Preface ........................................................ xi
Acknowledgments .............................................. xiii
1 Introduction ................................................. 1
1.1 Terminology and Notation ................................ 1
1.2 Classification .......................................... 2
1.3 Canonical Forms ......................................... 3
1.4 Common PDEs ............................................. 4
1.5 Cauchy-Kowalevski Theorem ............................... 5
1.6 Initial Boundary Value Problems ......................... 7
1.7 Solution Techniques ..................................... 8
1.8 Separation of Variables ................................. 9
Exercises ................................................... 15
2 Fourier Series .............................................. 19
2.1 Vector Spaces .......................................... 19
2.1.1 Subspaces ....................................... 21
2.1.2 Basis and Dimension ............................. 21
2.1.3 Inner Products .................................. 22
2.2 The Integral as an Inner Product ....................... 23
2.2.1 Piecewise Continuous Functions .................. 24
2.2.2 Inner Product on  p(α, b) ....................... 25
2.3 Principle of Superposition ............................. 26
2.3.1 Finite Case ..................................... 26
2.3.2 Infinite Case ................................... 28
2.3.3 Hilbert Spaces .................................. 30
2.4 General Fourier Series ................................. 30
2.5 Fourier Sine Series on (0, c) .......................... 31
2.5.1 Odd, Periodic Extensions ........................ 33
2.6 Fourier Cosine Series on (0, c) ........................ 35
2.6.1 Even, Periodic Extensions ....................... 36
2.7 Fourier Series on (—с, c) .............................. 37
2.7.1 2c-Periodic Extensions .......................... 40
2.8 Best Approximation ..................................... 40
2.9 Bessel's Inequality .................................... 45
2.10 Piecewise Smooth Functions ............................. 46
2.11 Fourier Series Convergence ............................. 50
2.11.1 Alternate Form .................................. 50
2.11.2 Riemann-Lebesgue Lemma .......................... 51
2.11.3 A Dirichlet Kernel Lemma ........................ 52
2.11.4 A Fourier Theorem ............................... 54
2.12 2c-Periodic Functions .................................. 58
2.13 Concluding Remarks ..................................... 61
Exercises ................................................... 62
3 Sturm-Liouville Problems .................................... 69
3.1 Basic Examples ......................................... 69
3.2 Regular Sturm-Liouville Problems ....................... 70
3.3 Properties ............................................. 71
3.3.1 Eigenfunction Orthogonality ..................... 72
3.3.2 Real Eigenvalues ................................ 74
3.3.3 Eigenfunction Uniqueness ........................ 75
3.3.4 Non-negative Eigenvalues ........................ 77
3.4 Examples ............................................... 79
3.4.1 Neumann Boundary Conditions on [0, c] ........... 79
3.4.2 Robin and Neumann BCs ........................... 80
3.4.3 Periodic Boundary Conditions .................... 83
3.5 Bessel's Equation ...................................... 85
3.6 Legendre's Equation .................................... 90
Exercises ................................................... 93
4 Heat Equation ............................................... 97
4.1 Heat Equation in 1D .................................... 97
4.2 Boundary Conditions ................................... 100
4.3 Heat Equation in 2D ................................... 101
4.4 Heat Equation in 3D ................................... 103
4.5 Polar-Cylindrical Coordinates ......................... 105
4.6 Spherical Coordinates ................................. 108
Exercises .................................................. 108
5 Heat Transfer in ID ........................................ 113
5.1 Homogeneous IBVP ...................................... 113
5.1.1 Exanfple: Insulated Ends ....................... 114
5.2 Semihomogeneous PDE ................................... 116
5.2.1 Variation of Parameters ........................ 117
5.2.2 Example: Semihomogeneous IBVP .................. 119
5.3 Nonhomogeneous Boundary Conditions .................... 120
5.3.1 Example: Nonhomogeneous Boundary Condition ..... 122
5.3.2 Example: Time-Dependent Boundary Condition ..... 125
5.3.3 Laplace Transforms ............................. 127
5.3.4 Duhamel's Theorem .............................. 128
5.4 Spherical Coordinate Example .......................... 131
Exercises .................................................. 133
6 Heat Transfer in 2D and 3D ................................. 139
6.1 Homogeneous 2D IBVP ................................... 139
6.1.1 Example: Homogeneous IBVP ...................... 142
6.2 Semihomogeneous 2D IBVP ............................... 143
6.2.1 Example: Internal Source or Sink ............... 146
