An introduction to nonlinear partial differential equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hoboken, NJ
Wiley
2008
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Pure and applied mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 397 S. graph. Darst. |
| ISBN: | 9780470225950 |
Internformat
MARC
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| 020 | |a 9780470225950 |9 978-0-470-22595-0 | ||
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| 100 | 1 | |a Logan, J. David |d 1944- |e Verfasser |0 (DE-588)120361000 |4 aut | |
| 245 | 1 | 0 | |a An introduction to nonlinear partial differential equations |c J. David Logan |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Hoboken, NJ |b Wiley |c 2008 | |
| 300 | |a XIII, 397 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Pure and applied mathematics | |
| 650 | 4 | |a Ecuaciones diferenciales no lineales | |
| 650 | 4 | |a Ecuaciones diferenciales parciales | |
| 650 | 0 | 7 | |a Nichtlineare partielle Differentialgleichung |0 (DE-588)4128900-6 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016538229 | ||
Datensatz im Suchindex
| _version_ | 1804137714186977280 |
|---|---|
| adam_text | Contents
Preface
......................................................... xi
1.
Introduction
to
Partial
Differential
Equations
............... 1
1.1
Partial
Differential
Equations
.............................. 2
1.1.1
Equations and Solutions
............................. 2
1.1.2
Classification
...................................... 5
1.1.3
Linear
versus
Nonlinear
............................. 8
1.1.4
Linear
Equations
................................... 11
1.2
Conservation
Laws
....................................... 20
1.2.1
One Dimension
.................................... 20
1.2.2
Higher
Dimensions
................................. 23
1.3
Constitutive Relations
.................................... 25
1.4
Initial
and Boundary
Value
Problems
....................... 35
1.5
Waves
.................................................. 45
1.5.1
Traveling
Waves
.................................... 45
1.5.2
Plane Waves
....................................... 50
1.5.3
Plane Waves
and Transforms
........................ 52
1.5.4
Nonlinear
Dispersion
................................ 54
2.
First-Order
Equations
and Characteristics
.................. 61
2.1
Linear First-Order
Equations
.............................. 62
2.1.1
Advection Equation
................................ 62
2.1.2
Variable Coefficients
................................ 64
2.2
Nonlinear
Equations
...................................... 68
2.3
Quasilinear Equations
..................................... 72
2.3.1
The General Solution
............................... 76
viü Contents
2.4
Propagation
of Singularities
................................ 81
2.5
General First-Order Equation
.............................. 86
2.5.1
Complete Integral
.................................. 91
2.6
A Uniqueness Result
...................................... 94
2.7
Models in Biology
........................................ 96
2.7.1
Age Structure
...................................... 96
2.7.2
Structured Predator-Prey Model
.....................101
2.7.3
Chemotherapy
.....................................103
2.7.4
Mass Structure
.....................................105
2.7.5
Size-Dependent
Prédation
...........................106
3.
Weak Solutions to Hyperbolic Equations
...................113
3.1
Discontinuous Solutions
...................................114
3.2
Jump Conditions
.........................................116
3.2.1
Rarefaction Waves
..................................118
3.2.2
Shock Propagation
.................................119
3.3
Shock Formation
.........................................125
3.4
Applications
.............................................131
3.4.1
Traffic Flow
.......................................132
3.4.2
Plug Flow Chemical Reactors
........................136
3.5
Weak Solutions: A Formal Approach
........................140
3.6
Asymptotic Behavior of Shocks
.............................148
3.6.1
Equal-Area Principle
...............................148
3.6.2
Shock Fitting
......................................152
3.6.3
Asymptotic Behavior
...............................154
4.
Hyperbolic Systems
........................................159
4.1
Shallow-Water Waves; Gas Dynamics
.......................160
4.1.1
Shallow-Water Waves
...............................160
4.1.2
Small-Amplitude Approximation
.....................163
4.1.3
Gas Dynamics
.....................................164
4.2
Hyperbolic Systems and Characteristics
.....................169
4.2.1
Classification
......................................170
4.3
The Riemann Method
.....................................179
4.3.1
Jump Conditions for Systems
........................179
4.3.2
Breaking Dam Problem
.............................181
4.3.3
Receding Wall Problem
.............................183
4.3.4
Formation of a Bore
................................187
4.3.5
Gas Dynamics
.....................................190
4.4
Hodographs and Wavefronts
...............................192
4.4.1
Hodograph Transformation
..........................192
4.4.2
Wavefront
Expansions
..............................193
Contents ix
4.5
Weakly Nonlinear Approximations
..........................201
4.5.1
Derivation of Burgers Equation
......................202
5.
