An introduction to second order partial differential equations: classical and variational solutions
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
2018
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xvii, 279 Seiten Illustrationen |
| ISBN: | 9789813229174 |
Internformat
MARC
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| 100 | 1 | |a Cioranescu, Doina |d ca. 20./21. Jh. |0 (DE-588)1089308787 |4 aut | |
| 245 | 1 | 0 | |a An introduction to second order partial differential equations |b classical and variational solutions |c Doina Cioranescu (Université Pierre et Marie Curie (Paris 6), France), Patrizia Donato (Université de Rouen, France), Marian P. Roque (University of the Philippines Diliman, Philippines) |
| 246 | 1 | 3 | |a Second order partial differential equations |
| 246 | 1 | 3 | |a Partial differential equations |
| 264 | 1 | |a New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo |b World Scientific |c 2018 | |
| 300 | |a xvii, 279 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Differential equations, Partial |x Study and teaching (Higher) | |
| 650 | 4 | |a Differential equations, Partial |x Study and teaching (Graduate) | |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Donato, Patrizia |4 aut | |
| 700 | 1 | |a Roque, Marian P. |0 (DE-588)1156197929 |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030265540&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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Datensatz im Suchindex
| _version_ | 1804178402071019520 |
|---|---|
| adam_text | Contents
Preface vii
List of Symbols xvii
Classical Partial Differential Equations 1
1. What is a Partial Differential Equation? 3
1.1 Introduction................................................ 3
1.2 General notation............................................ 7
1.3 Boundary and initial conditions............................. 9
1.4 Some classical partial differential equations ............. 10
1.4.1 Laplace and Poisson equations.................... 11
1.4.2 Modeling of heat transfer: The heat equation ... 11
1.4.3 The Black—Scholes equation......................... 13
1.4.4 Modeling of a vibrating string: The wave equation 14
1.4.5 The Helmholtz equation............................ 17
1.4.6 The Maxwell equations............................. 18
1.4.7 The telegraph equations.......................... 19
1.4.8 The Navier-Stokes equations....................... 20
1.5 The concept of well-posed equations....................... 21
1.6 The Cauchy—Kovalevskaya Theorem............................ 25
2. Classification of Partial Differential Equations 27
2.1 Characteristic curves...................................... 27
2.2 The Cauchy problem and existence theorems.................. 30
2.3 Canonical forms: Case of constant coefficients............. 31
XI
xii An Introduction to Second Order Partial Differential Equations
2.4 Canonical forms: Case of nonconstant coefficients....... 40
2.5 Case of more than two variables........................... 43
2.5.1 Concluding remarks................................. 46
3. Elliptic Equations 49
3.1 The Laplace equation...................................... 49
3.1.1 The Laplacian in polar coordinates .................. 50
3.2 Harmonic functions by the method of separation of variables 52
3.2.1 The two-dimensional case........................... 52
3.2.2 The three-dimensional case......................... 54
3.3 The Dirichlet problem in special geometries............. 55
3.3.1 The solution for the unit disc..................... 55
3.3.2 The solution of the exterior Dirichlet problem in R3 57
3.4 Green formulas and related properties of harmonic functions 59
3.4.1 Green formulas..................................... 59
3.4.2 Green representation theorem and consequences . 62
3.5 Green functions and the Laplace equation................ 66
3.5.1 Properties of Green functions and Poisson integral 70
3.5.2 The Neumann problem in the unit disc in R2 . . . 71
3.6 General elliptic equations . ............................... 72
4. Parabolic Equations 77
4.1 The one-dimensional heat equation........................... 78
4.1.1 The method of separation of variables.............. 82
4.1.2 A finite rod with Neumann conditions............... 83
4.1.3 An infinite rod . . ................................. 86
4.2 The two and three-dimensional heat equations............ 88
4.3 The general case of parabolic equations................. 88
5. Hyperbolic Equations 91
5.1 Wave propagation............................................ 92
5.1.1 Plane waves........................................ 92
5.1.2 Spherical waves ..................................... 93
5.2 The Cauchy problem for the wave equation................ 93
5.2.1 Conservation of the energy......................... 94
5.2.2 Poisson formula..................................... 96
5.2.3 The Huygens principle and wave diffusion.......... 100
5.3 The one-dimensional wave equation.......................... 101
Contents xiii
5.3.1 Separation of variables . ....................... 104
5.3.2 Eigenvalues and eigenfunctions................... 106
5.3.3 The d’Alembert solution.......................... 107
5.4 The two-dimensional wave equation...................... Ill
Variational Partial Differential Equations 113
6. Lp-spaces 115
6.1 Some properties of Banach spaces......................... 115
6.1.1 Linear operators in normed vector spaces......... 117
6.1.2 Dual of a normed space........................... 119
6.2 Hilbert spaces.......................................... 120
6.2.1 Orthogonal bases in Hilbert spaces . . 121
6.3 Lp-spaces and their properties......................... 125
6.3.1 The Holder inequality and applications........... 126
6.3.2 Main properties of Lp-spaces..................... 130
6.3.3 Dual of Lp-spaces................................ 132
6.4 Density in Lp-spaces..................................... 132
6.4.1 Convolution and modifiers........................ 135
6.4.2 Density results in i^R^) 136
6.4.3 Density results in Lp(0) 138
6.5 Weak and weak* convergence............................... 141
6.5.1 Weak convergence................................. 141
6.5.2 Weak* convergence ............................... 144
7. The Sobolev Spaces WlyP 147
7.1 A motivation............................................ 147
7.2 Distributions.......................................... 149
7.2.1 Regular distributions................. 150
7.2.2 Convergence in the sense of distributions........ 152
