Classical methods in ordinary differential equations: with applications to boundary value problems
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Mathematical Society
2012
|
| Series: | Graduate Studies in Mathematics
129 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVII, 373 Seiten graph. Darst. |
| ISBN: | 9780821846940 |
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MARC
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| 245 | 1 | 0 | |a Classical methods in ordinary differential equations |b with applications to boundary value problems |c Stuart P. Hastings, J. Bryce McLeod |
| 264 | 1 | |a Providence, RI |b American Mathematical Society |c 2012 | |
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Record in the Search Index
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| adam_text | Titel: Classical methods in ordinary differential equations
Autor: Hastings, Stuart P
Jahr: 2012
Contents
Preface xiii
Chapter 1. Introduction 1
§1.1. What are classical methods? 1
§1.2. Exercises 5
Chapter 2. An introduction to shooting methods 7
§2.1. Introduction 7
§2.2. A first order example 8
§2.3. Some second order examples 13
§2.4. Heteroclinic orbits and the FitzHugh-Nagumo equations 17
§2.5. Shooting when there are oscillations: A third order problem 27
§2.6. Boundedness on (-oo, oo) and two-parameter shooting 30
§2.7. Wazewski s principle, Conley index, and an n-dimensional
lemma 33
§2.8. Exercises 34
Chapter 3. Some boundary value problems for the Painleve
transcendents 37
§3.1. Introduction 37
§3.2. A boundary value problem for Painleve I 38
§3.3. Painleve II-shooting from infinity 44
§3.4. Some interesting consequences 52
§3.5. Exercises 53
vii
viii Contents
Chapter 4. Periodic Solutions of a higher order System 55
§4.1. Introduction, Hopf bifurcation approach 55
§4.2. A global approach via the Brouwer fixed point theorem 57
§4.3. Subsequent developments 61
§4.4. Exercises 62
Chapter 5. A linear example 63
§5.1. Statement of the problem and a basic lemma 63
§5.2. Uniqueness 65
§5.3. Existence using Schauder s fixed point theorem 66
§5.4. Existence using a continuation method 69
§5.5. Existence using linear algebra and finite dimensional
continuation 73
§5.6. A fourth proof 76
§5.7. Exercises 76
Chapter 6. Homoclinic orbits of the FitzHugh-Nagumo equations 77
§6.1. Introduction 77
§6.2. Existence of two bounded Solutions 81
§6.3. Existence of homoclinic orbits using geometric
perturbation theory 83
§6.4. Existence of homoclinic orbits by shooting 92
§6.5. Advantages of the two methods 99
§6.6. Exercises 101
Chapter 7. Singular perturbation problems-rigorous matching 103
§7.1. Introduction to the method of matched asymptotic
expansions 103
§7.2. A problem of Kaplun and Lagerstrom 109
§7.3. A geometric approach 116
§7.4. A classical approach 120
§7.5. The case n = 3 126
§7.6. The case n = 2 128
§7.7. A second application of the method 131
§7.8. A brief discussion of blow-up in two dimensions 137
§7.9. Exercises 139
Contents ix
Chapter 8. Asymptotics beyond all orders 141
§8.1. Introduction 141
§8.2. Proof of nonexistence 144
§8.3. Exercises 150
Chapter 9. Some Solutions of the Falkner-Skan equation 151
§9.1. Introduction 151
§9.2. Periodic Solutions 153
§9.3. Further periodic and other oscillatory Solutions 158
§9.4. Exercises 160
Chapter 10. Poiseuille flow: Perturbation and decay 163
§10.1. Introduction 163
§10.2. Solutions for small data 164
§10.3. Some details 166
§10.4. A classical eigenvalue approach 169
§10.5. On the spectrum of D^j^ for large R 171
§10.6. Exercises 176
Chapter 11. Bending of a tapered rod; variational methods and
shooting 177
§11.1. Introduction 177
§11.2. A calculus of variations approach in Hubert Space 180
§11.3. Existence by shooting for p 2 187
§11.4. Proof using Nehari s method 195
§11.5. More about the case p = 2 197
§11.6. Exercises 198
Chapter 12. Uniqueness and multiplicity 199
§12.1. Introduction 199
§12.2. Uniqueness for a third order problem 203
§12.3. A problem with exactly two Solutions 205
§12.4. A problem with exactly three Solutions 210
§12.5. The Gelfand and perturbed Gelfand equations in three
dimensions 217
§12.6. Uniqueness of the ground State for Au - u + u3 = 0 219
§12.7. Exercises 223
x Contents
Chapter 13. Shooting with more parameters 225
§13.1. A problem from the theory of compressible flow 225
§13.2. A result of Y.-H. Wan 231
§13.3. Exercise 232
