Differential equations: an introduction to basic concepts, results and applications
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey [u.a.]
World Scientific
2004
|
| Ausgabe: | Ioan I. Vrabie |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 401 S. graph. Darst. |
| ISBN: | 9812388389 |
Internformat
MARC
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| 245 | 1 | 0 | |a Differential equations |b an introduction to basic concepts, results and applications |
| 250 | |a Ioan I. Vrabie | ||
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2004 | |
| 300 | |a XVI, 401 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Datensatz im Suchindex
| _version_ | 1804135136301678592 |
|---|---|
| adam_text | Contents
Preface vii
List of Symbols
xi
1.
Generalities
1
1.1
Brief History
.......................... 1
1.1.1
The Birth of the Discipline
............... 1
1.1.2
Major Themes
...................... 3
1.2
Introduction
........................... 11
1.3
Elementary Equations
..................... 19
1.3.1
Equations with Separable Variables
.......... 19
1.3.2
Linear Equations
.................... 20
1.3.3
Homogeneous Equations
................ 21
1.3.4
Bernoulli Equations
................... 23
1.3.5
Riccati Equations
.................... 24
1.3.6
Exact Differential Equations
.............. 24
1.3.7
Equations Reducible to Exact Differential Equations
25
1.3.8 Lagrange
Equations
................... 26
1.3.9
Clairaut Equations
................... 27
1.3.10
Higher-Order Differential Equations
.......... 28
1.4
Some Mathematical Models
.................. 30
1.4.1
Radioactive Disintegration
............... 31
1.4.2
The Carbon Dating Method
.............. 31
1.4.3
Equations of Motion
.................. 32
1.4.4
The Harmonic Oscillator
................ 32
1.4.5
The Mathematical Pendulum
............. 33
xiv Contents
1.4.6
Two
Demographic Models............... 34
1.4.7
A Spatial
Model in
Ecology..............
36
1.4.8
The Prey-Predator
Model............... 36
1.4.9
The Spreading of a Disease
.............. 39
1.4.10
Lotka
Model
....................... 41
1.4.11
An Autocatalytic Generation Model
......... 42
1.4.12
An RLC Circuit Model
................ 43
1.5
Integral Inequalities
...................... 45
1.6
Exercises and Problems
.................... 47
2.
The Cauchy Problem
51
2.1
General Presentation
...................... 51
2.2
The Local Existence Problem
................. 57
2.3
The Uniqueness Problem
.................... 61
2.3.1
The Locally Lipschitz Case
.............. 62
2.3.2
The Dissipative Case
.................. 64
2.4
Saturated Solutions
....................... 66
2.4.1
Characterization of Continuable Solutions
...... 66
2.4.2
The Existence of Saturated Solutions
......... 68
2.4.3
Types of Saturated Solutions
............. 69
2.4.4
The Existence of Global Solutions
........... 73
2.5
Continuous Dependence on Data and Parameters
...... 75
2.5.1
The Dissipative Case
.................. 78
2.5.2
The Locally Lipschitz Case
.............. 80
2.5.3
Continuous Dependence on Parameters
........ 83
2.6
Problems of Differentiability
.................. 84
2.6.1
Differentiability with Respect to the Data
...... 85
2.6.2
Differentiability with Respect to the Parameters
... 89
2.7
The Cauchy Problem for the nth-Order Differential Equation
93
2.8
Exercises and Problems
.................... 95
3.
Approximation Methods
99
3.1
Power Series Method
...................... 99
3.1.1
An Example
....................... 99
3.1.2
The Existence of Analytic Solutions
.......... 101
3.2
The Successive Approximations Method
........... 105
3.3
The Method of Polygonal Lines
................ 109
3.4
Euler
Implicit Method. Exponential Formula
........ 113
Contents xv
3.4.1
The Semigroup Generated by
Л
............ 114
3.4.2
Two Auxiliary Lemmas
................. 115
3.4.3
The Exponential Formula
............... 118
3.5
Exercises and Problems
.................... 121
4.
Systems of Linear Differential Equations
125
4.1
Homogeneous Systems. The Space of Solutions
....... 125
4.2
Non-homogeneous Systems.
Variation of Constants Formula
................ 134
4.3
The Exponential of a Matrix
................. 137
4.4
A Method to Find etA
..................... 142
4.5
The nth-Order Linear Differential Equation
......... 145*
4.6
The nth-order Linear Differential Equation
with Constants Coefficients
.................. 150
4.7
Exercises and Problems
.................... 154
5.
