The theory of singular perturbations:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier
1996
|
| Series: | North-Holland series in applied mathematics and mechanics
42 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 340 S. graph. Darst. |
| ISBN: | 0444821708 |
Staff View
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| 100 | 1 | |a Jager, Eduardus M. de |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The theory of singular perturbations |c E. M. de Jager ; Jiang Furu |
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 1996 | |
| 300 | |a 340 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a North-Holland series in applied mathematics and mechanics |v 42 | |
| 650 | 7 | |a Differentiaalvergelijkingen |2 gtt | |
| 650 | 7 | |a Perturbations singulières (Mathématiques) |2 ram | |
| 650 | 7 | |a Singulariteiten |2 gtt | |
| 650 | 7 | |a Storingsrekening |2 gtt | |
| 650 | 4 | |a Singular perturbations (Mathematics) | |
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Record in the Search Index
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| adam_text | CONTENTS
Preface vii
Chapter 1 General Introduction 1
Chapter 2 Asymptotic Expansions 9
1 Order Symbols 9
2 Gauge Functions and Asymptotic Sequences 12
3 Asymptotic Series 13
4 Convergence versus Asymptotic Convergence 16
5 Elementary Operations on Asymptotic Expansions 18
6 Other Types of Estimates 20
7 Generalized Asymptotic Expansions 21
Exercises 22
Chapter 3 Regular Perturbations 27
1 Regular Perturbations 27
2 A Nonlinear Initial Value Problem Containing a Small Parameter ... 28
3 Applications 37
3.1 Duffing Equation 37
3.2 The Motion of a Planet around the Sun 38
Exercises 41
Chapter 4 The Method of the Strained Coordinate 43
1 Introduction 43
2 Applications of the Method of the Strained Coordinate 44
2.1 The Nonlinear Spring 44
2.2 The Perihelium Precession 46
3 The Method of the Strained Parameter 47
4 Lighthill s Method 50
5 Temple s Method 55
6 Limitations of the Lindstedt Poincare Method 57
Exercises 58
Chapter 5 The Method of Averaging 61
1 Introduction 61
2 The Krilov Bogoliubov Mitropolski Theorem 63
2.1 Introduction to First Order Averaging 64
2.2 Generalization of Theorem 2; K.B.M. Theorem Second Variant .. 66
2.3 The Krilov Bogoliubov Mitropolski Theorem for Nonperiodic Fields;
K.B.M. Theorem Third Variant 70
3 Weakly Nonlinear Free Oscillations 75
3.1 The General Case 75
3.2 The Duffing Equation 77
x Contents
3.3 The Perihelium Precession 77
3.4 The Linear Oscillator with Small Damping 78
3.5 The Free van der Pol Equation 79
4 Weakly Forced Nonlinear Oscillations 80
4.1 The Case without Damping 80
4.2 The Case with Damping 84
5 A Linear Oscillator with Increasing Damping 87
Exercises 89
Chapter 6 The Method of Multiple Scales 91
1 Introduction 91
2 Weakly Nonlinear Free Oscillations 93
2.1 The Duffing Equation 93
2.2 The Perihelium Precession 96
3 The Linear Oscillator with Damping 98
4 The Equation of Mathieu 101
4.1 Introduction 101
4.2 Floquet s Theory for Linear Equations with Periodic Coefficients 102
4.3 Application to Hill s Equation 103
4.4 Application to Mathieu s Equation 105
4.5 The Transition Curves for the Mathieu Equation 106
4.6 The Approximation of the Solution Outside the Transition Curves 110
5 The General Case and the Error Estimate 115
5.1 Introduction 115
5.2 The Formal Approximation 116
5.3 Estimate of the Remainder Term 117
6 Averaging and Multiple Scales for Perturbed Wave Equations 123
6.1 The Approximation by Chikwendu and Kevorkian 123
6.2 Examples 126
6.2.1 Wave Equation with Linear Damping 126
6.2.2 Wave Equation with Cubic Damping 127
6.3 Justification of the Chikwendu Kevorkian Procedure 130
Exercises 135
Chapter 7 Singular Perturbations of Linear Ordinary Differential Equations 137
1 The initial Value Problem 137
1.1 Introduction 137
1.2 The Formal Approximation 137
1.3 The A Priori Estimate of the Solution of a Singularly Perturbed
Ordinary Differential Equation with Given Initial Data 140
1.4 The Estimate of the Remainder Term and Final Results 142
2 The Boundary Value Problem 144
2.1 Introduction 144
2.2 The Maximum Principle for Ordinary Differential Operators ... 145
2.3 An A Priori Estimate of the Solution of the Boundary Value
Problem 146
2.4 The Formal Approximation 148
2.5 The A Priori Estimate of the Remainder Term and Final Results 151
3 Boundary Value Problems with Turning Points 158
3.1 Introduction 158
Contents xi
3.2 The Turning Point Problem with f (x) 0 158
3.3 The Asymptotic Approximation around the Turning Point and the
Case p ^ 2m, m = 0,1,2 161
3.4 The Asymptotic Approximation in the Case of Resonance 164
3.5 The Turning Point Problem with f (x) 0 168
Exercises 172
Chapter 8 Singular Perturbations of Second Order Elliptic Type. Linear Theory 175
1 Introduction 175
2 The Maximum Principle for Elliptic Operators 177
3 The Formal Approximation 179
4 Estimation of the Remainder Term and Final Results 185
5 Domains with Characteristic Boundaries 191
5.1 Introduction 191
5.2 The Singular Perturbation Problem in a Rectangle 194
6 Elliptic Boundary Value Problems with Turning Points 200
6.1 Introduction 200
6.2 Examples of Turning Point Problems 201
6.2.1 Curves of Turning Points 201
6.2.2 Isolated Turning Points; Nodes 202
6.2.3 A Saddle Turning Point 204
Exercises 207
Chapter 9 Singular Perturbations of Second Order Hyperbolic Type.