§13.4. Appendix: Proof of Wan s theorem 232
Chapter 14. Some problems of A. C. Lazer 237
§14.1. Introduction 237
§14.2. First Lazer-Leach problem 239
§14.3. The pde result of Landesman and Lazer 248
§14.4. Second Lazer-Leach problem 250
§14.5. Second Landesman-Lazer problem 252
§14.6. A problem of Littlewood, and the Moser twist technique 255
§14.7. Exercises 256
Chapter 15. Chaotic motion of a pendulum 257
§15.1. Introduction 257
§15.2. Dynamical Systems 258
§15.3. Melnikov s method 265
§15.4. Application to a forced pendulum 271
§15.5. Proof of Theorem 15.3 when 6 = 0 274
§15.6. Damped pendulum with nonperiodic forcing 277
§15.7. Final remarks 284
§15.8. Exercises 286
Chapter 16. Layers and spikes in reaction-diffusion equations, I 289
§16.1. Introduction 289
§16.2. A model of shallow water sloshing 291
§16.3. Proofs 293
§16.4. Complicated Solutions ( chaos ) 297
§16.5. Other approaches 299
§16.6. Exercises 300
Chapter 17. Uniform expansions for a class of second order
Problems 301
§17.1. Introduction 301
§17.2. Motivation 302
Contents xi
§17.3. Asymptotic expansion 304
§17.4. Exercise 313
Chapter 18. Layers and spikes in reaction-diffusion equations, II 315
§18.1. A basic existence result 316
§18.2. Variational approach to layers 317
§18.3. Three different existence proofs for a single layer in a
simple case 318
§18.4. Uniqueness and stability of a Single layer 327
§18.5. Further stable and unstable Solutions, including multiple
layers 332
§18.6. Single and multiple spikes 340
§18.7. A different type of result for the layer model 342
§18.8. Exercises 343
Chapter 19. Three unsolved problems 345
§19.1. Homoclinic orbit for the equation of a Suspension bridge 345
§19.2. The nonlinear Schrödinger equation 346
§19.3. Uniqueness of radial Solutions for an elliptic problem 346
§19.4. Comments on the Suspension bridge problem 346
§19.5. Comments on the nonlinear Schrödinger equation 347
§19.6. Comments on the elliptic problem and a new existence
proof 349
§19.7. Exercises 355
Bibliography 357
Index 371
|
| any_adam_object | 1 |
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| dewey-raw | 515/.352 |
| dewey-search | 515/.352 |
| dewey-sort | 3515 3352 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| spelling | Hastings, Stuart P. 1937- Verfasser (DE-588)1020118695 aut Classical methods in ordinary differential equations with applications to boundary value problems Stuart P. Hastings, J. Bryce McLeod Providence, RI American Mathematical Society 2012 XVII, 373 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate Studies in Mathematics 129 Includes bibliographical references and index Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s DE-604 McLeod, John Bryce 1929-2014 Verfasser (DE-588)172244951 aut Graduate Studies in Mathematics 129 (DE-604)BV009739289 129 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024845400&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hastings, Stuart P. 1937- McLeod, John Bryce 1929-2014 Classical methods in ordinary differential equations with applications to boundary value problems Graduate Studies in Mathematics Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4020929-5 |
| title | Classical methods in ordinary differential equations with applications to boundary value problems |
| title_auth | Classical methods in ordinary differential equations with applications to boundary value problems |
| title_exact_search | Classical methods in ordinary differential equations with applications to boundary value problems |
| title_full | Classical methods in ordinary differential equations with applications to boundary value problems Stuart P. Hastings, J. Bryce McLeod |
| title_fullStr | Classical methods in ordinary differential equations with applications to boundary value problems Stuart P. Hastings, J. Bryce McLeod |
| title_full_unstemmed | Classical methods in ordinary differential equations with applications to boundary value problems Stuart P. Hastings, J. Bryce McLeod |
| title_short | Classical methods in ordinary differential equations |
| title_sort | classical methods in ordinary differential equations with applications to boundary value problems |
| title_sub | with applications to boundary value problems |
| topic | Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Gewöhnliche Differentialgleichung |
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