This textbook provides beginning graduate students and advanced undergraduates with an accessible introduction to the rich subject of partial differential equations (PDEs). It presents a rigorous and clear explanation of the more elementary theoretical aspects of PDEs, while also drawing connections to deeper analysis and applications. The book serves as a needed bridge between basic undergraduate texts and more advanced books that require a significant background in functional analysis.
Topics include first order equations and the method of characteristics, second order linear equations, wave and heat equations, Laplace and Poisson equations, and separation of variables. The book also covers fundamental solutions, Green’s functions and distributions, beginning functional analysis applied to elliptic PDEs, traveling wave solutions of selected parabolic PDEs, and scalar conservation laws and systems of hyperbolic PDEs.
Provides an accessible yet rigorous introduction to partial differential equations
Draws connections to advanced topics in analysis
Covers applications to continuum mechanics
An electronic solutions manual is available only to professors
An online illustration package is available to professors
Michael Shearer is professor of mathematics at North Carolina State University. He is a fellow of the American Mathematical Society. Rachel Levy is associate professor of mathematics at Harvey Mudd College. She is a recipient of the 2013 Henry L. Alder Award for Distinguished Teaching by a Beginning College or University Mathematics Faculty Member and creator of the Grandma Got STEM project.
Author(s): Michael Shearer, Rachel Levy
Publisher: Princeton University Press
Year: 2015
Language: English
Pages: 299
City: Princeton, Oxford
Tags: Математика;Дифференциальные уравнения;Дифференциальные уравнения в частных производных;
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-OrderPDE 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 R n 35
3.4. Scalar Conservation Laws and the Formation of Shocks 38
Problems 40
4. TheWaveEquation 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 R 2 and R 3 59
Problems 61
5. TheHeatEquation 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. SeparationofVariablesandFourierSeries 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. EigenfunctionsandConvergenceofFourierSeries 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 L 2 110
7.6. Fourier Transform 114
Problems 117
8. Laplace’sEquationandPoisson’sEquation 119
8.1. The Fundamental Solution 119
8.2. Solving Poisson’s Equation in R n 120
8.3. Properties of Harmonic Functions 122
8.4. Separation of Variables for Laplace’s Equation 125
Problems 130
9. Green’sFunctionsandDistributions 133
9.1. Boundary Value Problems 133
9.2. Test Functions and Distributions 136
9.3. Green’s Functions 144
Problems 149
10. FunctionSpaces 153
10.1. Basic Inequalities and Definitions 153
10.2. Multi-Index Notation 157
10.3. Sobolev Spaces W k,p (U) 158
Problems 159
11. EllipticTheorywithSobolevSpaces 161
11.1. Poisson’s Equation 161
11.2. Linear Second-Order Elliptic Equations 167
Problems 173
12. TravelingWaveSolutionsofPDE 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. ScalarConservationLaws 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. SystemsofFirst-OrderHyperbolicPDE 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. TheEquationsofFluidMechanics 245
15.1. The Navier-Stokes and Stokes Equations 245
15.2. The Euler Equations 247
Problems 250
AppendixA.Multivariable Calculus 253
AppendixB.Analysis 259
AppendixC.Systems of Ordinary Differential Equations 263
References 265
Index 269