Elements of Stability
159
5.1
Types of Stability
........................ 159
5.2
Stability of Linear Systems
.................. 165
5.3
The Case of Perturbed Systems
................ 173
5.4
The Lyapunov Function Method
............... 178
5.5
The Case of Dissipative Systems
............... 186
5.6
The Case of Controlled Systems
................ 192
5.7
Unpredictability and Chaos
.................. 196
5.8
Exercises and Problems
.................... 203
6.
Prime Integrals
207
6.1
Prime Integrals for Autonomous Systems
.......... 207
6.2
Prime Integrals for Non-Autonomous Systems
........ 217
6.3
First Order Partial Differential Equations
.......... 219
6.4
The Cauchy Problem for Quasi-Linear Equations
...... 223
6.5
Conservation Laws
....................... 227
6.5.1
Some Examples
..................... 227
6.5.2
A Local Existence and Uniqueness Result
...... 230
6.5.3
Weak Solutions
..................... 231
6.6
Exercises and Problems
.................... 240
xvi Contents
7.
Extensions
and Generalizations
245
7.1
Distributions of One Variable
................. 245
7.2
The Convolution Product
................... 254
7.3
Generalized Solutions
..................... 258
7.4
Carathéodory
Solutions
.................... 265
7.5
Differential Inclusions
..................... 271
7.6
Variational Inequalities
..................... 280
7.7
Problems of Viability
...................... 287
7.8
Proof of Nagumo s Viability Theorem
............ 290
7.9
Sufficient Conditions for
Invariance
.............. 295
7.10
Necessary Conditions for
Invariance
............. 299
7.11
Gradient Systems. Probenius Theorem
............ 303
7.12
Exercises and Problems
.................... 310
8.
Auxiliary Results
313
8.1
Elements of Vector Analysis
.................. 313
8.2
Compactness in C([
а,Ъ];Шп)
................. 319
8.3
The Projection of
а
Point on
а
Convex Set
.......... 324
Solutions
329
Chapter
1............................... 329
Chapter
2............................... 339
Chapter
3............................... 349
Chapter
4............................... 358
Chapter
5............................... 366
Chapter
6............................... 375
Chapter
7............................... 387
Bibliography
393
Index
397
|
| adam_txt |
Contents
Preface vii
List of Symbols
xi
1.
Generalities
1
1.1
Brief History
. 1
1.1.1
The Birth of the Discipline
. 1
1.1.2
Major Themes
. 3
1.2
Introduction
. 11
1.3
Elementary Equations
. 19
1.3.1
Equations with Separable Variables
. 19
1.3.2
Linear Equations
. 20
1.3.3
Homogeneous Equations
. 21
1.3.4
Bernoulli Equations
. 23
1.3.5
Riccati Equations
. 24
1.3.6
Exact Differential Equations
. 24
1.3.7
Equations Reducible to Exact Differential Equations
25
1.3.8 Lagrange
Equations
. 26
1.3.9
Clairaut Equations
. 27
1.3.10
Higher-Order Differential Equations
. 28
1.4
Some Mathematical Models
. 30
1.4.1
Radioactive Disintegration
. 31
1.4.2
The Carbon Dating Method
. 31
1.4.3
Equations of Motion
. 32
1.4.4
The Harmonic Oscillator
. 32
1.4.5
The Mathematical Pendulum
. 33
xiv Contents
1.4.6
Two
Demographic Models. 34
1.4.7
A Spatial
Model in
Ecology.
36
1.4.8
The Prey-Predator
Model. 36
1.4.9
The Spreading of a Disease
. 39
1.4.10
Lotka
Model
. 41
1.4.11
An Autocatalytic Generation Model
. 42
1.4.12
An RLC Circuit Model
. 43
1.5
Integral Inequalities
. 45
1.6
Exercises and Problems
. 47
2.
The Cauchy Problem
51
2.1
General Presentation
. 51
2.2
The Local Existence Problem
. 57
2.3
The Uniqueness Problem
. 61
2.3.1
The Locally Lipschitz Case
. 62
2.3.2
The Dissipative Case
. 64
2.4
Saturated Solutions
. 66
2.4.1
Characterization of Continuable Solutions
. 66
2.4.2
The Existence of Saturated Solutions
. 68
2.4.3
Types of Saturated Solutions
. 69
2.4.4
The Existence of Global Solutions
. 73
2.5
Continuous Dependence on Data and Parameters
. 75
2.5.1
The Dissipative Case
. 78
2.5.2
The Locally Lipschitz Case
. 80
2.5.3
Continuous Dependence on Parameters
. 83
2.6
Problems of Differentiability
. 84
2.6.1
Differentiability with Respect to the Data
. 85
2.6.2
Differentiability with Respect to the Parameters
. 89
2.7
The Cauchy Problem for the nth-Order Differential Equation
93
2.8
Exercises and Problems
. 95
3.
