oscillators ..............................212
The WKB and Related Methods .........................223
4.1 Introduction...........................................223
4.2 Introductory Example...................................224
4.2.1 Second Term of Expansion ........................227
4.2.2 General Discussion...............................229
4.3 Turning Points.........................................236
4.3.1 The Case Where q (xt) 0........................236
4.3.2 The Case Where q {xt) 0........................242
4.3.3 Multiple Turning Points...........................243
4.3.4 Uniform Approximation...........................244
4.4 Wave Propagation and Energy Methods...................250
4.4.1 Energy Methods .................................252
4.5 Wave Propagation and Slender-Body Approximations.......256
4.5.1 Solution in Transition Region......................259
4.5.2 Matching........................................260
4.6 Ray Methods..........................................264
4.6.1 WKB Expansion.................................266
4.6.2 Surfaces and Wave Fronts.........................267
4.6.3 Solution of Eikonal Equation.......................268
4.6.4 Solution of Transport Equation ....................269
4.6.5 Ray Equations...................................270
4.6.6 Summary........................................271
4.7 Parabolic Approximations...............................281
4.7.1 Heuristic Derivation..............................282
4.7.2 Multiple-Scale Derivation..........................283
4.8 Discrete WKB Method..................................286
4.8.1 Turning Points...................................289
The Method of Homogenization..........................297
5.1 Introduction...........................................297
5.2 Introductory Example...................................297
5.2.1 Properties of the Averaging Procedure..............305
5.2.2 Summary........................................305
5.3 Multidimensional Problem: Periodic Substructure...........309
5.3.1 Implications of Periodicity.........................309
5.3.2 Homogenization Procedure........................311
5.4 Porous Flow........................................... 316
5.4.1 Reduction Using Homogenization................... 317
5.4.2 Averaging....................................... 319
5.4.3 Homogenized Problem............................ 320
6 Introduction to Bifurcation and Stability.................325
6.1 Introduction...........................................325
6.2 Introductory Example...................................326
6.3 Analysis of a Bifurcation Point...........................327
6.3.1 Lyapunov-Schmidt Method........................329
6.3.2 Linearized Stability...............................331
6.3.3 Example: Delay Equation .........................334
6.4 Quasi-Steady States and Relaxation......................341
6.4.1 Outer Expansion.................................343
6.4.2 Initial Layer Expansion...........................343
6.4.3 Corner-Layer Expansion...........................344
6.4.4 Interior-Layer Expansion..........................345
6.5 Bifurcation of Periodic Solutions .........................351
6.6 Systems of Ordinary Differential Equations................357
6.6.1 Linearized Stability Analysis.......................357
6.6.2 Limit Cycles.....................................362
6.7 Weakly Coupled Nonlinear Oscillators.....................369
6.8 An Example Involving a Nonlinear Partial Differential
Equation..............................................376
6.8.1 Steady State Solutions............................ 377
6.8.2 Linearized Stability Analysis....................... 379
6.8.3 Stability of Zero Solution.......................... 380
6.8.4 Stability of the Branches that Bifurcate
from the Zero Solution............................381
6.9 Metastability ..........................................386
A Taylor Series.............................................393
A.l Single Variable.........................................393
A.2 Two Variables .........................................393
A.3 Multivariable..........................................394
A.4 Useful Examples for x Near Zero.........................394
A.5 Power Functions........................................395
A.6 Trig Functions.........................................395
A.7 Exponential and Log Functions...........................396
A.8 Hyperbolic Functions...................................396
B Solution and Properties of Transition Layer Equations.... 397
B.l Airy Functions.........................................397
B.l.l Differential Equation..............................397
B.1.2 General Solution.................................397
B.1.3 Particular Values.................................398
B.1.4 Asymptotic Approximations.......................398
B.1.5 Connection with Bessel Functions..................399
B.2 Rummer s Function.....................................399
B.2.1 Differential Equation..............................399
B.2.2 General Solution.................................400
B.2.3 Particular Values.................................401
B.2.4 Useful Formulas..................................401
B.2.5 Special Cases....................................401
B.2.6 Polynomials.....................................402
B.2.7 Asymptotic Approximations.......................402
B.2.8 Related Special Functions.........................403
B.3 Higher-Order Turning Points.............................404
B.3.1 Differential Equation..............................404
B.3.2 General Solution.................................404
B.3.3 Asymptotic Approximations.......................404
C Asymptotic Approximations of Integrals..................407
C.l Introduction...........................................407
C.2 Watson s Lemma.......................................407
C.3 Laplace s Approximation................................408
C.4 Stationary Phase Approximation.........................409
D Second-Order Difference Equations.......................411
D.l Initial-Value Problems ..................................412
D.2 Boundary-Value Problems...............................413
E Delay Equations..........................................415
E.l Differential Delay Equations.............................415
E.2 Integrodifferential Delay Equations.......................416
E.2.1 Basis Function Approach..........................417
E.2.2 Differential Equation Approach....................418
References....................................................421
Index.........................................................433
|
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| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV040670888 |
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| indexdate | 2024-07-10T00:28:45Z |
| institution | BVB |
| isbn | 9781461454762 |
| language | German |
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| series2 | Texts in applied mathematics |
| spelling | Holmes, Mark H. Verfasser aut Introduction to perturbation methods Mark H. Holmes 2. ed. New York, NY [u.a.] Springer 2013 XVII, 436 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 20 Störungstheorie Störungstheorie (DE-588)4128420-3 gnd rswk-swf Störungstheorie (DE-588)4128420-3 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4614-5477-9 Texts in applied mathematics 20 (DE-604)BV002476038 20 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025497457&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Holmes, Mark H. Introduction to perturbation methods Texts in applied mathematics Störungstheorie Störungstheorie (DE-588)4128420-3 gnd |
| subject_GND | (DE-588)4128420-3 |
| title | Introduction to perturbation methods |
| title_auth | Introduction to perturbation methods |
| title_exact_search | Introduction to perturbation methods |
| title_full | Introduction to perturbation methods Mark H. Holmes |
| title_fullStr | Introduction to perturbation methods Mark H. Holmes |
| title_full_unstemmed | Introduction to perturbation methods Mark H. Holmes |
| title_short | Introduction to perturbation methods |
| title_sort | introduction to perturbation methods |
| topic | Störungstheorie Störungstheorie (DE-588)4128420-3 gnd |
| topic_facet | Störungstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025497457&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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