6.3 Nonhomogeneous 2D IBVP ................................ 147
6.4 2D BVP: Laplace and Poisson Equations ................. 150
6.4.1 Dirichlet Problems ............................. 150
6.4.2 Dirichlet Example .............................. 154
6.4.3 Neumann Problems ............................... 156
6.4.4 Neumann Example ................................ 159
6.4.5 Dirichlet, Neumann ВС Example .................. 163
6.4.6 Poisson Problems ............................... 166
6.5 Nonhomogeneous 2D Example ............................. 169
6.6 Time-Dependent BCs .................................... 170
6.7 Homogeneous 3D IBVP ................................... 173
Exercises .................................................. 176
7 Wave Equation .............................................. 181
7.1 Wave Equation in ID ................................... 181
7.1.1 d'Alembert's Solution .......................... 184
7.1.2 Homogeneous IBVP: Series Solution .............. 187
7.1.3 Semihomogeneous IBVP ........................... 190
7.1.4 Nonhomogeneous IBVP ............................ 193
7.1.5 Homogeneous IBVP in Polar Coordinates .......... 195
7.2 Wave Equation in 2D ................................... 199
7.2.1 2D Homogeneous Solution ......................... 199
Exercises .................................................. 202
8 Numerical Methods: an Overview ............................. 207
8.1 Grid Generation ....................................... 208
8.1.1 Adaptive Grids ................................. 210
8.1.2 Multilevel Methods ............................. 212
8.2 Numerical Methods ..................................... 214
8.2.1 Finite Difference Method ....................... 214
8.2.2 Finite Element Method .......................... 216
8.2.3 Finite Analytic Method ......................... 217
8.3 Consistency and Convergence ........................... 218
9 The Finite Difference Method ............................... 219
9.1 Discretization ........................................ 219
9.2 Finite Difference Formulas ............................ 222
9.2.1 First Partials ................................. 222
9.2.2 Second Partials ................................ 222
9.3 ID Heat Equation ...................................... 223
9.3.1 Explicit Formulation ........................... 223
9.3.2 Implicit Formulation ........................... 224
9.4 Crank-Nicolson Method ................................. 226
9.5 Error and Stability ................................... 226
9.5.1 Error Types .................................... 226
9.5.2 Stability ...................................... 227
9.6 Convergence in Practice ............................... 231
9.7 ID Wave Equation ...................................... 231
9.7.1 Implicit Formulation ........................... 231
9.7.2 Initial Conditions ............................. 233
9.8 2D Heat Equation in Cartesian Coordinates ............. 234
9.9 Two-Dimensional Wave Equation ......................... 239
9.10 2D Heat Equation in Polar Coordinates ................. 239
Exercises .................................................. 244
10 Finite Element Method ...................................... 249
10.1 General Framework ..................................... 250
10.2 ID Elliptical Example ................................. 252
10.2.1 Reformulations ................................. 252
10.2.2 Equivalence in Forms ........................... 253
10.2.3 Finite Element Solution ........................ 255
10.3 2D Elliptical Example ................................. 257
10.3.1 Weak Formulation ............................... 257
10.3.2 Finite Element Approximation ................... 258
10.4 Error Analysis ........................................ 261
10.5 ID Parabolic Example .................................. 264
10.5.1 Weak Formulation ............................... 264
10.5.2 Method of Lines ................................ 265
10.5.3 Backward Euler's Method ........................ 266
Exercises .................................................. 268
11 Finite Analytic Method ..................................... 271
11.1 ID Transport Equation ................................. 272
11.1.1 Finite Analytic Solution ....................... 273
11.1.2 FA and FD Coefficient Comparison ............... 275
11.1.3 Hybrid Finite Analytic Solution ................ 279
11.2 2D Transport Equation ................................. 280
11.2.1 FA Solution on Uniform Grids ................... 282
11.2.2 The Poisson Equation ........................... 287
11.3 Convergence and Accuracy .............................. 290
Exercises .................................................. 291
Appendix A: ID Case .......................................... 295
Appendix B: FA 2D Case ....................................... 303
B.l The Case  = 1 ................................... 308
B.2 The Case = -Bx + Ay ............................ 309
References .................................................... 311
Index ......................................................... 315
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