Cover
Half-title
Title
Copyright
Contents
Preface
Part One Linear Equations
CHAPTER ONE Variable Coefficient, Second Order, Linear, Ordinary Differential Equations
1.1 The Method of Reduction of Order
1.2 The Method of Variation of Parameters
1.2.1 The Wronskian
1.3 Solution by Power Series: The Method of Frobenius
1.3.1 The Roots of the Indicial Equation Differ by an Integer
1.3.2 The Roots of the Indicial Equation Differ by a Noninteger Quantity
1.3.3 The Roots of the Indicial Equation are Equal
1.3.4 Singular Points of Differential Equations
1.3.5 An outline proof of Theorem 1.3
1.3.6 The point at infinity
CHAPTER TWO Legendre Functions
2.1 Definition of the Legendre Polynomials, p(x)
2.2 The Generating Function for P(x)
2.3 Differential and Recurrence Relations Between Legendre Polynomials
2.4 Rodrigues’ Formula
2.5 Orthogonality of the Legendre Polynomials
2.6 Physical Applications of the Legendre Polynomials
2.6.1 Heat Conduction
2.6.2 Fluid Flow
2.7 The Associated Legendre Equation
CHAPTER THREE Bessel Functions
3.1 The Gamma Function and the Pockhammer Symbol
3.2 Series Solutions of Bessel’s Equation
3.3 The Generating Function for J(x), n an integer
3.4 Differential and Recurrence Relations Between Bessel Functions
3.5 Modifled Bessel Functions
3.6 Orthogonality of the Bessel Functions
3.7 Inhomogeneous Terms in Bessel’s Equation
3.8 Solutions Expressible as Bessel Functions
3.9 Physical Applications of the Bessel Functions
3.9.1 Vibrations of an Elastic Membrane
3.9.2 Frequency Modulation (FM)
CHAPTER FOUR Boundary Value Problems, Green’s Functions and Sturm–Liouville Theory
4.1 Inhomogeneous Linear Boundary Value Problems
4.1.1 Solubility
4.1.2 The Green’s Function
4.2 The Solution of Boundary Value Problems by Eigenfunction Expansions
4.2.1 Self-Adjoint Operators
4.2.2 Boundary Conditions
4.2.3 Eigenvalues and Eigenfunctions of Hermitian Linear
4.2.4 Eigenfunction Expansions
4.3 Sturm–Liouville Systems
4.3.1 The Sturm–Liouville Equation
4.3.2 Boundary Conditions
4.3.3 Properties of the Eigenvalues and Eigenfunctions
4.3.4 Bessel’s Inequality, Approximation in the Mean and Completeness
4.3.5 Further Properties of Sturm–Liouville Systems
4.3.6 Two Examples from Quantum Mechanics
CHAPTER FIVE Fourier Series and the Fourier Transform
5.1 General Fourier Series
5.2 The Fourier Transform
5.2.1 Generalized Functions
5.2.2 Derivatives of Generalized Functions
5.2.3 Fourier Transforms of Generalized Functions
5.2.4 The Inverse Fourier Transform
5.2.5 Transforms of Derivatives and Convolutions
5.3 Green’s Functions Revisited
5.4 Solution of Laplace’s Equation Using Fourier Transforms
5.5 Generalization to Higher Dimensions
5.5.1 The Delta Function in Higher Dimensions
5.5.2 Fourier Transforms in Higher Dimensions
CHAPTER SIX Laplace Transforms
6.1 Definition and Examples
6.1.1 The Existence of Laplace Transforms
6.2 Properties of the Laplace Transform
6.3 The Solution of Ordinary Differential Equations Using Laplace Transforms
6.3.1 The Convolution Theorem
6.4 The Inversion Formula for Laplace Transforms
CHAPTER SEVEN Classification, Properties and Complex Variable Methods for Second Order Partial Differential Equations
7.1 Classification and Properties of Linear, Second Order Partial Differential Equations in Two Independent Variables
7.1.1 Classification
7.1.2 Canonical Forms
7.1.3 Properties of Hyperbolic Equations
7.1.4 Properties of Elliptic Equations
7.1.5 Properties of Parabolic Equations
7.2 Complex Variable Methods for Solving Laplace’s Equation
7.2.1 The Complex Potential
7.2.2 Simple Flows Around Blunt Bodies
7.2.3 Conformal Transformations
Part Two Nonlinear Equations and Advanced Techniques
CHAPTER EIGHT Existence, Uniqueness, Continuity and Comparison of Solutions of Ordinary Differential Equations
8.1 Local Existence of Solutions
8.2 Uniqueness of Solutions
8.3 Dependence of the Solution on the Initial Conditions
8.4 Comparison Theorems
CHAPTER NINE Nonlinear Ordinary Differential Equations: Phase Plane Methods
9.1 Introduction: The Simple Pendulum
9.2 First Order Autonomous Nonlinear Ordinary Differential Equations
9.2.1 The Phase Line
9.2.2 Local Analysis at an Equilibrium Point
9.3 Second Order Autonomous Nonlinear Ordinary Differential Equations
