Applied partial differential equations:
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Oxford
Oxford Univ. Press
2003
|
| Edition: | Rev. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 449 S. graph. Darst. |
| ISBN: | 0198527713 0198527705 |
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| 245 | 1 | 0 | |a Applied partial differential equations |c John Ockendon ... |
| 250 | |a Rev. ed. | ||
| 264 | 1 | |a Oxford |b Oxford Univ. Press |c 2003 | |
| 300 | |a XI, 449 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | Titel: Applied partial differential equations
Autor: Ockendon, John R
Jahr: 2003
Contents
Introduction 1
1 First-order scalar quasilinear equations 6
1.1 Introduction 6
1.2 Cauchy data 8
1.3 Characteristics 9
1.3.1 Linear and semilinear equations 11
1.4 Domain of definition and blow-up 13
1.5 Quasilinear equations 15
1.6 Solutions with discontinuities 10
*1.7 Weak solutions 22
*1.8 More independent variables 25
1.9 Postscript 28
Exercises 29
2 First-order quasilinear systems 35
2.1 Motivation and models 35
2.2 Cauchy data and characteristics 41
2.3 The Cauchy-Kowalevski theorem 45
2.4 Hyperbolicity 48
2.4.1 Two-by-two systems 49
2.4.2 Systems of dimension n 50
2.4.3 Examples 52
*2.5 Weak solutions and shock waves 55
2.5.1 Causality 56
2.5.2 Viscosity and entropy 59
2.5.3 Other discontinuities 62
*2.6 Systems with more than two independent variables 63
Exercises 68
3 Introduction to second-order scalar equations 76
3.1 Preamble 76
3.2 The Cauchy problem for semilinear equations 78
3.3 Characteristics 80
3.4 Canonical forms for semilinear equations 83
3.4.1 Hyperbolic equations 83
3.4.2 Elliptic equations 84
3.4.3 Parabolic equations 86
3.5 Some general remarks 87
Exercises
vii
viii CONTENTS
4 Hyperbolic equations ^
4.1 Introduction
4.2 Linear equations: the solution to the Cauchy problem 94
4.2.1 An ad hoc approach to Riemann functions 94
4.2.2 The rationale for Riemann functions 96
4.2.3 Implications of the Riemann function representation 100
4.3 Non-Cauchy data for the wave equation 102
*4.3.1 Strongly discontinuous boundary data 105
4.4 Transforms and eigenfunction expansions 106
4.5 Applications to wave equations 113
4.5.1 The wave equation in one space dimension 113
4.5.2 Circular and spherical symmetry 116
*4.5.3 The telegraph equation 118
*4.5.4 Waves in periodic media 119
*4.5.5 General remarks 119
4.6 Wave equations with more than two independent variables 120
4.6.1 The method of descent and Huygens principle 120
4.6.2 Hyperbolicity and time-likeness 125
*4.7 Higher-order systems 128
4.7.1 Linear elasticity 128
4.7.2 Maxwell s equations of electromagnetism 131
4.8 Nonlinearity 135
4.8.1 Simple waves 135
4.8.2 Hodograph methods 137
4.8.3 Liouville s equation 139
*4.8.4 Another method 141
Exercises 141
5 Elliptic equations 151
5.1 Models 151
5.1.1 Gravitation 151
5.1.2 Electromagnetism 152
5.1.3 Heat transfer 153
5.1.4 Mechanics 155
5.1.5 Acoustics 100
5.1.6 Aerofoil theory and fracture 161
5.2 Well-posed boundary data 163
o.2.1 The Laplace and Poisson equations 163
^ o.2.2 More general elliptic equations 166
5.3 The maximum principle 1^7
5.4 Variational principles
5.5 Green s functions ^
5.5.1 The classical formulation Igg
5.5.2 Generalised function formulation 171
5.6 Explicit representations of Green s functions 174
5.6.1 Laplace s equation and Poisson s equation 174
CONTENTS ix
5.6.2 Helmholtz equation 180
5.6.3 The modified Helmholtz equation 182
*5.7 Green s functions, eigenfunction expansions and transforms 183
5.7.1 Eigenvalues and eigenfunctions 183
5.7.2 Green s functions and transforms 184
5.8 Transform solutions of elliptic problems 186
5.8.1 Laplace s equation with cylindrical symmetry: Hankel
transforms 187
5.8.2 Laplace s equation in a wedge geometry; the Mellin
transform 190
*5.8.3 Helmholtz equation 191
*5.8.4 Higher-order problems 194
5.9 Complex variable methods 195
5.9.1 Conformal maps 197
*5.9.2 Riemann-Hilbert problems 199
*5.9.3 Mixed boundary value problems and singular integral
equations 204
*5.9.4 The Wiener-Hopf method 206
*5.9.5 Singularities and index 209
*5.10 Localised boundary data 211
5.11 Nonlinear problems 212
5.11.1 Nonlinear models 212
5.11.2 Existence and uniqueness 213
5.11.3 Parameter dependence and singular behaviour 215
5.12 Liouville s equation again 221
5.13 Postscript: V2 or -A? 222
Exercises 223
6 Parabolic equations 241
6.1 Linear models of diffusion 241
6.1.1 Heat and mass transfer 241
6.1.2 Probability and finance 243
6.1.3 Electromagnetism 245
