Singular nonlinear partial differential equations:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Braunschweig [u.a.]
Vieweg
1996
|
| Schriftenreihe: | Aspects of mathematics / E
28 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | VIII, 269 S. |
| ISBN: | 3528066598 |
Internformat
MARC
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| 245 | 1 | 0 | |a Singular nonlinear partial differential equations |c Raymond Gérard ; Hidetoshi Tahara |
| 264 | 1 | |a Braunschweig [u.a.] |b Vieweg |c 1996 | |
| 300 | |a VIII, 269 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Aspects of mathematics / E |v 28 | |
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Datensatz im Suchindex
| _version_ | 1804125023870386176 |
|---|---|
| adam_text | VI
Contents
Preface V
1 Operators with regular singularities: One variable case 1
Introduction 1
1.1 Notations, definitions, examples 1
1.2 The good operators 4
1.3 A class of operators with a regular singularity 7
1.4 Applications to differential equations 19
1.4.1 Non linear differential equations 19
1.4.2 Some particular cases 20
1.5 The Maillet theorem 31
1.5.1 A generalization of theorem 1.3.5 31
1.5.2 Application: The theorem of E. Maillet 36
1.5.3 Some notes on the theorem of E. Maillet 40
2 Operators with regular singularities: Several variables case 42
Introduction 42
A Formal theory 42
2.1 Notations 42
2.2 Linear operators on C[[x]] 43
2.3 Non linear operators on 971/ 46
2.4 Solutions of linear equations 48
2.5 Solutions of non linear equations 51
2.5.1 Non linear equations with an affine part in (Op.l.f)btt • ¦ ¦ 51
2.5.2 Non linear equations without affine part 54
B Analytic theory 54
2.6 Notations and definitions 54
2.7 The good operators and the notion of domination 55
2.7.1 In (Op.l.f)t 55
2.7.2 In (Op.l.f)btt 56
2.7.3 In (Op.l.f)b,d 57
2.8 A class of operators having a regular singularity 57
2.8.1 In (Op.l.f)d 58
2.8.2 In (Op.l.f)b,d 63
2.8.3 In (Op.l.f)b,t 69
2.9 Applications to partial differential equations 70
2.9.1 With a Poincare vector field 70
2.9.2 The partial differential equations of the hypergeometric func¬
tions of P. Appell 70
VII
2.9.3 The systems of partial differential equations of the confluent
hypergeometric functions in two variables 73
2.9.4 The systems of partial differential equations of the hypergeo¬
metric functions in n variables of G. Lauricella 74
2.9.5 A theorem of M. Kashiwara, T. Kawai, and J. Sjostrand ... 76
3 Formal and convergent solutions of singular partial differential equa¬
tions 77
Introduction 77
3.1 Notations and definitions 77
3.2 Holomorphic solutions of certain equations 81
3.2.1 Solution of the linear equation 82
3.2.2 Solution of the non linear equation 88
3.3 Equations with parameters 92
3.4 An application: A theorem of S. Kaplan 96
3.5 The case of small denominators 100
4 Local study of differential equations of the form xy = f(x,y) near
z = 0 111
Introduction Ill
4.1 Coupling of two differential equations 111
4.2 Behavior of solutions of a differential equation near a regular point . 113
4.3 Local study of a differential equation near a singular point of regular
type 116
4.3.1 Singular point of regular type 116
4.3.2 The singularities of solutions of the Briot Bouquet equation . 119
4.4 Study of the Hukuhara equation and of the Hukuhara function . . . 135
5 Holomorphic and singular solutions of non linear singular first order
partial differential equations 138
Introduction 138
5.1 Notations and definitions 139
5.2 Statement of the main theorem 140
5.3 Holomorphic solutions 141
5.4 Singular solutions 147
5.5 Uniqueness of the solution 153
5.6 Proof of the main theorem 5.2.3 156
5.7 Remarks 159
5.8 Supplementary result 159
6 Maillet s type theorems for non linear singular partial differential
equations 161
Introduction 161
6.1 Implicit function theorem 163
6.2 Non linear equations with first order linear part 164
6.3 Non linear equations with higher order linear part 179
VIII
6.4 Formal Gevrey index for a particular type of equations — Examples 181
6.5 Supplementary results 184
7 Maillet s type theorems for non linear singular partial differential
equations without linear part 187
Introduction 187
7.1 Notations and definitions 187
7.2 Assumptions and results 189
7.3 A basic lemma 192
7.4 Proof of theorem 7.2.5 195
7.5 Complementary results 198
7.6 A remark 200
8 Holomorphic and singular solutions of non linear singular partial
differential equations 203