Linear Theory 209
1 Introduction 209
2 Characteristics and Subcharacteristics 210
3 The Formal Approximation 213
4 A Priori Estimates of Solutions of Initial Value Problems for Partial
Differential Equations with a Singular Perturbation of
Hyperbolic Type 215
5 The Estimate of the Remainder Term and Final Results 223
Exercises 227
Chapter 10 Singular Perturbations in Nonlinear Initial Value Problems of
Second Order 229
1 Introduction 229
2 A Fixed Point Theorem 230
3 The Quasilinear Initial Value Problem 232
3.1 Introduction 232
3.2 The Formal Approximation 232
3.3 The Estimate of the Remainder Term and Final Results 235
4 A General Nonlinear Initial Value Problem 239
4.1 Introduction 239
4.2 The Formal Approximation 240
4.3 The Estimate of the Remainder Term and Final Results 244
5 Quasilinear Initial Value Problems with a Singular Perturbation of
Second Order Hyperbolic Type 250
5.1 Introduction 250
5.2 The Formal Approximation 250
5.3 The Estimate of the Remainder Term and Final Results 253
xii Contents
Exercises 260
Chapter 11 Singular Perturbations in Nonlinear Boundary Value Problems of
Second Order 261
1 Introduction 261
2 Boundary Value Problems for Quasilinear Ordinary Differential
Equations 263
2.1 The Formal Approximation 263
2.2 The Estimate of the Remainder Term and Final Results 267
3 Transition Layers 273
4 Autonomous Conservative Equations 283
5 A More General Case 288
6 Boundary Value Problems for Quasilinear Partial Differential
Equations of Elliptic Type 291
6.1 Introduction 291
6.2 The Nonlinear Generalization of the Maximum Principle 291
6.3 Elliptic Equations without First Derivatives 293
6.4 Elliptic Equations with First Derivatives 300
Exercises 304
Chapter 12 Perturbations of Higher Order 307
1 Introduction 307
2 Perturbations of Higher Order in Ordinary Differential Equations .. 308
2.1 Introduction 308
2.2 The Formal Approximtion 309
3 Elliptic Perturbations of Elliptic Equations 315
3.1 Introduction 315
3.2 Elliptic Partial Differential Equations 315
3.2.1 Sobolev Spaces 315
3.2.2 Elliptic Operators, Bilinear Forms and Garding s Inequality 317
3.2.3 Generalized Dirichlet Problems 318
3.2.4 Existence and Generalized Solutions 319
4 Elliptic Singular Perturbations of Higher Order 323
4.1 The Boundary Value Problem 323
4.2 Existence and A Priori Estimate 324
4.3 The approximation of the Solution 325
4.4 The Estimate of the Remainder and Final Results 328
Exercises 330
References 331
Subject Index 339
|
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| isbn | 0444821708 |
| language | English |
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| series2 | North-Holland series in applied mathematics and mechanics |
| spelling | Jager, Eduardus M. de Verfasser aut The theory of singular perturbations E. M. de Jager ; Jiang Furu Amsterdam [u.a.] Elsevier 1996 340 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier North-Holland series in applied mathematics and mechanics 42 Differentiaalvergelijkingen gtt Perturbations singulières (Mathématiques) ram Singulariteiten gtt Storingsrekening gtt Singular perturbations (Mathematics) Singuläre Störung (DE-588)4055100-3 gnd rswk-swf Partieller Differentialoperator (DE-588)4173439-7 gnd rswk-swf Störungstheorie (DE-588)4128420-3 gnd rswk-swf Störungstheorie (DE-588)4128420-3 s DE-604 Singuläre Störung (DE-588)4055100-3 s Partieller Differentialoperator (DE-588)4173439-7 s Furu, Jiang Verfasser aut North-Holland series in applied mathematics and mechanics 42 (DE-604)BV001900898 42 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007871201&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jager, Eduardus M. de Furu, Jiang The theory of singular perturbations North-Holland series in applied mathematics and mechanics Differentiaalvergelijkingen gtt Perturbations singulières (Mathématiques) ram Singulariteiten gtt Storingsrekening gtt Singular perturbations (Mathematics) Singuläre Störung (DE-588)4055100-3 gnd Partieller Differentialoperator (DE-588)4173439-7 gnd Störungstheorie (DE-588)4128420-3 gnd |
| subject_GND | (DE-588)4055100-3 (DE-588)4173439-7 (DE-588)4128420-3 |
| title | The theory of singular perturbations |
| title_auth | The theory of singular perturbations |
| title_exact_search | The theory of singular perturbations |
| title_full | The theory of singular perturbations E. M. de Jager ; Jiang Furu |
| title_fullStr | The theory of singular perturbations E. M. de Jager ; Jiang Furu |
| title_full_unstemmed | The theory of singular perturbations E. M. de Jager ; Jiang Furu |
| title_short | The theory of singular perturbations |
| title_sort | the theory of singular perturbations |
| topic | Differentiaalvergelijkingen gtt Perturbations singulières (Mathématiques) ram Singulariteiten gtt Storingsrekening gtt Singular perturbations (Mathematics) Singuläre Störung (DE-588)4055100-3 gnd Partieller Differentialoperator (DE-588)4173439-7 gnd Störungstheorie (DE-588)4128420-3 gnd |
| topic_facet | Differentiaalvergelijkingen Perturbations singulières (Mathématiques) Singulariteiten Storingsrekening Singular perturbations (Mathematics) Singuläre Störung Partieller Differentialoperator Störungstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007871201&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001900898 |
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