Ordinary differential equations: principles and applications
"Discusses the relevant topics including first and second order linear equations, initial value problems and qualitative theory"...
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2017
|
| Schriftenreihe: | Cambridge-IISc Series
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "Discusses the relevant topics including first and second order linear equations, initial value problems and qualitative theory"... |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xviii, 328 Seiten Diagramme |
| ISBN: | 9781108416412 |
Internformat
MARC
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| 245 | 1 | 0 | |a Ordinary differential equations |b principles and applications |c A.K. Nandakumaran, P.S. Datti, Raju K. George |
| 264 | 1 | |a Cambridge |b Cambridge University Press |c 2017 | |
| 300 | |a xviii, 328 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cambridge-IISc Series | |
| 500 | |a Includes bibliographical references and index | ||
| 520 | |a "Discusses the relevant topics including first and second order linear equations, initial value problems and qualitative theory"... | ||
| 650 | 4 | |a Differential equations | |
| 650 | 0 | 7 | |a Gewöhnliche Differentialgleichung |0 (DE-588)4020929-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Gewöhnliche Differentialgleichung |0 (DE-588)4020929-5 |D s |
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| 700 | 1 | |a Datti, P. S. |4 aut | |
| 700 | 1 | |a George, Raju K. |0 (DE-588)1135938474 |4 aut | |
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Datensatz im Suchindex
| _version_ | 1804177749597749248 |
|---|---|
| adam_text | Contents
Figures xiii
Preface xv
Acknowledgement xix
1 Introduction and Examples: Physical Models
1.1 A Brief General Introduction 1
1.2 Physical and Other Models 3
1.2.1 Population growth model 3
1.2.2 An atomic waste disposal problem 6
1.2.3 Mechanical vibration model 8
1.2.4 Electrical circuit 9
1.2.5 Satellite problem 10
1.2.6 Flight trajectory problem 13
1.2.7 Other examples 14
1.3 Exercises 19
1.4 Notes 21
2 Preliminaries
2.1 Introduction 22
2.2 Preliminaries from Real Analysis 22
2.2.1 Convergence and uniform convergence 22
2.3 Fixed Point Theorem 34
2.4 Some Topics in Linear Algebra 38
2.4.1 Euclidean space 1R” 41
2.4.2 Points versus vectors 42
2.4.3 Linear operators 42
viii Contents
2.5 Matrix Exponential and its Properties 43
2.5.1 Diagonalizability and block diagonalizability 46
2.5.2 Spectral analysis of A 48
2.5.3 Computation of for a Jordan block J 51
2.6 Linear Dependence and Independence of Functions 53
2.7 Exercises 54
2.8 Notes 55
3 First and Second Order Linear Equations
3.1 First Order Equations 56
3.1.1 Initial and boundary value problems 57
3.1.2 Concept of a solution 59
3.1.3 First order linear equations 60
3.1.4 Variable separable equations 65
3.2 Exact Differential Equations 66
3.3 Second Order Linear Equations 72
3.3.1 Homogeneous SLDE (HSLDE) 74
3.3.2 Linear equation with constant coefficients 77
3.3.3 Non֊homogeneous equation 79
3.3.4 Green’s functions 88
3.4 Partial Differential Equations and ODE 89
3.5 Exercises 93
3.6 Notes 97
4 General Theory of Initial Value Problems
4.1 Introduction 99
4.1.1 Well-posed problems 99
4.1.2 Examples 100
4.2 Sufficient Condition for Uniqueness of Solution 103
4.2.1 A basic lemma 103
4.2.2 Uniqueness theorem 106
4.3 Sufficient Condition for Existence of Solution 107
4.3.1 Cauchy—Peano existence theorem 112
4.3.2 Existence and uniqueness by fixed point theorem 116
4.4 Continuous Dependence of the Solution on
Initial Data and Dynamics 119
Contents ix
4.5 Continuation of a Solution into Larger Intervals and Maximal
Interval of Existence 120
4.5.1 Continuation of the solution outside the
interval t — to h 121
4.5.2 Maximal interval of existence 123
4.6 Existence and Uniqueness of a System of Equations 125
4.6.1 Existence and uniqueness results for systems 127
4.7 Exercises 130
4.8 Notes 133
5 Linear Systems and Qualitative Analysis
5.1 General nth Order Equations and Linear Systems 134
5.2 Autonomous Homogeneous Systems 136
5.2.1 Computation of etA in special cases 137
5.3 Two-dimensional Systems 139
5.3.1 Computation of eBJ and etBJ 140
5.4 Stability Analysis 143
5.4.1 Phase plane and phase portrait 143
5.4.2 Dynamical system, flow, vector fields 145