Introduction 203
8.1 Holomorphic solutions 206
8.2 Singular solutions: Special case 208
8.3 Singular solutions: General case 213
8.4 Asymptotic study 221
8.5 Completion of the proof of the main theorem 228
9 On the existence of holomorphic solutions of the Cauchy problem
for non linear partial differential equations 234
Introduction 234
9.1 Notations and definitions 234
9.2 Results 236
9.3 Proof of theorem 9.2.1 238
9.4 Proof of theorem 9.2.3 242
lOMaillet s type theorems for non linear singular integro difFerential
equations 246
Introduction 246
10.1 Notations and definitions 246
10.2 The main theorem 248
10.3 Construction of the formal solution 250
10.4 Some discussions 253
10.5 Convergence of the formal solution in the case si = 1 258
10.6 Convergence of the formal solution in the case s; 1 261
10.7 Supplementary results and remark 263
Bibliography 264
Index 268
|
| any_adam_object | 1 |
| author | Gérard, Raymond Tahara, Hidetoshi |
| author_facet | Gérard, Raymond Tahara, Hidetoshi |
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| indexdate | 2024-07-09T17:54:58Z |
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| isbn | 3528066598 |
| language | English |
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| physical | VIII, 269 S. |
| publishDate | 1996 |
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| series2 | Aspects of mathematics / E |
| spelling | Gérard, Raymond Verfasser aut Singular nonlinear partial differential equations Raymond Gérard ; Hidetoshi Tahara Braunschweig [u.a.] Vieweg 1996 VIII, 269 S. txt rdacontent n rdamedia nc rdacarrier Aspects of mathematics / E 28 Niet-lineaire vergelijkingen gtt Partiële differentiaalvergelijkingen gtt Differential equations, Nonlinear Numerical solutions Singuläre Gleichung (DE-588)4181517-8 gnd rswk-swf Reihenlösung (DE-588)4406554-1 gnd rswk-swf Holomorphe Lösung (DE-588)4406551-6 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Formale Potenzreihe (DE-588)4204495-9 gnd rswk-swf Singuläre Lösung (DE-588)4290969-7 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s Singuläre Gleichung (DE-588)4181517-8 s DE-604 Formale Potenzreihe (DE-588)4204495-9 s Reihenlösung (DE-588)4406554-1 s Singuläre Lösung (DE-588)4290969-7 s Holomorphe Lösung (DE-588)4406551-6 s Tahara, Hidetoshi Verfasser aut E Aspects of mathematics 28 (DE-604)BV000018737 28 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007036211&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gérard, Raymond Tahara, Hidetoshi Singular nonlinear partial differential equations Niet-lineaire vergelijkingen gtt Partiële differentiaalvergelijkingen gtt Differential equations, Nonlinear Numerical solutions Singuläre Gleichung (DE-588)4181517-8 gnd Reihenlösung (DE-588)4406554-1 gnd Holomorphe Lösung (DE-588)4406551-6 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd Formale Potenzreihe (DE-588)4204495-9 gnd Singuläre Lösung (DE-588)4290969-7 gnd |
| subject_GND | (DE-588)4181517-8 (DE-588)4406554-1 (DE-588)4406551-6 (DE-588)4128900-6 (DE-588)4204495-9 (DE-588)4290969-7 |
| title | Singular nonlinear partial differential equations |
| title_auth | Singular nonlinear partial differential equations |
| title_exact_search | Singular nonlinear partial differential equations |
| title_full | Singular nonlinear partial differential equations Raymond Gérard ; Hidetoshi Tahara |
| title_fullStr | Singular nonlinear partial differential equations Raymond Gérard ; Hidetoshi Tahara |
| title_full_unstemmed | Singular nonlinear partial differential equations Raymond Gérard ; Hidetoshi Tahara |
| title_short | Singular nonlinear partial differential equations |
| title_sort | singular nonlinear partial differential equations |
| topic | Niet-lineaire vergelijkingen gtt Partiële differentiaalvergelijkingen gtt Differential equations, Nonlinear Numerical solutions Singuläre Gleichung (DE-588)4181517-8 gnd Reihenlösung (DE-588)4406554-1 gnd Holomorphe Lösung (DE-588)4406551-6 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd Formale Potenzreihe (DE-588)4204495-9 gnd Singuläre Lösung (DE-588)4290969-7 gnd |
| topic_facet | Niet-lineaire vergelijkingen Partiële differentiaalvergelijkingen Differential equations, Nonlinear Numerical solutions Singuläre Gleichung Reihenlösung Holomorphe Lösung Nichtlineare partielle Differentialgleichung Formale Potenzreihe Singuläre Lösung |
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