Advanced topics in applied mathematics

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1.1 Heaviside Step Function......Page 13
1.2 Dirac Delta Function......Page 15
1.2.1 Macaulay Brackets......Page 18
1.2.3 Test Functions, Linear Functionals, and Distributions......Page 19
1.2.4 Examples: Delta Function......Page 20
1.3.1 Example: Boundary Conditions......Page 22
1.4 Inner Product and Norm......Page 23
1.5 Green's Operator and Green's Function......Page 24
1.5.1 Examples: Direct Integrations......Page 25
1.6 Adjoint Operators......Page 28
1.6.1 Example: Adjoint Operator......Page 29
1.7 Green's Function and Adjoint Green's Function......Page 30
1.8 Green's Function for L......Page 31
1.9 Sturm-Liouville Operator......Page 32
1.9.1 Method of Variable Constants......Page 34
1.9.2 Example: Self-Adjoint Problem......Page 35
1.9.3 Example: Non-Self-Adjoint Problem......Page 36
1.10 Eigenfunctions and Green's Function......Page 38
1.11 Higher-Dimensional Operators......Page 40
1.11.2 Example: Poisson’s Equation in a Rectangle......Page 44
1.11.3 Steady-State Waves and the Helmholtz Equation......Page 45
1.12 Method of Images......Page 46
1.13 Complex Variables and the Laplace Equation......Page 48
1.13.2 Example: Laplace Equation in a Semi-infinite Region......Page 50
1.14 Generalized Green's Function......Page 51
1.14.1 Examples: Generalized Green’s Functions......Page 54
1.14.2 A Récipé for Generalized Green’s Function......Page 55
1.15 Non-Self-Adjoint Operator......Page 56
Suggested Reading......Page 59
Exercises......Page 60
2.1 Classification......Page 68
2.2 Integral Equation from Differentival Equations......Page 70
2.3 Example: Converting Differential Equation......Page 71
2.4 Separable kernel......Page 72
2.5 Eigenvalue Problem......Page 74
2.5.1 Example: Eigenvalues......Page 75
2.5.2 Nonhomogeneous Equation with a Parameter......Page 76
2.6 Hilbert-Schmidt Theory......Page 77
2.7 Iterations, Neumann Series, and Resolvent Kernel......Page 79
2.7.1 Example: Neumann Series......Page 80
2.7.2 Example: Direct Calculation of the Resolvent Kernel......Page 81
2.8 Quadratic Forms......Page 82
2.9 Expansion Theorems for Symmetric Kernels......Page 83
2.10 Eigenfunctions by Iteration......Page 84
2.11 Bound Relations......Page 85
2.12.2 Approximate Solution......Page 86
2.12.3 Numerical Solution......Page 87
2.13 Volterra Equation......Page 88
2.13.1 Example: Volterra Equation......Page 89
2.14 Equations of the First Kind......Page 90
2.15 Dual Integral Equations......Page 92
2.16 Singular Integral Equations......Page 93
2.17 Abel Integral Equation......Page 94
2.18 Boundary Element Method......Page 96
2.18.1 Example: Laplace Operator......Page 98
2.19 Proper Orthogonal Decomposition (POD)......Page 100
Suggested Reading......Page 104
Exercises......Page 105
3.1 Fourier Series......Page 110
3.2 Fourier Transform......Page 111
3.2.1 Riemann-Lebesgue Lemma......Page 114
3.2.2 Localization Lemma......Page 115
3.3 Fourier Integral Theorem......Page 116
3.4 Fourier Cosine and Sine Transforms......Page 117
3.5.1 Derivatives of F......Page 120
3.5.5 Derivatives of f......Page 121
3.6.3 Derivatives of f......Page 122
3.7.1 Exponential Functions......Page 123
3.7.2 Gaussian Function......Page 125
3.7.3 Powers......Page 129
3.8 Convolution Integral......Page 131
3.8.1 Inner Products and Norms......Page 132
3.8.2 Convolution for Trigonometric Transforms......Page 133
3.9 Mixed Trigonometric Transform......Page 134
3.9.1 Example: Mixed Transform......Page 135
1. Laplace Equation in the Half Space......Page 136
2. Steady-State Temperature Distribution in a Strip......Page 139
3. Transient Heat Conduction in a Semi-infinite Rod......Page 142
4. Transient Heat Conduction in a Rod with Radiation Condition......Page 144
5. Transient Heat Conduction in an Infinite Plate......Page 146
6. Laplace Equation in a Semi-infinite 3D Domain......Page 147
1. The Fourier Integral......Page 149
2. Equations of Convolution Type......Page 150
4. Evaluation of Integrals......Page 152
3.12 Hilbert Transform......Page 154
3.13 Cauchy Principal Value......Page 155
3.14 Hilbert Transform on a Unit Circle......Page 157
3.15.1 Cauchy Integral......Page 158
3.15.2 Plemelj Formulas......Page 161
3.16 Complex Fourier Transform......Page 163
3.16.2 Example: Complex Fourier Transform of e|x|......Page 166
3.17.1 Example: Integral Equation......Page 167
3.17.2 Example: Factoring the Kernel......Page 171
3.18 Discrete Fourier Transforms......Page 174
Suggested Reading......Page 177
Exercises......Page 178
4 Laplace Transforms......Page 186
4.1 Inversion Formula......Page 187
4.2.1 Linearity......Page 188
4.2.4 Phase Factor......Page 189
4.2.6 Integral......Page 190
4.3 Transforms of Elementary Functions......Page 191
4.4 Convolution Integral......Page 192
4.5 Inversion Using Elementary Properties......Page 193
4.6 Inversion Using the Residue Theorem......Page 194
4.7 Inversion Requiring Branch Cuts......Page 195
4.8.1 Behavior of f (t) as t →0......Page 198
4.9.1 Ordinary Differential Equations......Page 199
4.9.3 Partial Differential Equations......Page 203
4.9.4 Integral Equations......Page 208
4.9.5 Cagniard–De Hoop Method......Page 210
4.10 Sequences and the Z-Transform......Page 215
4.10.1 Difference Equations......Page 217
4.10.2 First-Order Difference Equation......Page 218
4.10.3 Second-Order Difference Equation......Page 219
4.10.4 Brilluoin Approximation for Crystal Acoustics......Page 222
Exercises......Page 224
Author Index......Page 231
Subject Index......Page 232