Preface ........................................................ ix
1 Introduction ................................................. 1
1.1 Linear PDE .............................................. 2
1.2 Solutions; Initial and Boundary Conditions .............. 3
1.3 Nonlinear PDE ........................................... 4
1.4 Beginning Examples with Explicit Wave-like Solutions .... 6
Problems ..................................................... 8
2 Beginnings .................................................. 11
2.1 Four Fundamental Issues in PDE Theory .................. 11
2.2 Classification of Second-Order PDE ..................... 12
2.3 Initial Value Problems and the Cauchy-Kovalevskaya
Theorem ................................................ 17
2.4 PDE from Balance Laws .................................. 21
Problems .................................................... 26
3 First-Order PDE ............................................. 29
3.1 The Method of Characteristics for Initial Value
Problems ............................................... 29
3.2 The Method of Characteristics for Cauchy Problems in
Two Variables .......................................... 32
3.3 The Method of Characteristics in  n .................... 35
3.4 Scalar Conservation Laws and the Formation of Shocks ... 38
Problems ............................................... 40
4 The Wave Equation ........................................... 43
4.1 The Wave Equation in Elasticity ........................ 43
4.2 D'Alembert's Solution .................................. 48
4.3 The Energy E(t) and Uniqueness of Solutions ............ 56
4.4 Duhamel's Principle for the Inhomogeneous Wave
Equation ............................................... 57
4.5 The Wave Equation on 2 and 3 ......................... 59
Problems .................................................... 61
5 The Heat Equation ........................................... 65
5.1 The Fundamental Solution ............................... 66
5.2 The Cauchy Problem for the Heat Equation ............... 68
5.3 The Energy Method ...................................... 73
5.4 The Maximum Principle .................................. 75
5.5 Duhamel's Principle for the Inhomogeneous Heat
Equation .................................................... 77
Problems .................................................... 78
6 Separation of Variablesand Fourier Series ................... 81
6.1 Fourier Series ......................................... 81
6.2 Separation of Variables for the Heat Equation .......... 82
6.3 Separation of Variables for the Wave Equation .......... 91
6.4 Separation of Variables for a Nonlinear Heat Equation .. 93
6.5 The Beam Equation ...................................... 94
Problems .................................................... 96
7 Eigenfunctions and Convergence of Fourier Series ............ 99
7.1 Eigenfunctions for ODE ................................. 99
7.2 Convergence and Completeness .......................... 102
7.3 Pointwise Convergence of Fourier Series ............... 105
7.4 Uniform Convergence of Fourier Series ................. 108
7.5 Convergence in L2 ..................................... 110
7.6 Fourier Transform ..................................... 114
Problems ................................................... 117
8 Laplace's Equation and Poisson's Equation .................. 119
8.1 The Fundamental Solution .............................. 119
8.2 Solving Poisson's Equation in n ...................... 120
8.3 Properties of Harmonic Functions ...................... 122
8.4 Separation of Variables for Laplace's Equation ........ 125
Problems ................................................... 130
9 Green's Functions and Distributions ........................ 133
9.1 Boundary Value Problems ............................... 133
9.2 Test Functions and Distributions ...................... 136
9.3 Green's Functions ..................................... 144
Problems ................................................... 149
10 Function Spaces ............................................ 153
10.1 Basic Inequalities and Definitions .................... 153
10.2 Multi-Index Notation .................................. 157
10.3 Sobolev Spaces Wk,p(U) ................................ 158
Problems ................................................... 159
11 Elliptic Theory with Sobolev Spaces ........................ 161
11.1 Poisson's Equation .................................... 161
11.2 Linear Second-Order Elliptic Equations ................ 167
Problems ................................................... 173
12 Traveling Wave Solutions of PDE ............................ 175
12.1 Burgers'Equation ...................................... 175
12.2 The Korteweg-deVries Equation ......................... 176
12.3 Fisher's Equation ..................................... 179
12.4 The Bistable Equation ................................. 181
Problems ................................................... 186
13 Scalar Conservation Laws ................................... 189
13.1 The Inviscid Burgers Equation ......................... 189
13.2 Scalar Conservation Laws .............................. 196
13.3 The Lax Entropy Condition Revisited ................... 201
13.4 Undercompressive Shocks ............................... 204
13.5 The (Viscous) Burgers Equation ........................ 206
13.6 Multidimensional Conservation Laws .................... 208
Problems ................................................... 211
14 Systems of First-Order Hyperbolic PDE ...................... 215
14.1 Linear Systems of First-Order PDE ..................... 215
14.2 Systems of Hyperbolic Conservation Laws ............... 219
14.3 The Dam-Break Problem Using Shallow Water Equations ... 239
14.4 Discussion ............................................ 241
Problems ................................................... 242
15 The Equations of Fluid Mechanics ........................... 245
15.1 The Navier-Stokes and Stokes Equations ................ 245
15.2 The Euler Equations ................................... 247
Problems ................................................... 250
Appendix A. Multivariable Calculus ............................ 253
Appendix B. Analysis .......................................... 259
Appendix C. Systems of Ordinary Differential Equations ........ 263
References .................................................... 265
Index ......................................................... 269
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