Introduction to Partial Differential Equations: A Computational Approach

Contents......Page 12
1.1 What Is a Differential Equation?......Page 18
1.1.1 Concepts......Page 19
1.2.1 An Ordinary Differential Equation......Page 21
1.3 A Numerical Method......Page 23
1.4.1 First-Order Homogeneous Equations......Page 27
1.4.2 First-Order Nonhomogeneous Equations......Page 30
1.4.3 The Wave Equation......Page 32
1.4.4 The Heat Equation......Page 35
1.5 Exercises......Page 37
1.6 Projects......Page 45
2 Two-Point Boundary Value Problems......Page 56
2.1 Poisson's Equation in One Dimension......Page 57
2.1.1 Green's Function......Page 59
2.1.2 Smoothness of the Solution......Page 60
2.1.3 A Maximum Principle......Page 61
2.2 A Finite Difference Approximation......Page 62
2.2.1 Taylor Series......Page 63
2.2.2 A System of Algebraic Equations......Page 64
2.2.3 Gaussian Elimination for Tridiagonal Linear Systems......Page 67
2.2.4 Diagonal Dominant Matrices......Page 70
2.2.5 Positive Definite Matrices......Page 72
2.3.1 Difference and Differential Equations......Page 74
2.3.2 Symmetry......Page 75
2.3.4 A Maximum Principle for the Discrete Problem......Page 78
2.3.5 Convergence of the Discrete Solutions......Page 80
2.4.1 The Continuous Eigenvalue Problem......Page 82
2.4.2 The Discrete Eigenvalue Problem......Page 85
2.5 Exercises......Page 89
2.6 Projects......Page 99
3 The Heat Equation......Page 104
3.1 A Brief Overview......Page 105
3.2 Separation of Variables......Page 107
3.3 The Principle of Superposition......Page 109
3.4 Fourier Coefficients......Page 112
3.5 Other Boundary Conditions......Page 114
3.6 The Neumann Problem......Page 115
3.6.1 The Eigenvalue Problem......Page 116
3.6.2 Particular Solutions......Page 117
3.6.3 A Formal Solution......Page 118
3.7 Energy Arguments......Page 119
3.8 Differentiation of Integrals......Page 123
3.9 Exercises......Page 125
3.10 Projects......Page 130
4 Finite Difference Schemes For The Heat Equation......Page 134
4.1 An Explicit Scheme......Page 136
4.2 Fourier Analysis of the Numerical Solution......Page 139
4.2.1 Particular Solutions......Page 140
4.2.2 Comparison of the Analytical and Discrete Solution......Page 144
4.2.3 Stability Considerations......Page 146
4.2.4 The Accuracy of the Approximation......Page 147
4.2.5 Summary of the Comparison......Page 148
4.3 Von Neumann's Stability Analysis......Page 149
4.3.1 Particular Solutions: Continuous and Discrete......Page 150
4.3.2 Examples......Page 151
4.3.3 A Nonlinear Problem......Page 154
4.4 An Implicit Scheme......Page 157
4.4.1 Stability Analysis......Page 160
4.5 Numerical Stability by Energy Arguments......Page 162
4.6 Exercises......Page 165
5 The Wave Equation......Page 176
5.1 Separation of Variables......Page 177
5.2 Uniqueness and Energy Arguments......Page 180
5.3 A Finite Difference Approximation......Page 182
5.3.1 Stability Analysis......Page 185
5.4 Exercises......Page 187
6.1 A Two-Point Boundary Value Problem......Page 192
6.2 The Linear Heat Equation......Page 195
6.2.1 The Continuous Case......Page 197
6.2.2 Uniqueness and Stability......Page 200
6.2.3 The Explicit Finite Difference Scheme......Page 201
6.2.4 The Implicit Finite Difference Scheme......Page 203
6.3 The Nonlinear Heat Equation......Page 205
6.3.1 The Continuous Case......Page 206
6.3.2 An Explicit Finite Difference Scheme......Page 207
6.4 Harmonic Functions......Page 208
6.4.1 Maximum Principles for Harmonic Functions......Page 210
6.5 Discrete Harmonic Functions......Page 212
6.6 Exercises......Page 218
7.1 Rectangular Domains......Page 226
7.2 Polar Coordinates......Page 229
7.2.1 The Disc......Page 230
7.2.2 A Wedge......Page 233
7.2.3 A Corner Singularity......Page 234
7.3 Applications of the Divergence Theorem......Page 235
7.4 The Mean Value Property for Harmonic Functions......Page 239
7.5.1 The Five-Point Stencil......Page 242
7.5.2 An Error Estimate......Page 245
7.6.1 Upper Triangular Systems......Page 247
7.6.2 General Systems......Page 248
7.6.3 Banded Systems......Page 251
7.6.4 Positive Definite Systems......Page 253
7.7 Exercises......Page 254
8 Orthogonality and General Fourier Series......Page 262
8.1 The Full Fourier Series......Page 263
8.1.1 Even and Odd Functions......Page 266
8.1.2 Differentiation of Fourier Series......Page 269
8.1.3 The Complex Form......Page 272
8.1.4 Changing the Scale......Page 273
8.2.1 Other Boundary Conditions......Page 274
8.2.2 Sturm-Liouville Problems......Page 278
8.3 The Mean Square Distance......Page 281
8.4 General Fourier Series......Page 284
8.5 A Poincaré Inequality......Page 290
8.6 Exercises......Page 293
9.1 Different Notions of Convergence......Page 302
9.2 Pointwise Convergence......Page 307
9.3 Uniform Convergence......Page 313
9.4 Mean Square Convergence......Page 317
9.5 Smoothness and Decay of Fourier Coefficients......Page 319
9.6 Exercises......Page 324
10 The Heat Equation Revisited......Page 330
10.1 Compatibility Conditions......Page 331
10.2.1 The Smoothing Property......Page 336
10.2.2 The Differential Equation......Page 338
10.2.3 The Initial Condition......Page 340
10.2.4 Smooth and Compatible Initial Functions......Page 342
10.3 Convergence of Finite Difference Solutions......Page 344
10.4 Exercises......Page 348
11.1 The Logistic Model of Population Growth......Page 354
11.1.1 A Numerical Method for the Logistic Model......Page 356
11.2 Fisher's Equation......Page 357
11.3 A Finite Difference Scheme for Fisher's Equation......Page 359
11.4 An Invariant Region......Page 360
11.5 The Asymptotic Solution......Page 363
11.6 Energy Arguments......Page 366
11.6.1 An Invariant Region......Page 367
11.6.2 Convergence Towards Equilibrium......Page 368
11.6.3 Decay of Derivatives......Page 369
11.7 Blowup of Solutions......Page 371
11.8 Exercises......Page 374
11.9 Projects......Page 377
12 Applications of the Fourier Transform......Page 382
12.1 The Fourier Transform......Page 383
12.2 Properties of the Fourier Transform......Page 385
12.3 The Inversion Formula......Page 389
12.4 The Convolution......Page 392
12.5.1 The Heat Equation......Page 394
12.5.2 Laplace's Equation in a Half-Plane......Page 397
12.6 Exercises......Page 399
References......Page 402
D......Page 406
I......Page 407
R......Page 408
Z......Page 409