7.2.3 Derivative of a distribution.................... 152
7.3 The space WliP and its properties ....................... 154
7.3.1 Density in Sobolev spaces on R^................. 161
7.3.2 Lipschitz-continuous boundaries.................. 164
7.3.3 Extension operators and density in W1,P(Q) . . . 166
7.3.4 Chain rule and applications..................... 167
7.3.5 One-dimensional Sobolev inclusions............... 171
7.4 The notion of trace...................................... 174
xiv An Introduction to Second Order Partial Differential Equations
7.4.1 Integral on Lipschitz-continuous boundaries .... 174
7.4.2 The Trace Theorem................................ 175
7.5 The space Hq and its properties.......................... 178
7.5.1 A characterization of Hq(Q).................... 180
7.5.2 The Poincaré inequality in Æq(Q)............... 182
7.5.3 Dual of fig(Î2) ................................. 183
7.6 A characterization of H1 (R^)............................ 185
8. Sobolev Embedding Theorems 189
8.1 Continuous embedding theorems............................ 189
8.1.1 The case 1 p N for RN....................... 190
8.1.2 The case p = N for ............................. 195
8.1.3 The case p N for R^........................... 196
8.1.4 Embedding results for open sets................. 199
8.2 Compact embedding theorems................................202
8.2.1 The cases N = 1 and N 2 with p N ..........203
8.2.2 The Rellich-Kondrachov Theorem (case 1 p N) 204
8.2.3 Compactness for the case p = N..................208
8.3 Some consequences of compactness........................ 210
9. Variational Elliptic Problems 215
9.1 Setting of the problems.................................. 215
9.2 Bilinear forms on Hilbert spaces . .......................217
9.3 The Lax-Milgram Theorem...................................218
9.4 Dirichlet boundary conditions............................ 222
9.4.1 Homogeneous Dirichlet boundary conditions . . . 222
9.4.2 Nonhomogeneous Dirichlet boundary conditions . 224
9.5 Neumann boundary conditions.............................. 226
9.6 Robin boundary conditions ................................233
9.7 The Dirichlet eigenvalue problem..........................235
9.8 The eigenvalue problem for the Laplacian..................238
9.9 Regularity............................................... 241
10. Variational Evolution Problems 245
10.1 Setting of the problems . . . .......................... 245
10.2 Some classes of vector-valued functions................. 247
10.3 Variational weak formulations............................ 252
10.3.1 Variational formulation of the heat equation ... 252
Contents
xv
10.3.2 Variational formulation of the wave equation . . . 255
10.4 Existence and uniqueness theorems.........................256
10.4.1 The heat equation................................. 257
10.4.2 The wave equation................................. 264
Bibliography 273
Index 277
|
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| indexdate | 2024-07-10T08:03:24Z |
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| isbn | 9789813229174 |
| language | English |
| lccn | 017042805 |
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| spelling | Cioranescu, Doina ca. 20./21. Jh. (DE-588)1089308787 aut An introduction to second order partial differential equations classical and variational solutions Doina Cioranescu (Université Pierre et Marie Curie (Paris 6), France), Patrizia Donato (Université de Rouen, France), Marian P. Roque (University of the Philippines Diliman, Philippines) Second order partial differential equations Partial differential equations New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific 2018 xvii, 279 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Differential equations, Partial Study and teaching (Higher) Differential equations, Partial Study and teaching (Graduate) Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Donato, Patrizia aut Roque, Marian P. (DE-588)1156197929 aut Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030265540&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cioranescu, Doina ca. 20./21. Jh Donato, Patrizia Roque, Marian P. An introduction to second order partial differential equations classical and variational solutions Differential equations, Partial Study and teaching (Higher) Differential equations, Partial Study and teaching (Graduate) Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 |
| title | An introduction to second order partial differential equations classical and variational solutions |
| title_alt | Second order partial differential equations Partial differential equations |
| title_auth | An introduction to second order partial differential equations classical and variational solutions |
| title_exact_search | An introduction to second order partial differential equations classical and variational solutions |
| title_full | An introduction to second order partial differential equations classical and variational solutions Doina Cioranescu (Université Pierre et Marie Curie (Paris 6), France), Patrizia Donato (Université de Rouen, France), Marian P. Roque (University of the Philippines Diliman, Philippines) |
| title_fullStr | An introduction to second order partial differential equations classical and variational solutions Doina Cioranescu (Université Pierre et Marie Curie (Paris 6), France), Patrizia Donato (Université de Rouen, France), Marian P. Roque (University of the Philippines Diliman, Philippines) |
| title_full_unstemmed | An introduction to second order partial differential equations classical and variational solutions Doina Cioranescu (Université Pierre et Marie Curie (Paris 6), France), Patrizia Donato (Université de Rouen, France), Marian P. Roque (University of the Philippines Diliman, Philippines) |
| title_short | An introduction to second order partial differential equations |
| title_sort | an introduction to second order partial differential equations classical and variational solutions |
| title_sub | classical and variational solutions |
| topic | Differential equations, Partial Study and teaching (Higher) Differential equations, Partial Study and teaching (Graduate) Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Differential equations, Partial Study and teaching (Higher) Differential equations, Partial Study and teaching (Graduate) Partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030265540&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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