Applied asymptotic analysis:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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Providence, RI
American Mathematical Society
2006
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| Series: | Graduate studies in mathematics
75 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 467 S. graph. Darst. |
| ISBN: | 9780821840788 0821840789 |
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| 245 | 1 | 0 | |a Applied asymptotic analysis |c Peter D. Miller |
| 264 | 1 | |a Providence, RI |b American Mathematical Society |c 2006 | |
| 300 | |a XV, 467 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Graduate studies in mathematics |v 75 | |
| 650 | 4 | |a Asymptotic expansions | |
| 650 | 4 | |a Differential equations |x Asymptotic theory | |
| 650 | 4 | |a Approximation theory | |
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| adam_text | Contents
Preface xiii
Part 1. Fundamentals
Chapter 0. Themes of Asymptotic Analysis 3
§0.1. Theme: Asymptotics, Convergent and Divergent
Asymptotic Series 3
§0.2. Theme: Other Parameters and Nonuniformity 8
0.2.1. First example. Oscillations 8
0.2.2. Second example. Boundary layers 10
§0.3. Theme: Differential Equations 12
§0.4. Theme: Universal Partial Differential Equations and
Canonical Physical Models 13
Chapter 1. The Nature of Asymptotic Approximations 15
§1.1. Asymptotic Approximations and Errors 15
1.1.1. Order relations among functions 15
1.1.2. Statements following from the order relations 20
1.1.3. Absolute and relative errors 23
§1.2. Convergent versus Asymptotic Series: Concepts 24
1.2.1. Convergent power series 24
1.2.2. Introduction to asymptotic series 26
§1.3. Asymptotic Sequences and Series: General Definitions 28
§1.4. How to Sum an Asymptotic Series 32
§1.5. Asymptotic Root Finding 30
1.5.1. A regular perturbation problem 38
vii
1.5.2. A singular perturbation problem. Rescaling and the
principle of dominant balance 40
§1.6. Notes and References 43
Part 2. Asymptotic Analysis of Exponential Integrals
Chapter 2. Fundamental Techniques for Integrals 47
§2.1. Review of Basic Methods 47
§2.2. Exponential Integrals and Watson s Lemma 52
§2.3. Elementary Generalizations of Watson s Lemma 56
Chapter 3. Laplace s Method for Asymptotic Expansions of
Integrals 61
§3.1. Introduction 61
§3.2. Nonlocal Contributions 62
§3.3. Contributions from Endpoints 64
§3.4. Contributions from Interior Maxima 67
§3.5. Summary of Generic Leading order Behavior 70
§3.6. Application: Weakly Diffusive Regularization of Shock
Waves 73
3.6.1. The method of characteristics 75
3.6.2. Regularization of shocks by diffusion. Burgers equation 78
3.6.3. The Cole Hopf transformation and the solution of the
initial value problem for Burgers equation 80
3.6.4. Analysis of the solution in the limit of vanishing
diffusion 82
§3.7. Multidimensional Integrals 87
§3.8. Notes and References 93
Chapter 4. The Method of Steepest Descents for Asymptotic
Expansions of Integrals 95
§4.1. Introduction 95
§4.2. Contour Deformation 97
§4.3. Paths of Steepest Descent 98
§4.4. Saddle Points 103
§4.5. Paranietrization indepeiident Local Contributions 107
§4.6. Application: Long time Asymptotic Behavior of Diffusion
Processes 108
4.6.1. A derivation of the diffusion equation 109
4.6.2. Solution of the diffusion equation and the corresponding
initial value problem 110
4.6.3. Long time asymptotics via the method of steepest
descents 112
§4.7. Application: Asymptotic Behavior of Special Functions,
Airy Functions and the Stokes Phenomenon 116
4.7.1. Integral representations for Airy functions 116