9.3.1 The Phase Plane
9.3.2 Equilibrium Points
9.3.3 An Example from Mechanics
9.3.4 Example: Population Dynamics
9.3.5 The Poincaré Index
9.3.6 Bendixson’s Negative Criterion and Dulac’s Extension
9.3.7 The Poincaré–Bendixson Theorem
9.3.8 The Phase Portrait at Infinity
9.3.9 A Final Example: Hamiltonian Systems
9.4 Third Order Autonomous Nonlinear Ordinary Differential Equations
CHAPTER TEN Group Theoretical Methods
10.1 Lie Groups
10.1.1 The Infinitesimal Transformation
10.1.2 Infinitesimal Generators and the Lie Series
10.2 Invariants Under Group Action
10.3 The Extended Group
10.4 Integration of a First Order Equation with a Known Group Invariant
10.5 Towards the Systematic Determination of Groups Under Which a First Order Equation is Invariant
10.6 Invariants for Second Order Differential Equations
10.7 Partial Differential Equations
CHAPTER ELEVEN Asymptotic Methods: Basic Ideas
11.1 Asymptotic Expansions
11.1.1 Gauge Functions
11.1.2 Example: Series Expansions of the Exponential Integral, Ei(x)
11.1.3 Asymptotic Sequences of Gauge Functions
11.2 The Asymptotic Evaluation of Integrals
11.2.1 Laplace’s Method
11.2.2 The Method of Stationary Phase
11.2.3 The Method of Steepest Descents
CHAPTER TWELVE Asymptotic Methods: Differential Equations
12.1 An Instructive Analogy: Algebraic Equations
12.1.1 Example: A Regular Perturbation
12.1.2 Example: A Singular Perturbation
12.2 Ordinary Differential Equations
12.2.1 Regular Perturbations
12.2.2 The Method of Matched Asymptotic Expansions
Van Dyke’s Matching Principle
Composite Expansions
Interior Layers
12.2.3 Nonlinear Problems
12.2.4 The Method of Multiple Scales
12.2.5 Slowly Damped Nonlinear Oscillations: Kuzmak’s Method
12.2.6 The Effect of Fine Scale Structure on Reaction–Diffusion Processes
12.2.7 The WKB Approximation
12.3 Partial Differential Equations
CHAPTER THIRTEEN Stability, Instability and Bifurcations
13.1 Zero Eigenvalues and the Centre Manifold Theorem
13.1.1 Construction of the Centre Manifold
13.1.2 The Stable, Unstable and Centre Manifolds
13.2 Lyapunov’s Theorems
13.3 Bifurcation Theory
13.3.1 First Order Ordinary Differential Equations
13.3.2 Second Order Ordinary Differential Equations
13.3.3 Global Bifurcations
CHAPTER FOURTEEN Time-Optimal Control in the Phase Plane
14.1 Definitions
14.2 First Order Equations
14.3 Second Order Equations
14.3.1 Properties of sets of points in the plane
14.3.2 Matrix solution of systems of constant coefficient ordinary differential equations
14.4 Examples of Second Order Control Problems
14.5 Properties of the Controllable Set
14.6 The Controllability Matrix
14.7 The Time-Optimal Maximum Principle (TOMP)
CHAPTER FIFTEEN An Introduction to Chaotic Systems
15.1 Three Simple Chaotic Systems
15.1.1 A Mechanical Oscillator
15.1.2 A Chemical Oscillator
15.1.3 The Lorenz Equations
15.2 Mappings
15.2.1 Fixed and Periodic Points of Maps
15.2.2 Tents and Horseshoes
15.3 The Poincaré Return Map
15.4 Homoclinic Tangles
15.4.1 Mel’nikov Theory
15.4.2 Unperturbed System (Epsilon = 0)
15.4.3 Perturbed System…
15.5 Quantifying Chaos: Lyapunov Exponents and the Lyapunov Spectrum
15.5.1 Lyapunov Exponents of Systems of Ordinary Differential Equations
15.5.2 The Lyapunov Spectrum
APPENDIX 1 Linear Algebra
A1.1 Vector Spaces Over the Real Numbers
A1.2 Inner Product Spaces
A1.3 Linear Transformations and Matrices
A1.4 The Eigenvalues and Eigenvectors of a Matrix
APPENDIX 2 Continuity and Differentiability
APPENDIX 3 Power Series
A3.1 Maclaurin Series
A3.2 Taylor Series
A3.3 Convergence of Power Series
A3.4 Taylor Series for Functions of Two Variables
APPENDIX 4 Sequences of Functions
APPENDIX 5 Ordinary Differential Equations
A5.1 Variables Separable
A5.2 Integrating Factors
A5.3 Second Order Equations with Constant Coefficients
APPENDIX 6 Complex Variables
A6.1 Analyticity and the Cauchy–Riemann Equations
A6.2 Cauchy’s Theorem, Cauchy’s Integral Formula and Taylor’s Theorem
A6.3 The Laurent Series and Residue Calculus
A6.4 Jordan’s Lemma
A6.5 Linear Ordinary Differential Equations in the Complex Plane
APPENDIX 7 A Short Introduction to MATLAB
A7.1 Getting Started
A7.2 Variables, Vectors and Matrices
A7.3 User-Defined Functions
A7.4 Graphics
A7.5 Programming in MATLAB
Bibliography
Index