Singular perturbation in the physical sciences:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2015]
|
| Series: | Graduate studies in mathematics
volume 167 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 326 Seiten Illustrationen, Diagramme |
| ISBN: | 9781470425555 |
Staff View
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| 650 | 4 | |a Approximations and expansions | |
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Record in the Search Index
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| adam_text | Titel: Singular perturbation in the physical sciences
Autor: Neu, John C
Jahr: 2015
Contents Acknowledgments ix Introduction xi Chapter 1. What is a singular perturbation? 1 Prototypical examples 1 Singularly perturbed polynomial equations 1 Radiation reaction 4 Problem 1.1: Bad truncations 7 Problem 1.2: Harmonic oscillator with memory, and even worse truncations 9 Convection-diffusion boundary layer 11 Problem 1.3: A simple boundary layer 16 Problem 1.4: Pileup near x = 0 17 Modulated oscillations 19 Problem 1.5: Secular terms 23 Problem 1.6: Approach to limit cycle 25 Problem 1.7: Adiabatic invariant for particle in a box 28 Guide to bibliography 29 Chapter 2. Asymptotic expansions 31 Problem 2.1: Uniqueness 33 A divergent but asymptotic series 34 Problem 2.2: Divergent outer expansion 35 Problem 2.3: Another outrageous example 37 Asymptotic expansions of integrals — the usual suspects 38 Problem 2.4: Simple endpoint examples 42 Problem 2.5: Stirling approximation to n! 43 v
VI Contents Problem 2.6: Endpoint and minimum both contribute 43 Problem 2.7: Central limit theorem 44 Steepest descent method 48 Chasing the waves with velocity v 0 51 No waves for v 0 53 Problem 2.8: Steepest descent asymptotics 54 A primer on linear waves 57 Problem 2.9: Amplitude transport 60 Problem 2.10: How far was that meteor? 61 Problem 2.11: Wave asymptotics in non-uniform medium 62 A hard logarithmic expansion 66 Problem 2.12: Logarithmic expansion 69 Guide to bibliography 71 Chapter 3. Matched asymptotic expansions 73 Problem 3.1: Physical scaling analysis of boundary layer thickness 74 Problem 3.2: Higher-order matching 81 Problem 3.3: Absorbing boundary condition 83 Matched asymptotic expansions in practice 84 Problem 3.4: Derivative layer 85 Corner layers and internal layers 88 Problem 3.5: Phase diagram 93 Problem 3.6: Internal derivative layer 97 Problem 3.7: Where does the kink go? 101 Guide to bibliography 103 Chapter 4. Matched asymptotic expansions in PDE’s 105 Moving internal layers 105 Chapman-Enskog asymptotics 109 Problem 4.1: Relaxation of kink position 113 Problem 4.2: Hamilton -Jacobi equation from front motion 114 Problem 4.3: Chapman-Enskog asymptotics 120 Projected Lagrangian 123 Problem 4.4: Circular fronts in nonlinear wave equation 127 Problem 4.5: Solitary wave dynamics in two dimensions 131 Problem 4.6: Solitary wave diffraction 135 Singularly perturbed eigenvalue problem 138 Homogenization of swiss cheese 142 Problem 4.7: Neumann boundary conditions and effective dipoles 144 Problem 4.8: Two dimensions 147 Guide to bibliography 149
Contents vii Chapter 5. Prandtl boundary layer theory 151 Stream function and vorticity 155 Preliminary non-dimensionalization 157 Outer expansion and “dry water” 157 Inner expansion 158 Problem 5.1: Vector calculus of boundary layer coordinates 161 Leading order matching and a first integral 163 Problem 5.2: The body surface is a source of vorticity 164 Problem 5.3: Downstream evolution 166 Displacement thickness 167 Solutions based on scaling symmetry 168 Blasius flow over flat plate 171 Nonzero wedge angles (m ^ 0) 172 Precursor of boundary layer separation 173 Problem 5.4: Wedge flows with source 174 Problem 5.5: Mixing by vortex 177 Guide to bibliography 180 Chapter 6. Modulated oscillations 183 Physical flavors of modulated oscillations 184 Problem 6.1: Beats 185 Problem 6.2: The beat goes on 186 Problem 6.3: Wave packets as beats in spacetime 188 Problem 6.4: Adiabatic invariant of harmonic oscillator 189 Problem 6.5: Passage through resonance for harmonic oscillator 192 Problem 6.6: Internal resonance between waves on a ring 193 Method of two scales 197 Problem 6.7: Nonlinear parametric resonance 203 Problem 6.8: Forced van der Pol ODE 206 Problem 6.9: Inverted pendulum 214 Strongly nonlinear oscillations and action 216 Problem 6.10: Energy, action and frequency 220 Problem 6.11: Hamiltonian analysis of the adiabatic invariant 222 Problem 6.12: Poincaré analysis of nonlinear oscillations 223 A primer on nonlinear waves 227 Modulation Lagrangian 230 Problem 6.13: Nonlinear geometric attenuation 230 Problem 6.14: Modulational instability 233 A primer on homogenization theory 238 Problem 6.15: Direct homogenization 242 Guide to bibliography 245 Chapter 7. Modulation theory by transforming variables 247