Approximation Methods
99
3.1
Power Series Method
. 99
3.1.1
An Example
. 99
3.1.2
The Existence of Analytic Solutions
. 101
3.2
The Successive Approximations Method
. 105
3.3
The Method of Polygonal Lines
. 109
3.4
Euler
Implicit Method. Exponential Formula
. 113
Contents xv
3.4.1
The Semigroup Generated by
Л
. 114
3.4.2
Two Auxiliary Lemmas
. 115
3.4.3
The Exponential Formula
. 118
3.5
Exercises and Problems
. 121
4.
Systems of Linear Differential Equations
125
4.1
Homogeneous Systems. The Space of Solutions
. 125
4.2
Non-homogeneous Systems.
Variation of Constants Formula
. 134
4.3
The Exponential of a Matrix
. 137
4.4
A Method to Find etA
. 142
4.5
The nth-Order Linear Differential Equation
. 145*
4.6
The nth-order Linear Differential Equation
with Constants Coefficients
. 150
4.7
Exercises and Problems
. 154
5.
Elements of Stability
159
5.1
Types of Stability
. 159
5.2
Stability of Linear Systems
. 165
5.3
The Case of Perturbed Systems
. 173
5.4
The Lyapunov Function Method
. 178
5.5
The Case of Dissipative Systems
. 186
5.6
The Case of Controlled Systems
. 192
5.7
Unpredictability and Chaos
. 196
5.8
Exercises and Problems
. 203
6.
Prime Integrals
207
6.1
Prime Integrals for Autonomous Systems
. 207
6.2
Prime Integrals for Non-Autonomous Systems
. 217
6.3
First Order Partial Differential Equations
. 219
6.4
The Cauchy Problem for Quasi-Linear Equations
. 223
6.5
Conservation Laws
. 227
6.5.1
Some Examples
. 227
6.5.2
A Local Existence and Uniqueness Result
. 230
6.5.3
Weak Solutions
. 231
6.6
Exercises and Problems
. 240
xvi Contents
7.
Extensions
and Generalizations
245
7.1
Distributions of One Variable
. 245
7.2
The Convolution Product
. 254
7.3
Generalized Solutions
. 258
7.4
Carathéodory
Solutions
. 265
7.5
Differential Inclusions
. 271
7.6
Variational Inequalities
. 280
7.7
Problems of Viability
. 287
7.8
Proof of Nagumo's Viability Theorem
. 290
7.9
Sufficient Conditions for
Invariance
. 295
7.10
Necessary Conditions for
Invariance
. 299
7.11
Gradient Systems. Probenius Theorem
. 303
7.12
Exercises and Problems
. 310
8.
Auxiliary Results
313
8.1
Elements of Vector Analysis
. 313
8.2
Compactness in C([
а,Ъ];Шп)
. 319
8.3
The Projection of
а
Point on
а
Convex Set
. 324
Solutions
329
Chapter
1. 329
Chapter
2. 339
Chapter
3. 349
Chapter
4. 358
Chapter
5. 366
Chapter
6. 375
Chapter
7. 387
Bibliography
393
Index
397 |
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| author_facet | Vrabie, Ioan I. |
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| building | Verbundindex |
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| discipline | Mathematik |
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| edition | Ioan I. Vrabie |
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| id | DE-604.BV021325491 |
| illustrated | Illustrated |
| index_date | 2024-07-02T14:00:14Z |
| indexdate | 2024-07-09T20:35:42Z |
| institution | BVB |
| isbn | 9812388389 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014645840 |
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| physical | XVI, 401 S. graph. Darst. |
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| spelling | Vrabie, Ioan I. Verfasser aut Differential equations an introduction to basic concepts, results and applications Ioan I. Vrabie New Jersey [u.a.] World Scientific 2004 XVI, 401 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Gewöhnliche Differentialgleichung (DE-588)4020929-5 s 2\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014645840&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Vrabie, Ioan I. Differential equations an introduction to basic concepts, results and applications Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4020929-5 (DE-588)4123623-3 |
| title | Differential equations an introduction to basic concepts, results and applications |
| title_auth | Differential equations an introduction to basic concepts, results and applications |
| title_exact_search | Differential equations an introduction to basic concepts, results and applications |
| title_exact_search_txtP | Differential equations an introduction to basic concepts, results and applications |
| title_full | Differential equations an introduction to basic concepts, results and applications |
| title_fullStr | Differential equations an introduction to basic concepts, results and applications |
| title_full_unstemmed | Differential equations an introduction to basic concepts, results and applications |
| title_short | Differential equations |
| title_sort | differential equations an introduction to basic concepts results and applications |
| title_sub | an introduction to basic concepts, results and applications |
| topic | Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Gewöhnliche Differentialgleichung Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014645840&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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