Partial Differential Equations (eBook)

Preface 7
Contents 8
Part I Background 13
1 Vector Fields and Ordinary Differential Equations 14
1.1 Introduction 14
1.2 Curves and Surfaces in mathbbRn 15
1.2.1 Cartesian Products, Affine Spaces 15
1.2.2 Curves in mathbbRn 17
1.2.3 Surfaces in mathbbR3 19
1.3 The Divergence Theorem 20
1.3.1 The Divergence of a Vector Field 20
1.3.2 The Flux of a Vector Field over an Orientable Surface 21
1.3.3 Statement of the Theorem 22
1.3.4 A Particular Case 22
1.4 Ordinary Differential Equations 23
1.4.1 Vector Fields as Differential Equations 23
1.4.2 Geometry Versus Analysis 24
1.4.3 An Example 25
1.4.4 Autonomous and Non-autonomous Systems 27
1.4.5 Higher-Order Equations 28
1.4.6 First Integrals and Conserved Quantities 29
1.4.7 Existence and Uniqueness 32
1.4.8 Food for Thought 33
References 35
2 Partial Differential Equations in Engineering 36
2.1 Introduction 36
2.2 What is a Partial Differential Equation? 37
2.3 Balance Laws 38
2.3.1 The Generic Balance Equation 39
2.3.2 The Case of Only One Spatial Dimension 42
2.3.3 The Need for Constitutive Laws 45
2.4 Examples of PDEs in Engineering 47
2.4.1 Traffic Flow 47
2.4.2 Diffusion 48
2.4.3 Longitudinal Waves in an Elastic Bar 49
2.4.4 Solitons 50
2.4.5 Time-Independent Phenomena 51
2.4.6 Continuum Mechanics 52
References 58
Part II The First-Order Equation 59
3 The Single First-Order Quasi-linear PDE 60
3.1 Introduction 60
3.2 Quasi-linear Equation in Two Independent Variables 62
3.3 Building Solutions from Characteristics 65
3.3.1 A Fundamental Lemma 65
3.3.2 Corollaries of the Fundamental Lemma 66
3.3.3 The Cauchy Problem 67
3.3.4 What Else Can Go Wrong? 69
3.4 Particular Cases and Examples 70
3.4.1 Homogeneous Linear Equation 70
3.4.2 Non-homogeneous Linear Equation 71
3.4.3 Quasi-linear Equation 73
3.5 A Computer Program 80
References 83
4 Shock Waves 84
4.1 The Way Out 84
4.2 Generalized Solutions 85
4.3 A Detailed Example 87
4.4 Discontinuous Initial Conditions 91
4.4.1 Shock Waves 91
4.4.2 Rarefaction Waves 94
References 97
5 The Genuinely Nonlinear First-Order Equation 98
5.1 Introduction 98
5.2 The Monge Cone Field 99
5.3 The Characteristic Directions 101
5.4 Recapitulation 105
5.5 The Cauchy Problem 107
5.6 An Example 108
5.7 More Than Two Independent Variables 110
5.7.1 Quasi-linear Equations 110
5.7.2 Non-linear Equations 113
5.8 Application to Hamiltonian Systems 114
5.8.1 Hamiltonian Systems 114
5.8.2 Reduced Form of a First-Order PDE 115
5.8.3 The Hamilton--Jacobi Equation 116
5.8.4 An Example 117
References 121
Part III Classification of Equations and Systems 122
6 The Second-Order Quasi-linear Equation 123
6.1 Introduction 123
6.2 The First-Order PDE Revisited 125
6.3 The Second-Order Case 126
6.4 Propagation of Weak Singularities 129
6.4.1 Hadamard's Lemma and Its Consequences 129
6.4.2 Weak Singularities 131
6.4.3 Growth and Decay 133
6.5 Normal Forms 135
References 138
7 Systems of Equations 139
7.1 Systems of First-Order Equations 139
7.1.1 Characteristic Directions 139
7.1.2 Weak Singularities 141
7.1.3 Strong Singularities in Linear Systems 142
7.1.4 An Application to the Theory of Beams 143
7.1.5 Systems with Several Independent Variables 145
7.2 Systems of Second-Order Equations 148
7.2.1 Characteristic Manifolds 148
7.2.2 Variation of the Wave Amplitude 150
7.2.3 The Timoshenko Beam Revisited 152
7.2.4 Air Acoustics 155
7.2.5 Elastic Waves 158
References 161
Part IV Paradigmatic Equations 162
8 The One-Dimensional Wave Equation 163
8.1 The Vibrating String 163
8.2 Hyperbolicity and Characteristics 164
8.3 The d'Alembert Solution 165
8.4 The Infinite String 166
8.5 The Semi-infinite String 169
8.5.1 D'Alembert Solution 169
8.5.2 Interpretation in Terms of Characteristics 171
8.5.3 Extension of Initial Data 173
8.6 The Finite String 174
8.6.1 Solution 174
8.6.2 Uniqueness and Stability 177
8.6.3 Time Periodicity 179
8.7 Moving Boundaries and Growth 180
8.8 Controlling the Slinky? 181
8.9 Source Terms and Duhamel's Principle 183
References 188
9 Standing Waves and Separation of Variables 189
9.1 Introduction 189
9.2 A Short Review of the Discrete Case 190
9.3 Shape-Preserving Motions of the Vibrating String 195
9.4 Solving Initial-Boundary Value Problems by Separation of Variables 198
9.5 Shape-Preserving Motions of More General Continuous Systems 204
9.5.1 String with Variable Properties 204
9.5.2 Beam Vibrations 207
9.5.3 The Vibrating Membrane 209
References 214
10 The Diffusion Equation 215
10.1 Physical Considerations 215
10.1.1 Diffusion of a Pollutant 215
10.1.2 Conduction of Heat 218
10.2 General Remarks on the Diffusion Equation 220
10.3 Separating Variables 221
10.4 The Maximum--Minimum Theorem and Its Consequences 222
10.5 The Finite Rod 225
10.6 Non-homogeneous Problems 227
10.7 The Infinite Rod 229
10.8 The Fourier Series and the Fourier Integral 231
10.9 Solution of the Cauchy Problem 234
10.10 Generalized Functions 236
10.11 Inhomogeneous Problems and Duhamel's Principle 240
References 244
11 The Laplace Equation 245
11.1 Introduction 245
11.2 Green's Theorem and the Dirichlet and Neumann Problems 246
11.3 The Maximum-Minimum Principle 249
11.4 The Fundamental Solutions 250
11.5 Green's Functions 252
11.6 The Mean-Value Theorem 254
11.7 Green's Function for the Circle and the Sphere 255
References 258
Index 259