Contents viii Transformations in classical mechanics 247 Problem 7.1: Geometry of action-angle variables 250 Problem 7.2: Stokes expansion for quadratically nonlinear oscillator 254 Problem 7.3: Frequency-action relation 257 Problem 7.4: Follow the bouncing ball 258 Near-identity transformations 262 Problem 7.5: van der Pol ODE by near-identity transformations 267 Problem 7.6: Subtle balance between positive and negative damping 270 Problem 7.7: Adiabatic invariants again 273 Dissipative perturbations of the Kepler problem 275 Modulation theory of damped orbits 279 Guide to bibliography 285 Chapter 8. Nonlinear resonance 287 Problem 8.1: Modulation theory of resonance 290 A prototype example 292 What resonance looks like 295 Problem 8.2: Resonance of the bouncing ball 299 Problem 8.3: Resonance by rebounds off a vibrating wall 302 Generalized resonance 307 Energy beats 308 Modulation theory of generalized resonance 309 Problem 8.4: Modulation theory for generalized resonance 312 Thickness of the resonance annulus 313 Asymptotic isolation of resonances 315 Guide to bibliography 316 Bibliography 319 Index 321
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| any_adam_object | 1 |
| author | Neu, John C. 1952- |
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| isbn | 9781470425555 |
| language | English |
| lccn | 015023206 |
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| physical | xiv, 326 Seiten Illustrationen, Diagramme |
| publishDate | 2015 |
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| spelling | Neu, John C. 1952- Verfasser (DE-588)14059647X aut Singular perturbation in the physical sciences John C. Neu Providence, Rhode Island American Mathematical Society [2015] © 2015 xiv, 326 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics volume 167 Ordinary differential equations msc Partial differential equations msc Dynamical systems and ergodic theory msc Approximations and expansions msc Fluid mechanics msc Mathematics education msc Singular perturbations (Mathematics) Asymptotic expansions Ordinary differential equations Partial differential equations Dynamical systems and ergodic theory Approximations and expansions Fluid mechanics Mathematics education Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Asymptotische Entwicklung (DE-588)4112609-9 gnd rswk-swf Singuläre Störung (DE-588)4055100-3 gnd rswk-swf Singuläre Störung (DE-588)4055100-3 s Mathematische Physik (DE-588)4037952-8 s DE-604 Asymptotische Entwicklung (DE-588)4112609-9 s Erscheint auch als Online-Ausgabe 978-1-4704-2733-7 Graduate studies in mathematics volume 167 (DE-604)BV009739289 167 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028788611&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Neu, John C. 1952- Singular perturbation in the physical sciences Graduate studies in mathematics Ordinary differential equations msc Partial differential equations msc Dynamical systems and ergodic theory msc Approximations and expansions msc Fluid mechanics msc Mathematics education msc Singular perturbations (Mathematics) Asymptotic expansions Ordinary differential equations Partial differential equations Dynamical systems and ergodic theory Approximations and expansions Fluid mechanics Mathematics education Mathematische Physik (DE-588)4037952-8 gnd Asymptotische Entwicklung (DE-588)4112609-9 gnd Singuläre Störung (DE-588)4055100-3 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4112609-9 (DE-588)4055100-3 |
| title | Singular perturbation in the physical sciences |
| title_auth | Singular perturbation in the physical sciences |
| title_exact_search | Singular perturbation in the physical sciences |
| title_full | Singular perturbation in the physical sciences John C. Neu |
| title_fullStr | Singular perturbation in the physical sciences John C. Neu |
| title_full_unstemmed | Singular perturbation in the physical sciences John C. Neu |
| title_short | Singular perturbation in the physical sciences |
| title_sort | singular perturbation in the physical sciences |
| topic | Ordinary differential equations msc Partial differential equations msc Dynamical systems and ergodic theory msc Approximations and expansions msc Fluid mechanics msc Mathematics education msc Singular perturbations (Mathematics) Asymptotic expansions Ordinary differential equations Partial differential equations Dynamical systems and ergodic theory Approximations and expansions Fluid mechanics Mathematics education Mathematische Physik (DE-588)4037952-8 gnd Asymptotische Entwicklung (DE-588)4112609-9 gnd Singuläre Störung (DE-588)4055100-3 gnd |
| topic_facet | Ordinary differential equations Partial differential equations Dynamical systems and ergodic theory Approximations and expansions Fluid mechanics Mathematics education Singular perturbations (Mathematics) Asymptotic expansions Mathematische Physik Asymptotische Entwicklung Singuläre Störung |
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