Diffusion Processes
.........................................209
5.1
Diffusion and Random Motion
.............................210
5.2
Similarity Methods
.......................................217
5.3
Nonlinear Diffusion Models
................................224
5.4
Reaction- Diffusion; Fisher s Equation
.......................234
5.4.1
Traveling Wave Solutions
............................235
5.4.2
Perturbation Solution
...............................238
5.4.3
Stability of Traveling Waves
.........................240
5.4.4
Nagumo s Equation
.................................242
5.5
Advection-Diffusion; Burgers Equation
.....................245
5.5.1
Traveling Wave Solution
.............................246
5.5.2
Initial Value Problem
...............................247
5.6
Asymptotic Solution to Burgers Equation
...................250
5.6.1
Evolution of a Point Source
..........................252
Appendix: Dynamical Systems
..................................257
6.
Reaction—Diffusion Systems
.................................267
6.1
Reaction-Diffusion Models
.................................268
6.1.1
Predator-Prey Model
...............................270
6.1.2
Combustion
.......................................271
6.1.3
Chemotaxis
........................................274
6.2
Traveling Wave Solutions
..................................277
6.2.1
Model for the Spread of a Disease
....................278
6.2.2
Contaminant Transport in Groundwater
...............284
6.3
Existence of Solutions
.....................................292
6.3.1
Fixed-Point Iteration
...............................293
6.3.2
Semilinear Equations
...............................297
6.3.3
Normed Linear Spaces
..............................300
6.3.4
General Existence Theorem
..........................303
6.4
Maximum Principles and Comparison Theorems
..............309
6.4.1
Maximum Principles
................................309
6.4.2
Comparison Theorems
..............................314
6.5
Energy Estimates and Asymptotic Behavior
.................317
6.5.1
Calculus Inequalities
................................318
6.5.2
Energy Estimates
..................................320
6.5.3
Invariant Sets
......................................326
6.6
Pattern Formation
........................................333
x
Contents
7.
Equilibrium Models
........................................345
7.1
Elliptic Models
...........................................346
7.2
Theoretical Results
.......................................352
7.2.1
Maximum Principle
.................................353
7.2.2
Existence Theorem
.................................355
7.3
Eigenvalue Problems
......................................358
7.3.1
Linear Eigenvalue Problems
.........................358
7.3.2
Nonlinear Eigenvalue Problems
.......................361
7.4
Stability and Bifurcation
..................................364
7.4.1
Ordinary Differential Equations
......................364
7.4.2
Partial Differential Equations
........................368
References
......................................................387
Index
...........................................................395
|
| adam_txt |
Contents
Preface
. xi
1.
Introduction
to
Partial
Differential
Equations
. 1
1.1
Partial
Differential
Equations
. 2
1.1.1
Equations and Solutions
. 2
1.1.2
Classification
. 5
1.1.3
Linear
versus
Nonlinear
. 8
1.1.4
Linear
Equations
. 11
1.2
Conservation
Laws
. 20
1.2.1
One Dimension
. 20
1.2.2
Higher
Dimensions
. 23
1.3
Constitutive Relations
. 25
1.4
Initial
and Boundary
Value
Problems
. 35
1.5
Waves
. 45
1.5.1
Traveling
Waves
. 45
1.5.2
Plane Waves
. 50
1.5.3
Plane Waves
and Transforms
. 52
1.5.4
Nonlinear
Dispersion
. 54
2.
First-Order
Equations
and Characteristics
. 61
2.1
Linear First-Order
Equations
. 62
2.1.1
Advection Equation
. 62
2.1.2
Variable Coefficients
. 64
2.2
Nonlinear
Equations
. 68
2.3
Quasilinear Equations
. 72
2.3.1
The General Solution
. 76
viü Contents
2.4
Propagation
of Singularities
. 81
2.5
General First-Order Equation
. 86
2.5.1
Complete Integral
. 91
2.6
A Uniqueness Result
. 94
2.7
Models in Biology
. 96
2.7.1
Age Structure
. 96
2.7.2
Structured Predator-Prey Model
.101
2.7.3
Chemotherapy
.103
2.7.4
Mass Structure
.105
2.7.5
Size-Dependent
Prédation
.106
3.
Weak Solutions to Hyperbolic Equations
.113
3.1
Discontinuous Solutions
.114
3.2
Jump Conditions
.116
3.2.1
Rarefaction Waves
.118
3.2.2
Shock Propagation
.119
3.3
Shock Formation
.125
3.4
Applications
.131
3.4.1
Traffic Flow
.132
3.4.2
Plug Flow Chemical Reactors
.136
3.5
Weak Solutions: A Formal Approach
.140
3.6
Asymptotic Behavior of Shocks
.148
3.6.1
Equal-Area Principle
.148
3.6.2
Shock Fitting
.152
3.6.3
Asymptotic Behavior
.154
4.
