Billingham J., King A., Otto S. - Differential Equations - Linear, Nonlinear, Ordinary, Partial (Cambridge, 2003).pdf

(6920 KB) Pobierz
134263059 UNPDF
Differential Equations
Linear, Nonlinear, Ordinary, Partial
A.C. King, J. Billingham and S.R. Otto
134263059.001.png
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge
, United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
© Cambridge University Press 2003
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format
2003
- - - - -
eBook (NetLibrary)
eBook (NetLibrary)
- - - -
- - - - -
hardback
- - - -
hardback
- - - - -
paperback
paperback
- - - -
s for external or third-party internet websites referred to in this book, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Cambridge University Press has no responsibility for the persistence or accuracy of
134263059.002.png
Contents
Preface
page ix
PartOne:LinearEquations
1
1 Variable Coecient, Second Order, Linear, Ordinary
DifferentialEquations
3
1.1 The Method of Reduction of Order
5
1.2 The Method of Variation of Parameters
7
1.3 Solution by Power Series: The Method of Frobenius
11
2LegendreFunctions
31
2.1DefinitionoftheLegendrePolynomials, P n ( x )
31
2.2 The Generating Function for P n ( x )
35
2.3 Differential and Recurrence Relations Between Legendre
Polynomials
38
2.4Rodrigues’Formula
39
2.5 Orthogonality of the Legendre Polynomials
41
2.6 Physical Applications of the Legendre Polynomials
44
2.7 The Associated Legendre Equation
52
3BesselFunctions 58
3.1 The Gamma Function and the Pockhammer Symbol 58
3.2 Series Solutions of Bessel’s Equation 60
3.3 The Generating Function for J n ( x ), n an integer 64
3.4 Differential and Recurrence Relations Between Bessel Functions 69
3.5 Modified Bessel Functions
71
3.6 Orthogonality of the Bessel Functions
71
3.7 Inhomogeneous Terms in Bessel’s Equation
77
3.8 Solutions Expressible as Bessel Functions
79
3.9 Physical Applications of the Bessel Functions
80
4 Boundary Value Problems, Green’s Functions and
Sturm–Liouville Theory
93
4.1 Inhomogeneous Linear Boundary Value Problems
96
4.2 The Solution of Boundary Value Problems by Eigenfunction
Expansions
100
4.3 Sturm–Liouville Systems
107
vi CONTENTS
5 Fourier Series and the Fourier Transform
123
5.1 General Fourier Series
127
5.2 The Fourier Transform
133
5.3 Green’s Functions Revisited
141
5.4 Solution of Laplace’s Equation Using Fourier Transforms
143
5.5 Generalization to Higher Dimensions
145
6 Laplace Transforms
152
6.1 Definition and Examples
152
6.2 Properties of the Laplace Transform
154
6.3 The Solution of Ordinary Differential Equations using Laplace
Transforms
157
6.4 The Inversion Formula for Laplace Transforms
162
7 Classification, Properties and Complex Variable Methods for
Second Order Partial Differential Equations
175
7.1 Classification and Properties of Linear, Second Order Partial
Differential Equations in Two Independent Variables
175
7.2 Complex Variable Methods for Solving Laplace’s Equation
186
Part Two: Nonlinear Equations and Advanced Techniques
201
8 Existence, Uniqueness, Continuity and Comparison of
Solutions of Ordinary Differential Equations
203
8.1 Local Existence of Solutions
204
8.2 Uniqueness of Solutions
210
8.3 Dependence of the Solution on the Initial Conditions
211
8.4 Comparison Theorems
212
9 Nonlinear Ordinary Differential Equations: Phase Plane
Methods
217
9.1 Introduction: The Simple Pendulum
217
9.2 First Order Autonomous Nonlinear Ordinary Differential
Equations
222
9.3 Second Order Autonomous Nonlinear Ordinary Differential
Equations
224
9.4 Third Order Autonomous Nonlinear Ordinary Differential
Equations
249
10 Group Theoretical Methods
256
10.1 Lie Groups
257
10.2 Invariants Under Group Action
261
10.3 The Extended Group
262
10.4 Integration of a First Order Equation with a Known Group
Invariant
263
CONTENTS vii
10.5 Towards the Systematic Determination of Groups Under Which
a First Order Equation is Invariant
265
10.6 Invariants for Second Order Differential Equations
266
10.7 Partial Differential Equations
270
11 Asymptotic Methods: Basic Ideas
274
11.1 Asymptotic Expansions
275
11.2 The Asymptotic Evaluation of Integrals
280
12 Asymptotic Methods: Differential Equations
303
12.1 An Instructive Analogy: Algebraic Equations
303
12.2 Ordinary Differential Equations
306
12.3 Partial Differential Equations
351
13 Stability, Instability and Bifurcations
372
13.1 Zero Eigenvalues and the Centre Manifold Theorem
372
13.2 Lyapunov’s Theorems
381
13.3 Bifurcation Theory
388
14 Time-Optimal Control in the Phase Plane
417
14.1 Definitions
418
14.2 First Order Equations
418
14.3 Second Order Equations
422
14.4 Examples of Second Order Control Problems
426
14.5 Properties of the Controllable Set
429
14.6 The Controllability Matrix
433
14.7 The Time-Optimal Maximum Principle (TOMP)
436
15 An Introduction to Chaotic Systems
447
15.1 Three Simple Chaotic Systems
447
15.2 Mappings
452
15.3 The Poincare Return Map
467
15.4 Homoclinic Tangles
472
15.5 Quantifying Chaos: Lyapunov Exponents and the Lyapunov
Spectrum
484
Appendix 1 Linear Algebra
495
Appendix 2 Continuity and Differentiability
502
Appendix 3 Power Series
505
Appendix 4 Sequences of Functions
509
Appendix 5 Ordinary Differential Equations
511
Appendix 6 Complex Variables
517
Appendix 7 A Short Introduction to MATLAB
526
Bibliography
534
Index
536

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin