Hyperbolic partial differential equations and wave phenomena:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Japanese |
| Published: |
Providence, RI
American Math. Soc.
2000
|
| Series: | Translations of mathematical monographs
189 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 190 S. |
| ISBN: | 0821810219 |
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MARC
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| 245 | 1 | 0 | |a Hyperbolic partial differential equations and wave phenomena |c Mitsuru Ikawa |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2000 | |
| 300 | |a XXI, 190 S. | ||
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| 490 | 1 | |a Translations of mathematical monographs |v 189 | |
| 490 | 0 | |a Iwanami series in modern mathematics | |
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| 650 | 4 | |a Boundary value problems | |
| 650 | 4 | |a Differential equations, Hyperbolic | |
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Record in the Search Index
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| adam_text | Contents
Preface to the English Edition xi
Preface to the Japanese Edition xiii
Outline of the Theory and Objectives xvii
Nomenclature used in this book xx
Chapter 1. Wave Phenomena and Hyperbolic Equations 1
1.1 Equations of wave phenomena 1
(a) The vibration of a string 1
(b) The equation of oscillation of a spring 5
(c) The equation for the propagation of sound 8
(d) Maxwell s equations, elastic equations 12
1.2 Hyperbolic partial differential operators 16
(a) Hyperbolic differential operators 16
(b) Problems that we will consider 18
1.3 Formulae for solutions of an initial value problem for
a wave equation 19
Chapter summary 27
Exercises 27
Chapter 2. The Existence of a Solution for a Hyperbolic Equa¬
tion and its Properties 31
2.1 Finite propagation speed, domains of dependence
and influence 31
(a) Energy estimate of the solution 34
(b) The domain of dependence 44
(c) The domain of influence 46
(d) Finite speed of propagation 47
2.2 An a priori estimate of the solution 48
(a) On the premise of results for elliptic equations 48
vii
viii CONTENTS
(b) Energy estimate 51
(c) An estimate of partial derivatives of higher order 52
(d) Elimination of the extra assumption, approxima¬
tion due a mollifier 58
(e) Energy conservation 61
2.3 Existence of the solution 63
(a) The Hille Yosida theorem 63
(b) The operator A 67
(c) The existence of a solution for the initial boundary
value problem 75
(d) Smoothness of the solution 77
Chapter summary 81
Exercises 81
Chapter 3. The Construction of Asymptotic Solutions 83
3.1 An asymptotic solution of the hyperbolic equation 83
3.2 The Eikonal equation 86
(a) Canonical equations 87
(b) The construction of the solution 91
3.3 The transport equation and the behaviour of the as¬
ymptotic solution 97
(a) The transport equation 97
(b) The behaviour of the asymptotic solution 100
(c) The propagation of sound in air in which the tem¬
perature is not constant 102
3.4 The propagation of singularities 103
(a) The composition of a smooth function with a dis¬
tribution 104
(b) Propagation of discontinuities 107
(c) The propagation of a general singularity 112
(d) The solution for a discontinuous initial value 113
3.5 The asymptotic solution of the wave equation and
geometrical optics 116
(a) The properties of the solution of the Eikonal equa¬
tion 116
(b) The transport equation of the wave equation 118
(c) Curvature 120
(d) The change of variable and the Laplacian 123
(e) The value of Ap(x) 127
CONTENTS ix
(f) An expression of the solution of the transport
equation and the behaviour of the principal term 129
(g) The solution when the wave equation incorporates
friction 131
(h) Reflection in the wave equation 132
(i) Refraction 136
3.6 The asymptotic solution of an elastic wave 145
Chapter summary 150
Exercises 150
Chapter 4. Local Energy of the Wave Equation 153
4.1 Local energy 153
4.2 A uniform decay of the local energy 159
4.3 The exponential decay of the local energy 167
Chapter summary 172
Exercises 173
Perspectives on Current Research in Mathematics 175
Bibliography 179
Cited references 179
General references 179
Solutions to the Exercises 181
Index 189
|
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| id | DE-604.BV013265796 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:42:46Z |
| institution | BVB |
| isbn | 0821810219 |
| language | English Japanese |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009043680 |
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| physical | XXI, 190 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs Iwanami series in modern mathematics |
| spelling | Ikawa, Mitsuru Verfasser aut Hyperbolic partial differential equations and wave phenomena Mitsuru Ikawa Providence, RI American Math. Soc. 2000 XXI, 190 S. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 189 Iwanami series in modern mathematics Equacoes diferenciais parciais larpcal Boundary value problems Differential equations, Hyperbolic Wave equation Wellengleichung (DE-588)4065315-8 gnd rswk-swf Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Hyperbolische Differentialgleichung (DE-588)4131213-2 s Randwertproblem (DE-588)4048395-2 s Wellengleichung (DE-588)4065315-8 s DE-604 Translations of mathematical monographs 189 (DE-604)BV000002394 189 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009043680&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ikawa, Mitsuru Hyperbolic partial differential equations and wave phenomena Translations of mathematical monographs Equacoes diferenciais parciais larpcal Boundary value problems Differential equations, Hyperbolic Wave equation Wellengleichung (DE-588)4065315-8 gnd Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Randwertproblem (DE-588)4048395-2 gnd |
| subject_GND | (DE-588)4065315-8 (DE-588)4131213-2 (DE-588)4048395-2 |
| title | Hyperbolic partial differential equations and wave phenomena |
| title_auth | Hyperbolic partial differential equations and wave phenomena |
| title_exact_search | Hyperbolic partial differential equations and wave phenomena |
| title_full | Hyperbolic partial differential equations and wave phenomena Mitsuru Ikawa |
| title_fullStr | Hyperbolic partial differential equations and wave phenomena Mitsuru Ikawa |
| title_full_unstemmed | Hyperbolic partial differential equations and wave phenomena Mitsuru Ikawa |
| title_short | Hyperbolic partial differential equations and wave phenomena |
| title_sort | hyperbolic partial differential equations and wave phenomena |
| topic | Equacoes diferenciais parciais larpcal Boundary value problems Differential equations, Hyperbolic Wave equation Wellengleichung (DE-588)4065315-8 gnd Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Randwertproblem (DE-588)4048395-2 gnd |
| topic_facet | Equacoes diferenciais parciais Boundary value problems Differential equations, Hyperbolic Wave equation Wellengleichung Hyperbolische Differentialgleichung Randwertproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009043680&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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