4.7.2. Preliminary transformations necessary for asymptotic
analysis of Ai(x) for large x 117
4.7.3. Determination of the path. Dependence of the path on k 119
4.7.4. Asymptotic behavior of Ai(x) for large x. The Stokes
phenomenon 122
§4.8. The Effect of Branch Points 125
4.8.1. Application: Asymptotics of transform integrals 135
4.8.2. Application: Selection of particular solutions of linear
differential equations admitting integral representations 142
§4.9. Notes and References 147
Chapter 5. The Method of Stationary Phase for Asymptotic
Analysis of Oscillatory Integrals 149
§5.1. Introduction 149
§5.2. Nonlocal Contributions 151
§5.3. Contributions from Interior Stationary Phase Points 156
5.3.1. Putting the exponent in normal form by a change of
variables 156
5.3.2. Analysis of Ji(A) by the method of steepest descents 158
5.3.3. Analysis of J2W using integration by parts 160
5.3.4. The asymptotic contribution of a stationary phase point 161
§5.4. Summary of Generic Leading order Behavior 162
§5.5. Application: Long time Behavior of Linear Dispersive
Waves 164
5.5.1. Partial differential equations for linear dispersive waves 164
5.5.2. Analysis of the solution formula. Long¬
time asymptotics using the method of stationary
phase 167
5.5.3. Structure of the wave field for large time. Modulated
wavetrains and group velocity 169
§5.6. Application: Semiclassical Dynamics of Free Particles in
Quantum Mechanics 171
5.6.1. Derivation of the dispersion relation for matter waves 171
5.6.2. The Schrodinger equation for a free particle.
Interpretation of the Schrodinger wave function 173
5.6.3. The semiclassical limit. Heuristic reasoning 174
5.6.4. Rigorous semiclassical asymptotics using the method of
stationary phase 177
§5.7. Multidimensional Integrals 181
§5.8. Notes and References 193
Part 3. Asymptotic Analysis of Differential Equations
Chapter 6. Asymptotic Behavior of Solutions of Linear Second
order Differential Equations in the Complex
Plane 197
§6.1. Qualitative Theory of Solutions 198
6.1.1. Reduction to canonical form 198
6.1.2. Solutions viewed as analytic functions of the complex
variable z 200
6.1.3. Reduction of order 213
§6.2. Asymptotic Behavior near Ordinary and Regular Singular
Points 214
6.2.1. Series solutions at ordinary points 215
6.2.2. Series solutions at regular singular points. The method
of Frobenius 216
§6.3. Asymptotic Behavior near Irregular Singular Points 223
6.3.1. Formal asymptotic series 223
6.3.2. Existence of true solutions described by the formal
asymptotic series. The Stokes phenomenon 229
6.3.3. Another approach to the existence of true solutions and
the Stokes phenomenon. Borel summation 246
§6.4. Notes and References 251
Chapter 7. Introduction to Asymptotics of Solutions of Ordinary
Differential Equations with Respect to Parameters 253
§7.1. Regular Perturbation Problems 254
7.1.1. Formal power series expansions 255
7.1.2. Solving for yn(x). Variation of parameters 256
7.1.3. Justification of the formal expansion 260
§7.2. Singular Asymptotics 263
7.2.1. The WKB method 263
7.2.2. The special case of an asymptotic power series for
fix: A) 268
7.2.3. Turning points 277
7.2.4. Problems with more than one turning point. The Bohr
Sommcrfeld quantization rule 300
7.2.5. Uniform asymptotics near turning points. Langer
transformations 304
§7.3. Notes and References 310
Chapter 8. Asymptotics of Linear Boundary value Problems 311
§8.1. Asymptotic Existence of Solutions 312
8.1.1. Case I: a(x) ^ 0 on [a,0] and f is positive but
sufficiently small 314
8.1.2. Case II: b(x) a (x)/2 0 on [a. if and e is positive 314
§8.2. An Exactly Solvable Boundary value Problem:
Phenomenology of Boundary Layers 315
§8.3. Outer Asymptotics 318
§8.4. Rescaling and Inner Asymptotics for Boundary Layers and
Internal Layers 321
§8.5. Matching of Asymptotic Expansions. Intermediate
Variables, and Uniformly Valid Asymptotics 325
§8.6. Examples 328
§8.7. Proving the Validity of Uniform Approximations 342
§8.8. The Method of Multiple Scales 350
§8.9. Notes and References 353
Chapter 9. Asymptotics of Oscillatory Phenomena 355
§9.1. Perturbation Theory in Linear Algebra and Eigenvalue
Problems 356
9.1.1. Nondegenerate theory 357
9.1.2. Degenerate theory 362
9.1.3. More on solvability conditions. Inner products and
adjoints 365
§9.2. Periodic Boundary Conditions and Mathicu s Equation 368
9.2.1. Floquet theory 368
9.2.2. Periodic and antiperiodic solutions. Formal asymptotics 371
9.2.3. Justification of the expansions 377
§9.3. Weakly Nonlinear Oscillations 382
9.3.1. Periodic solutions near equilibrium 383
9.3.2. A perturbative approach to weak cubic nonlinearity.
Secular terms 384
9.3.3. Removal of secular terms. Strained coordinates and the
Poincare Lindstedt method 388
9.3.4. The method of multiple scales 391
9.3.5. Justification of the expansions 397
§9.4. Notes and References 400
Chapter 10. Weakly Nonlinear Waves 401
§10.1. Derivation of Universal Partial Differential Equations
Using the Method of Multiple Scales 401
10.1.1. Modulated wavetrains with dispersion and nonlinear
effects. The cubic nonlinear Schrodinger equation 402
10.1.2. Spontaneous excitation of a mean flow 410
10.1.3. Multiple wave resonances 417
10.1.4. Long wave asymptotics. The Boussinesq equation and
the Korteweg de Vries equation 423
§10.2. Waves in Molecular Chains 425
10.2.1. The Fermi Pasta Ulam model 426
10.2.2. Derivation of the cubic nonlinear Schrodinger equation 427
10.2.3. Derivation of the Boussinesq and Korteweg
de Vries equations 432
§10.3. Water Waves 433
10.3.1. Derivation of the cubic nonlinear Schrodinger equation 436
10.3.2. Derivation of the Korteweg de Vries equation 444
§10.4. Notes and References 447
Appendix: Fundamental Inequalities 451
Triangle Inequalities 451
Minkowski Inequalities 452
Holder Inequalities 452
Bibliography 453
Index of Names 455
Subject Index 457
|
| adam_txt |
Contents
Preface xiii
Part 1. Fundamentals
Chapter 0. Themes of Asymptotic Analysis 3
§0.1. Theme: Asymptotics, Convergent and Divergent
Asymptotic Series 3
§0.2. Theme: Other Parameters and Nonuniformity 8
0.2.1. First example. Oscillations 8
0.2.2. Second example. Boundary layers 10
§0.3. Theme: Differential Equations 12
§0.4. Theme: Universal Partial Differential Equations and
Canonical Physical Models 13
Chapter 1. The Nature of Asymptotic Approximations 15
§1.1. Asymptotic Approximations and Errors 15
1.1.1. Order relations among functions 15
1.1.2. Statements following from the order relations 20
1.1.3. Absolute and relative errors 23
§1.2. Convergent versus Asymptotic Series: Concepts 24
1.2.1. Convergent power series 24
1.2.2. Introduction to asymptotic series 26
§1.3. Asymptotic Sequences and Series: General Definitions 28
§1.4. How to "Sum" an Asymptotic Series 32
§1.5. Asymptotic Root Finding 30
1.5.1. A regular perturbation problem 38
vii
1.5.2. A singular perturbation problem. Rescaling and the
principle of dominant balance 40
§1.6. Notes and References 43
Part 2. Asymptotic Analysis of Exponential Integrals
Chapter 2. Fundamental Techniques for Integrals 47
§2.1. Review of Basic Methods 47
§2.2. Exponential Integrals and Watson's Lemma 52
§2.3. Elementary Generalizations of Watson's Lemma 56
Chapter 3. Laplace's Method for Asymptotic Expansions of
Integrals 61
§3.1. Introduction 61
§3.2. Nonlocal Contributions 62
§3.3. Contributions from Endpoints 64
§3.4. Contributions from Interior Maxima 67
§3.5. Summary of Generic Leading order Behavior 70
§3.6. Application: Weakly Diffusive Regularization of Shock
Waves 73
3.6.1. The method of characteristics 75
3.6.2. Regularization of shocks by diffusion. Burgers' equation 78
3.6.3. The Cole Hopf transformation and the solution of the
initial value problem for Burgers' equation 80
3.6.4. Analysis of the solution in the limit of vanishing
diffusion 82
§3.7. Multidimensional Integrals 87
§3.8. Notes and References 93
Chapter 4. The Method of Steepest Descents for Asymptotic
Expansions of Integrals 95
§4.1. Introduction 95
§4.2. Contour Deformation 97
§4.3. Paths of Steepest Descent 98
§4.4. Saddle Points 103
§4.5. Paranietrization indepeiident Local Contributions 107
§4.6. Application: Long time Asymptotic Behavior of Diffusion
Processes 108
4.6.1. A derivation of the diffusion equation 109
4.6.2. Solution of the diffusion equation and the corresponding
initial value problem 110
4.6.3. Long time asymptotics via the method of steepest
descents 112
§4.7. Application: Asymptotic Behavior of Special Functions,
Airy Functions and the Stokes Phenomenon 116
4.7.1. Integral representations for Airy functions 116
4.7.2. Preliminary transformations necessary for asymptotic
analysis of Ai(x) for large x 117
4.7.3. Determination of the path. Dependence of the path on k 119
4.7.4. Asymptotic behavior of Ai(x) for large x. The Stokes
phenomenon 122
§4.8. The Effect of Branch Points 125
4.8.1. Application: Asymptotics of transform integrals 135
4.8.2. Application: Selection of particular solutions of linear
differential equations admitting integral representations 142
§4.9. Notes and References 147
Chapter 5. The Method of Stationary Phase for Asymptotic
Analysis of Oscillatory Integrals 149
§5.1. Introduction 149
§5.2. Nonlocal Contributions 151
§5.3. Contributions from Interior Stationary Phase Points 156
5.3.1. Putting the exponent in normal form by a change of
variables 156
5.3.2. Analysis of Ji(A) by the method of steepest descents 158
5.3.3. Analysis of J2W using integration by parts 160
5.3.4. The asymptotic contribution of a stationary phase point 161
§5.4. Summary of Generic Leading order Behavior 162
§5.5. Application: Long time Behavior of Linear Dispersive
Waves 164
5.5.1. Partial differential equations for linear dispersive waves 164
5.5.2. Analysis of the solution formula. Long¬
time asymptotics using the method of stationary
phase 167
5.5.3. Structure of the wave field for large time. Modulated
wavetrains and group velocity 169
§5.6. Application: Semiclassical Dynamics of Free Particles in
Quantum Mechanics 171
5.6.1. Derivation of the dispersion relation for ''matter waves" 171
5.6.2. The Schrodinger equation for a free particle.
Interpretation of the Schrodinger wave function 173
5.6.3. The semiclassical limit. Heuristic reasoning 174
5.6.4. Rigorous semiclassical asymptotics using the method of
stationary phase 177
§5.7. Multidimensional Integrals 181
§5.8. Notes and References 193
Part 3. Asymptotic Analysis of Differential Equations
Chapter 6. Asymptotic Behavior of Solutions of Linear Second
order Differential Equations in the Complex
Plane 197
§6.1. Qualitative Theory of Solutions 198
6.1.1. Reduction to canonical form 198
6.1.2. Solutions viewed as analytic functions of the complex
variable z 200
6.1.3. Reduction of order 213
§6.2. Asymptotic Behavior near Ordinary and Regular Singular
Points 214
6.2.1. Series solutions at ordinary points 215
6.2.2. Series solutions at regular singular points. The method
of Frobenius 216
§6.3. Asymptotic Behavior near Irregular Singular Points 223
6.3.1. Formal asymptotic series 223
6.3.2. Existence of true solutions described by the formal
asymptotic series. The Stokes phenomenon 229
6.3.3. Another approach to the existence of true solutions and
the Stokes phenomenon. Borel summation 246
§6.4. Notes and References 251
Chapter 7. Introduction to Asymptotics of Solutions of Ordinary
Differential Equations with Respect to Parameters 253
§7.1. Regular Perturbation Problems 254
7.1.1. Formal power series expansions 255
7.1.2. Solving for yn(x). Variation of parameters 256
7.1.3. Justification of the formal expansion 260
§7.2. Singular Asymptotics 263
7.2.1. The WKB method 263
7.2.2. The special case of an asymptotic power series for
fix: A) 268
7.2.3. Turning points 277
7.2.4. Problems with more than one turning point. The Bohr
Sommcrfeld quantization rule 300
7.2.5. Uniform asymptotics near turning points. Langer
transformations 304
§7.3. Notes and References 310
Chapter 8. Asymptotics of Linear Boundary value Problems 311
§8.1. Asymptotic Existence of Solutions 312
8.1.1. Case I: a(x) ^ 0 on [a,0] and f is positive but
sufficiently small 314
8.1.2. Case II: b(x) a'(x)/2 0 on [a. if\ and e is positive 314
§8.2. An Exactly Solvable Boundary value Problem:
Phenomenology of Boundary Layers 315
§8.3. Outer Asymptotics 318
§8.4. Rescaling and Inner Asymptotics for Boundary Layers and
Internal Layers 321
§8.5. Matching of Asymptotic Expansions. Intermediate
Variables, and Uniformly Valid Asymptotics 325
§8.6. Examples 328
§8.7. Proving the Validity of Uniform Approximations 342
§8.8. The Method of Multiple Scales 350
§8.9. Notes and References 353
Chapter 9. Asymptotics of Oscillatory Phenomena 355
§9.1. Perturbation Theory in Linear Algebra and Eigenvalue
Problems 356
9.1.1. Nondegenerate theory 357
9.1.2. Degenerate theory 362
9.1.3. More on solvability conditions. Inner products and
adjoints 365
§9.2. Periodic Boundary Conditions and Mathicu's Equation 368
9.2.1. Floquet theory 368
9.2.2. Periodic and antiperiodic solutions. Formal asymptotics 371
9.2.3. Justification of the expansions 377
§9.3. Weakly Nonlinear Oscillations 382
9.3.1. Periodic solutions near equilibrium 383
9.3.2. A perturbative approach to weak cubic nonlinearity.
Secular terms 384
9.3.3. Removal of secular terms. Strained coordinates and the
Poincare Lindstedt method 388
9.3.4. The method of multiple scales 391
9.3.5. Justification of the expansions 397
§9.4. Notes and References 400
Chapter 10. Weakly Nonlinear Waves 401
§10.1. Derivation of Universal Partial Differential Equations
Using the Method of Multiple Scales 401
10.1.1. Modulated wavetrains with dispersion and nonlinear
effects. The cubic nonlinear Schrodinger equation 402
10.1.2. Spontaneous excitation of a mean flow 410
10.1.3. Multiple wave resonances 417
10.1.4. Long wave asymptotics. The Boussinesq equation and
the Korteweg de Vries equation 423
§10.2. Waves in Molecular Chains 425
10.2.1. The Fermi Pasta Ulam model 426
10.2.2. Derivation of the cubic nonlinear Schrodinger equation 427
10.2.3. Derivation of the Boussinesq and Korteweg
de Vries equations 432
§10.3. Water Waves 433
10.3.1. Derivation of the cubic nonlinear Schrodinger equation 436
10.3.2. Derivation of the Korteweg de Vries equation 444
§10.4. Notes and References 447
Appendix: Fundamental Inequalities 451
Triangle Inequalities 451
Minkowski Inequalities 452
Holder Inequalities 452
Bibliography 453
Index of Names 455
Subject Index 457 |
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| author | Miller, Peter D. 1967- |
| author_GND | (DE-588)140761829 |
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| ctrlnum | (OCoLC)64289450 (DE-599)BVBBV021699082 |
| dewey-full | 511/.4 |
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| dewey-ones | 511 - General principles of mathematics |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| id | DE-604.BV021699082 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:16:45Z |
| indexdate | 2024-07-09T20:41:57Z |
| institution | BVB |
| isbn | 9780821840788 0821840789 |
| language | English |
| lccn | 2006040794 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014913043 |
| oclc_num | 64289450 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-83 DE-11 DE-188 |
| owner_facet | DE-91G DE-BY-TUM DE-83 DE-11 DE-188 |
| physical | XV, 467 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Miller, Peter D. 1967- Verfasser (DE-588)140761829 aut Applied asymptotic analysis Peter D. Miller Providence, RI American Mathematical Society 2006 XV, 467 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 75 Asymptotic expansions Differential equations Asymptotic theory Approximation theory Integral equations Asymptotic theory Asymptotische Approximation (DE-588)4739184-4 gnd rswk-swf Asymptotische Approximation (DE-588)4739184-4 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1154-1 Graduate studies in mathematics 75 (DE-604)BV009739289 75 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014913043&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Miller, Peter D. 1967- Applied asymptotic analysis Graduate studies in mathematics Asymptotic expansions Differential equations Asymptotic theory Approximation theory Integral equations Asymptotic theory Asymptotische Approximation (DE-588)4739184-4 gnd |
| subject_GND | (DE-588)4739184-4 |
| title | Applied asymptotic analysis |
| title_auth | Applied asymptotic analysis |
| title_exact_search | Applied asymptotic analysis |
| title_exact_search_txtP | Applied asymptotic analysis |
| title_full | Applied asymptotic analysis Peter D. Miller |
| title_fullStr | Applied asymptotic analysis Peter D. Miller |
| title_full_unstemmed | Applied asymptotic analysis Peter D. Miller |
| title_short | Applied asymptotic analysis |
| title_sort | applied asymptotic analysis |
| topic | Asymptotic expansions Differential equations Asymptotic theory Approximation theory Integral equations Asymptotic theory Asymptotische Approximation (DE-588)4739184-4 gnd |
| topic_facet | Asymptotic expansions Differential equations Asymptotic theory Approximation theory Integral equations Asymptotic theory Asymptotische Approximation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014913043&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT millerpeterd appliedasymptoticanalysis |