Partial differential equations: an introduction
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English German |
| Published: |
Stuttgart
Teubner
1977
|
| Edition: | 2. ed. |
| Series: | Mathematische Leitfäden
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XI, 259 S. Ill., graph. Darst. |
| ISBN: | 3519122138 |
Staff View
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| 240 | 1 | 0 | |a Partielle Differentialgleichungen |
| 245 | 1 | 0 | |a Partial differential equations |b an introduction |c by Günter Hellwig |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Stuttgart |b Teubner |c 1977 | |
| 300 | |a XI, 259 S. |b Ill., graph. Darst. | ||
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| 500 | |a Literaturangaben | ||
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
Part 7. Examples
1. INTRODUCTION
1.1 Definitions 3
1.2 The Gauss integral theorem 4
1.3 Vector fields 5
1.4 The Green formulas 6
1.5 The Maxwell equations 7
1.6 The equations of gas dynamics 8
1.7 The heat equation 10
2. THE WAVE EQUATION
2.1 The wave equation in Ri 11
2.2 Domain of dependence; domain of detenninateness 13
2.3 The initial boundary problem 14
2.4 The wave equation in Ri 15
2.5 Finding the solution 17
2.6 The domains a, (B for the wave equation in R.t 19
2.7 The wave equation in Ri 20
2.8 The domains fl, (B for the wave equation in R 21
2.9 Dependence of the wave equation on dimension 21
2.10 Continuable initial conditions; determinism in nature 23
2.11 Waveforms 24
2.12 An initial boundary value problem in Rs 25
3. THE POTENTIAL EQUATION
3.1 The initial value problem for the potential equation 28
3.2 Singularity functions 29
3.3 The fundamental solution 30
3.4 Green s function of the first kind 31
3.5 Poisson s formula 32
3.6 The existence of the Green s function in R2 35
3.7 The mean value and the maximum minimum properties 36
viii Contents
3.8 Harnack s inequality 39
3.9 H. Weyl s lemma in the simplest case 40
3.10 Mean value properties for Anu = / and A3u + k2u = / 41
4. THE HEAT EQUATION
4.1 The existence theorem for the initial value problem 43
4.2 The uniqueness theorem for the initial value problem 47
4.3 Counterexamples 49
4.4 Remarks 51
4.5 Initial boundary value problems 52
Part 2. Classification into Types, Theory of
Characteristics, and Normal Form
1. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
1.1 Classification into types 59
1.2 Invariance properties of C 61
1.3 Characteristic directions 62
1.4 Normal form in the hyperbolic case for n = 2 63
1.5 Normal form in the elliptic case for n = 2 64
1.6 Normal form in the parabolic case for n = 2 67
1.7 Differential equations of mixed type for n = 2 68
2. SYSTEMS OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
2.1 Hyperbolic systems for two independent variables 69
2.2 Characteristic manifolds; normal form 71
2.3 Normal form for the quasi linear case 72
2.4 Theory of characteristics for general systems 75
2.5 Classification into types for simple systems 77
2.6 Normal form for elliptic systems 77
3. ON THE NECESSITY OF CLASSIFICATION INTO TYPES
3.1 The existence theorem of Cauchy—S. Kowalewski 81
3.2 The example of O. Perron 83
Part 3. Questions of Uniqueness
1. ELLIPTIC AND ELLIPTIC PARABOLIC TYPE
1.1 The maximum minimum principle 89
1.2 The energy integral method 93
Contents ix
1.3 Treatment of existence problems by means of the maximum minimum
principle 94
1.4 A priori estimates 95
1.5 The analyticity of the harmonic functions 96
2. PARABOLIC TYPE
2.1 The maximum minimum principle 99
2.2 Counterexample 100
3. HYPERBOLIC TYPE
3.1 The energy integral method for the wave equation 102
3.2 The energy integral method for general systems 103
3.3 The radiation problem 107
3.4 Proof of F. Rellich s first lemma 110
3.5 The radiation problem for the whole space 113
4. MIXED TYPE
4.1 The energy integral method for equations of elliptic parabolic hyperbolic type 115
4.2 The maximum minimum principle for equations of elliptic parabolic
hyperbolic type 117
4.3 Remarks 120
Part 4. Questions of Existence
1. EQUATIONS OF HYPERBOLIC TYPE IN TWO INDEPENDENT
VARIABLES
1.1 The initial value problem for linear systems in two unknown functions 125
1.2 Supplements 130
1.3 The characteristic initial value problem 133