5.4.3 Equilibrium points and stability 147
5.5 Higher Dimensional Systems 155
5.6 Invariant Subspaces under the Flow etA 165
5.7 Non-homogeneous, Autonomous Systems 167
5.7.1 Solution to non-homogeneous systems
(variation of parameters) 168
5.7.2 Non-autonomous systems 169
5.8 Exercises 175
5.9 Notes 179
6 Series Solutions: Frobenius Theory
6.1 Introduction 180
6.2 Real Analytic Functions 180
6.3 Equations with Analytic Coefficients 183
6.4 Regular Singular Points 189
6.4.1 Equations with regular singular points 190
6.5 Exercises 198
6.6 Notes 200
201
203
208
211
218
219
220
222
223
224
228
229
230
233
238
245
248
248
253
264
266
267
269
274
275
276
280
284
284
286
Contents
Regular Sturm—Liouville Theory
7.1 Introduction
7.2 Basic Result and Orthogonality
7.3 Oscillation Results
7.3.1 Comparison theorems
7.3.2 Location of zeros
7.4 Existence of Eigenfunctions
7.5 Exercises
7.6 Notes
Qualitative Theory
8.1 Introduction
8.2 General Definitions and Results
8.2.1 Examples
8.3 Liapunov Stability, Liapunov Function
8.3.1 Linearization
8.3.2 Examples
8.4 Liapunov Function
8.5 Invariant Subspaces and Manifolds
8.6 Phase Plane Analysis
8.6.1 Examples
8.7 Periodic Orbits
8.8 Exercises
8.9 Notes
Two Point Boundary Value Problems
9.1 Introduction
9.2 Linear Problems
9.2.1 BVP for linear systems
9.2.2 Examples
9.3 General Second Order Equations
9.3.1 Examples
9.4 Exercises
9.5 Notes
First Order Partial Differential Equations:
Method of Characteristics
10.1 Linear Equations
Contents xi
10.2 Quasi-linear Equations 290
10.3 General First Order Equation in Two Variables 295
10.4 Hamilton—Jacobi Equation 302
10.5 Exercises 304
10.6 Notes 306
Appendix A Poincare-Bendixon and Leinard’s Theorems
A.l Introduction 307
A.2 Poincare—Bendixon Theorems 310
A.2.1 Intersection with transversals 310
A.3 Leinard’s Theorem 316
Bibliography 321
Index 325
|
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| id | DE-604.BV044440428 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:53:01Z |
| institution | BVB |
| isbn | 9781108416412 |
| language | English |
| lccn | 017022841 |
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| owner_facet | DE-703 DE-19 DE-BY-UBM DE-739 DE-20 |
| physical | xviii, 328 Seiten Diagramme |
| publishDate | 2017 |
| publishDateSearch | 2017 |
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| publisher | Cambridge University Press |
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| series2 | Cambridge-IISc Series |
| spelling | Nandakumaran, A. K. ca. 20./21. Jh. (DE-588)1135937907 aut Ordinary differential equations principles and applications A.K. Nandakumaran, P.S. Datti, Raju K. George Cambridge Cambridge University Press 2017 xviii, 328 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Cambridge-IISc Series Includes bibliographical references and index "Discusses the relevant topics including first and second order linear equations, initial value problems and qualitative theory"... Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s DE-604 Datti, P. S. aut George, Raju K. (DE-588)1135938474 aut Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029841606&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nandakumaran, A. K. ca. 20./21. Jh Datti, P. S. George, Raju K. Ordinary differential equations principles and applications Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4020929-5 |
| title | Ordinary differential equations principles and applications |
| title_auth | Ordinary differential equations principles and applications |
| title_exact_search | Ordinary differential equations principles and applications |
| title_full | Ordinary differential equations principles and applications A.K. Nandakumaran, P.S. Datti, Raju K. George |
| title_fullStr | Ordinary differential equations principles and applications A.K. Nandakumaran, P.S. Datti, Raju K. George |
| title_full_unstemmed | Ordinary differential equations principles and applications A.K. Nandakumaran, P.S. Datti, Raju K. George |
| title_short | Ordinary differential equations |
| title_sort | ordinary differential equations principles and applications |
| title_sub | principles and applications |
| topic | Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Differential equations Gewöhnliche Differentialgleichung |
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