6.1.4 General remarks 245
6.2 Initial and boundary conditions 245
6.3 Maximum principles and well-posedness 247
*6.3.1 The strong maximum principle 248
6.4 Green s functions and transform methods for the heat equation 249
6.4.1 Green s functions: general remarks 249
6.4.2 The Green s function for the heat equation with no
boundaries 251
6.4.3 Boundary value problems 254
*6.4.4 Convection-diffusion problems 260
6.5 Similarity solutions and groups 262
6.5.1 Ordinary differential equations 264
6.5.2 Partial differential equations 265
x CONTENTS
*6.5.3 General remarks i1 f J®
6.6 Nonlinear equations
6.6.1 Models ^71
6.6.2 Theoretical remarks
6.6.3 Similarity solutions and travelling waves 275
6.6.4 Comparison methods and the maximum principle 281
*6.6.5 Blow-up 2^
*6.7 Higher-order equations and systems 286
6.7.1 Higher-order scalar problems 287
6.7.2 Higher-order systems ; ¦ 289
Exercises 291
Free boundary problems 305
7.1 Introduction and models 305
7.1.1 Stefan and related problems 306
7.1.2 Other free boundary problems in diffusion 310
7.1.3 Some other problems from mechanics 314
7.2 Stability and well-posedncss 318
7.2.1 Surface gravity waves 319
7.2.2 Vortex sheets 321
7.2.3 Hele-Shaw flow 322
7.2.4 Shock waves 324
7.3 Classical solutions 326
7.3.1 Comparison methods 326
7.3.2 Energy methods and conserved quantities 327
7.3.3 Green s functions and integral equations 328
* 7.4 Weak and variational methods 329
7.4.1 Variational methods 330
7.4.2 The enthalpy method 335
7.5 Explicit solutions 338
7.5.1 Similarity solutions 339
*7.5.2 Complex variable methods 341
*7.6 Regularisation 345
*7.7 Postscript 347
Exercises 349
Non-quasilinear equations 359
8.1 Introduction 359
8.2 Scalar first-order equations 360
8.2.1 Two independent variables 360
8.2.2 More independent variables 366
8.2.3 The eikonal equation 366
*8.2.4 Eigenvalue problems 374
8.2.5 Dispersion 376
8.2.6 Bicharacteristics 377
*8.3 Hamilton-Jacobi equations and quantum mechanics 378
CONTENTS xi
*8.4 Higher-order equations 380
Exercises 383
*9 Miscellaneous topics 393
9.1 Introduction 393
9.2 Linear systems revisited 395
9.2.1 Linear systems: Green s functions 396
9.2.2 Linear elasticity 398
9.2.3 Linear inviscid hydrodynamics 400
9.2.4 Wave propagation and radiation conditions 403
9.3 Complex characteristics and classification by type 405
9.4 Quasilinear systems with one real characteristic 407
9.4.1 Heat conduction with ohmic heating 407
9.4.2 Space charge 408
9.4.3 Fluid dynamics: the Navier-Stokes equations 409
9.4.4 Inviscid flow: the Euler equations 409
9.4.5 Viscous flow 412
9.5 Interaction between media 414
9.5.1 Fluid/solid acoustic interactions 414
9.5.2 Fluid/fluid gravity wave interaction 415
9.6 Gauges and invariance 415
9.7 Solitons 417
Exercises 426
Conclusion 434
References 436
Index 439
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| id | DE-604.BV019408347 |
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| indexdate | 2024-07-09T19:59:37Z |
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| language | English |
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| spelling | Applied partial differential equations John Ockendon ... Rev. ed. Oxford Oxford Univ. Press 2003 XI, 449 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Freies Randwertproblem Quasilineare partielle Differentialgleichung Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Ockendon, John Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012870395&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Applied partial differential equations Freies Randwertproblem Quasilineare partielle Differentialgleichung Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 |
| title | Applied partial differential equations |
| title_auth | Applied partial differential equations |
| title_exact_search | Applied partial differential equations |
| title_full | Applied partial differential equations John Ockendon ... |
| title_fullStr | Applied partial differential equations John Ockendon ... |
| title_full_unstemmed | Applied partial differential equations John Ockendon ... |
| title_short | Applied partial differential equations |
| title_sort | applied partial differential equations |
| topic | Freies Randwertproblem Quasilineare partielle Differentialgleichung Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Freies Randwertproblem Quasilineare partielle Differentialgleichung Partielle Differentialgleichung |
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