1.4 The initial value problem for quasi linear systems 136
1.5 Proof of the lemma 139
1.6 Hyperbolic systems in the form of conservation theorems 145
1.7 Riemann s method 146
1.8 An example 149
2. BOUNDARY AND INITIAL VALUE PROBLEMS FOR EQUATIONS OF
HYPERBOLIC AND PARABOLIC TYPE IN TWO INDEPENDENT
VARIABLES
2.1 Posing of the problem 151
2.2 The calculus of the Laplace transform 152
2.3 Solution of the transformed problem II 154
x Contents
2.4 Justification of the calculus 157
2.5 Auxiliary considerations 163
2.6 The formal calculus of Laplace transforms 168
3. EQUATIONS OF ELLIPTIC TYPE
3.1 Estimates for potentials 172
3.2 A solution of A,,« = f(x) 173
3.3 Formulation of the general boundary value problem 176
3.4 Outline of proof and notations 177
3.5 Existence of a W^ solution 179
3.6 Differentiability of the ^ solution 182
3.7 Continuous assumption of the boundary values 183
3.8 Tools 190
4. WEYL S LEMMA FOR EQUATIONS OF ELLIPTIC TYPE
4.1 Singular integrals 195
4.2 Weyl s lemma 199
Part 5. Simple Tools from Functional Analysis
Applied to Questions of Existence
1. AUXILIARY TOOLS
1.1 Banach space 209
1.2 Hilbert space 210
1.3 Bounded linear functionals in Hilbert space 214
2. SCHAUDER S TECHNIQUE OF PROOF FOR EXISTENCE PROBLEMS
IN ELLIPTIC DIFFERENTIAL EQUATIONS
2.1 Posing the problem 218
2.2 Outline of proof 219
3. THE REGULAR EIGENVALUE PROBLEM
3.1 Posing the problem 222
3.2 Equivalent formulation of the problem 222
3.3 Complete continuity of the operator 226
3.4 The expansion theorem 229
4. ELLIPTIC SYSTEMS OF DIFFERENTIAL EQUATIONS
4.1 Posing the problem 232
4.2 The Green s function of the second kind 232
Contents xi
4.3 Hilbert s lemma 235
4.4 Equivalent formulations of the problem 237
4.5 The homogeneous first boundary value problem 240
4.6 The inhomogeneous first boundary value problem 242
4.7 The general boundary value problem with characteristic zero 244
4.8 The general boundary value problem with arbitrary integer characteristic 246
solutions 251
bibliography 256
INDEX 257
I
|
| any_adam_object | 1 |
| author | Hellwig, Günter 1926- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 |
| dewey-search | 515/.353 |
| dewey-sort | 3515 3353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| genre_facet | Einführung |
| id | DE-604.BV001975319 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T15:38:13Z |
| institution | BVB |
| isbn | 3519122138 |
| language | English German |
| lccn | 78360245 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001288373 |
| oclc_num | 4329332 |
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| physical | XI, 259 S. Ill., graph. Darst. |
| publishDate | 1977 |
| publishDateSearch | 1977 |
| publishDateSort | 1977 |
| publisher | Teubner |
| record_format | marc |
| series2 | Mathematische Leitfäden |
| spelling | Hellwig, Günter 1926- Verfasser (DE-588)172133033 aut Partielle Differentialgleichungen Partial differential equations an introduction by Günter Hellwig 2. ed. Stuttgart Teubner 1977 XI, 259 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematische Leitfäden Literaturangaben Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001288373&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hellwig, Günter 1926- Partial differential equations an introduction Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4151278-9 |
| title | Partial differential equations an introduction |
| title_alt | Partielle Differentialgleichungen |
| title_auth | Partial differential equations an introduction |
| title_exact_search | Partial differential equations an introduction |
| title_full | Partial differential equations an introduction by Günter Hellwig |
| title_fullStr | Partial differential equations an introduction by Günter Hellwig |
| title_full_unstemmed | Partial differential equations an introduction by Günter Hellwig |
| title_short | Partial differential equations |
| title_sort | partial differential equations an introduction |
| title_sub | an introduction |
| topic | Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Differential equations, Partial Partielle Differentialgleichung Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001288373&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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