Hyperbolic Systems
.159
4.1
Shallow-Water Waves; Gas Dynamics
.160
4.1.1
Shallow-Water Waves
.160
4.1.2
Small-Amplitude Approximation
.163
4.1.3
Gas Dynamics
.164
4.2
Hyperbolic Systems and Characteristics
.169
4.2.1
Classification
.170
4.3
The Riemann Method
.179
4.3.1
Jump Conditions for Systems
.179
4.3.2
Breaking Dam Problem
.181
4.3.3
Receding Wall Problem
.183
4.3.4
Formation of a Bore
.187
4.3.5
Gas Dynamics
.190
4.4
Hodographs and Wavefronts
.192
4.4.1
Hodograph Transformation
.192
4.4.2
Wavefront
Expansions
.193
Contents ix
4.5
Weakly Nonlinear Approximations
.201
4.5.1
Derivation of Burgers' Equation
.202
5.
Diffusion Processes
.209
5.1
Diffusion and Random Motion
.210
5.2
Similarity Methods
.217
5.3
Nonlinear Diffusion Models
.224
5.4
Reaction- Diffusion; Fisher's Equation
.234
5.4.1
Traveling Wave Solutions
.235
5.4.2
Perturbation Solution
.238
5.4.3
Stability of Traveling Waves
.240
5.4.4
Nagumo's Equation
.242
5.5
Advection-Diffusion; Burgers' Equation
.245
5.5.1
Traveling Wave Solution
.246
5.5.2
Initial Value Problem
.247
5.6
Asymptotic Solution to Burgers" Equation
.250
5.6.1
Evolution of a Point Source
.252
Appendix: Dynamical Systems
.257
6.
Reaction—Diffusion Systems
.267
6.1
Reaction-Diffusion Models
.268
6.1.1
Predator-Prey Model
.270
6.1.2
Combustion
.271
6.1.3
Chemotaxis
.274
6.2
Traveling Wave Solutions
.277
6.2.1
Model for the Spread of a Disease
.278
6.2.2
Contaminant Transport in Groundwater
.284
6.3
Existence of Solutions
.292
6.3.1
Fixed-Point Iteration
.293
6.3.2
Semilinear Equations
.297
6.3.3
Normed Linear Spaces
.300
6.3.4
General Existence Theorem
.303
6.4
Maximum Principles and Comparison Theorems
.309
6.4.1
Maximum Principles
.309
6.4.2
Comparison Theorems
.314
6.5
Energy Estimates and Asymptotic Behavior
.317
6.5.1
Calculus Inequalities
.318
6.5.2
Energy Estimates
.320
6.5.3
Invariant Sets
.326
6.6
Pattern Formation
.333
x
Contents
7.
Equilibrium Models
.345
7.1
Elliptic Models
.346
7.2
Theoretical Results
.352
7.2.1
Maximum Principle
.353
7.2.2
Existence Theorem
.355
7.3
Eigenvalue Problems
.358
7.3.1
Linear Eigenvalue Problems
.358
7.3.2
Nonlinear Eigenvalue Problems
.361
7.4
Stability and Bifurcation
.364
7.4.1
Ordinary Differential Equations
.364
7.4.2
Partial Differential Equations
.368
References
.387
Index
.395 |
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| author | Logan, J. David 1944- |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| id | DE-604.BV023354669 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:06:32Z |
| indexdate | 2024-07-09T21:16:41Z |
| institution | BVB |
| isbn | 9780470225950 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016538229 |
| oclc_num | 427504970 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-29T DE-20 DE-91G DE-BY-TUM DE-824 DE-19 DE-BY-UBM |
| owner_facet | DE-355 DE-BY-UBR DE-29T DE-20 DE-91G DE-BY-TUM DE-824 DE-19 DE-BY-UBM |
| physical | XIII, 397 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Wiley |
| record_format | marc |
| series2 | Pure and applied mathematics |
| spelling | Logan, J. David 1944- Verfasser (DE-588)120361000 aut An introduction to nonlinear partial differential equations J. David Logan 2. ed. Hoboken, NJ Wiley 2008 XIII, 397 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics Ecuaciones diferenciales no lineales Ecuaciones diferenciales parciales Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016538229&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Logan, J. David 1944- An introduction to nonlinear partial differential equations Ecuaciones diferenciales no lineales Ecuaciones diferenciales parciales Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| subject_GND | (DE-588)4128900-6 |
| title | An introduction to nonlinear partial differential equations |
| title_auth | An introduction to nonlinear partial differential equations |
| title_exact_search | An introduction to nonlinear partial differential equations |
| title_exact_search_txtP | An introduction to nonlinear partial differential equations |
| title_full | An introduction to nonlinear partial differential equations J. David Logan |
| title_fullStr | An introduction to nonlinear partial differential equations J. David Logan |
| title_full_unstemmed | An introduction to nonlinear partial differential equations J. David Logan |
| title_short | An introduction to nonlinear partial differential equations |
| title_sort | an introduction to nonlinear partial differential equations |
| topic | Ecuaciones diferenciales no lineales Ecuaciones diferenciales parciales Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| topic_facet | Ecuaciones diferenciales no lineales Ecuaciones diferenciales parciales Nichtlineare partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016538229&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT loganjdavid anintroductiontononlinearpartialdifferentialequations |