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ОбложкаShearer M. Partial differential equations: an introduction to theory and applications / M.Shearer, R.Levy. - Princeton; Oxford: Princeton university press, 2015. - x, 274 p.: ill. - Bibliogr.: p.265-267. - Ind.: p.269-274. - ISBN 978-0-691-16129-7
Шифр: (И/В16-S53) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
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 fig.1n .................... 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 fig.12 and fig.13 ......................... 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 fig.1n ...................... 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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