Page 12 -

xiii
Contents
159
5 The Wave Equation
Separation of Variables ..................... 160
5.1
Uniqueness and Energy Arguments .............. 163
5.2
A Finite Difference Approximation .............. 165
5.3
5.3.1
............................ 170
Exercises
5.4
175
6 Maximum Principles
A Two-Point Boundary Value Problem ............ 175
6.1
The Linear Heat Equation ................... 178
6.2
6.2.1 Stability Analysis .................... 168
The Continuous Case ................. 180
6.2.2 Uniqueness and Stability ............... 183
6.2.3 The Explicit Finite Difference Scheme ........ 184
6.2.4 The Implicit Finite Difference Scheme ........ 186
6.3 The Nonlinear Heat Equation ................. 188
6.3.1 The Continuous Case ................. 189
6.3.2 An Explicit Finite Difference Scheme ......... 190
6.4 Harmonic Functions ...................... 191
6.4.1 Maximum Principles for Harmonic Functions .... 193
6.5 Discrete Harmonic Functions ................. 195
6.6 Exercises ............................ 201
7 Poisson’s Equation in Two Space Dimensions 209
7.1 Rectangular Domains ..................... 209
7.2 Polar Coordinates ....................... 212
7.2.1 The Disc ........................ 213
7.2.2 A Wedge ........................ 216
7.2.3 A Corner Singularity .................. 217
7.3 Applications of the Divergence Theorem .......... 218
7.4 The Mean Value Property for Harmonic Functions ..... 222
7.5 A Finite Difference Approximation .............. 225
7.5.1 The Five-Point Stencil ................. 225
7.5.2 An Error Estimate ................... 228
7.6 Gaussian Elimination for General Systems .......... 230
7.6.1 Upper Triangular Systems ............... 230
7.6.2 General Systems .................... 231
7.6.3 Banded Systems .................... 234
7.6.4 Positive Definite Systems ............... 236
7.7 Exercises ............................ 237
8 Orthogonality and General Fourier Series 245
8.1 The Full Fourier Series ..................... 246
8.1.1 Even and Odd Functions ............... 249
8.1.2 Differentiation of Fourier Series ............ 252
8.1.3 The Complex Form ................... 255