Handbook of Sinc Numerical Methods

Cover......Page 1
Series: Chapman & Hall/CRC Numerical Analysis And Scientific Computing......Page 3
Published Titles......Page 4
Handbook Of Sinc Numerical Methods......Page 6
Copyright......Page 7
Contents......Page 10
Preface......Page 16
1.1 Introduction and Summary......Page 23
1.1.1 Some Introductory Remarks......Page 24
1.1.2 Uses and Misuses of Sinc......Page 27
1.2 Sampling over the Real Line......Page 29
Problems for Section 1.2......Page 33
1.3.1 Infinite Term Sinc Approximation on ΙR......Page 40
1.3.2 Finite Term Sinc Approximation on IR......Page 47
Problems for Section 1.3......Page 53
1.4 Sinc, Wavelets, Trigonometric and Algebraic Polynomials and Quadratures......Page 54
1.4.1 A General Theorem......Page 57
1.4.2 Explicit Special Cases on [0, 2π]......Page 58
1.4.3 Wavelets and Trigonometric Polynomials......Page 62
1.4.4 Error of Approximation......Page 64
1.4.5 Algebraic Interpolation and Quadrature......Page 70
1.4.6 Wavelet Differentiation......Page 82
1.4.7 Wavelet Indefinite Integration......Page 84
1.4.8 Hilbert Transforms......Page 85
1.4.9 Discrete Fourier Transform......Page 87
Problems for Section 1.4......Page 90
1.5.1 Sinc Approximation on a Finite Interval......Page 92
1.5.2 Sinc Spaces for Intervals and Arcs......Page 94
1.5.3 Important Explicit Transformations......Page 101
1.5.4 Interpolation on Γ......Page 104
1.5.5 Sinc Approximation of Derivatives......Page 109
1.5.6 Sinc Collocation......Page 111
1.5.7 Sinc Quadrature......Page 112
1.5.8 Sinc Indefinite Integration......Page 114
1.5.9 Sinc Indefinite Convolution......Page 115
1.5.10 Laplace Transform Inversion......Page 122
1.5.11 More General 1 − d Convolutions......Page 123
1.5.12 Hilbert and Cauchy Transforms......Page 127
1.5.13 Analytic Continuation......Page 135
1.5.14 Initial Value Problems......Page 138
1.5.15 Wiener– Hopf Equations......Page 140
Problems for Section 1.5......Page 142
1.6 Rational Approximation at Sinc Points......Page 147
1.6.1 Rational Approximation in Μ_{α,β,d}(φ)......Page 148
1.6.2 Thiele– Like Algorithms......Page 149
Problems for Section 1.6......Page 150
1.7 Polynomial Methods at Sinc Points......Page 151
1.7.1 Sinc Polynomial Approximation on (0,1)......Page 152
1.7.2 Polynomial Approximation on Γ......Page 155
1.7.3 Approximation of the Derivative on Γ......Page 156
Problems for Section 1.7......Page 159
2.1 Introduction and Summary......Page 161
2.2.1 Directional Derivatives......Page 163
2.2.3 Surface Integrals......Page 164
2.2.4 Some Green\'s Identities......Page 165
2.3 Free– Space Green\'s Functions for PDE......Page 172
2.3.2 Wave Problems......Page 173
2.3.3 Helmholtz Equations......Page 174
2.3.4 Biharmonic Green\'s Functions......Page 176
2.4 Laplace Transforms of Green\'s Functions......Page 177
2.4.1 Transforms for Poisson Problems......Page 180
2.4.2 Transforms for Helmholtz Equations......Page 185
2.4.3 Transforms for Hyperbolic Problems......Page 190
2.4.4 Wave Equation in ΙR^3 × (0, T)......Page 191
2.4.5 Transforms for Parabolic Problems......Page 194
2.4.6 Navier– Stokes Equations......Page 195
2.4.7 Transforms for Biharmonic Green\'s Functions......Page 202
Problems for Section 2.4......Page 206
2.5.1 Rectangular Region in 2 - d......Page 209
2.5.2 Rectangular Region in 3 - d......Page 213
2.5.3 Curvilinear Region in 2 - d......Page 214
2.5.4 Curvilinear Region in 3 - d......Page 221
2.5.5 Boundary Integral Convolutions......Page 229
2.6 Theory of Separation of Variables......Page 231
2.6.1 Regions and Function Spaces......Page 232
2.6.2 Analyticity and Separation of Variables......Page 244
Problems for Section 2.6......Page 264
3.1 Introduction and Summary......Page 265
3.2 Sinc Interpolation......Page 267
3.2.1 Sinc Points Programs......Page 268
3.2.2 Sinc Basis Programs......Page 270
3.2.3 Interpolation and Approximation......Page 273
Problems for Section 3.2......Page 279
3.3 Approximation of Derivatives......Page 280
Problems for Section 3.3......Page 283
3.4 Sinc Quadrature......Page 284
Problems for Section 3.4......Page 287
3.5 Sinc Indefinite Integration......Page 288
Problems for Section 3.5......Page 290
3.6 Sinc Indefinite Convolution......Page 292
Problems for Section 3.6......Page 296
3.7 Laplace Transform Inversion......Page 297
Problems for Section 3.7......Page 300
3.8 Hilbert and Cauchy Transforms......Page 302
Problems for Section 3.8......Page 305
3.9 Sinc Solution of ODE......Page 306
3.9.1 Nonlinear ODE– IVP on (0, T) via Picard......Page 307
3.9.2 Linear ODE– IVP on (0, T) via Picard......Page 308
3.9.3 Linear ODE– IVP on (0, T) via Direct Solution......Page 311
3.9.4 Second- Order Equations......Page 314
3.9.5 Wiener– Hopf Equations......Page 320
3.10 Wavelet Examples......Page 322
3.10.1 Wavelet Approximations......Page 324
3.10.2 Wavelet Sol\'n of a Nonlinear ODE via Picard......Page 331
4.2.1 Harmonic Sinc Approximation......Page 337
4.2.2 A Poisson- Dirichlet Problem over IR^2......Page 342
4.2.3 A Poisson– Dirichlet Problem over a Square......Page 345
4.2.4 Neumann to a Dirichlet Problem on Lemniscate......Page 354
4.2.5 A Poisson Problem over a Curvilinear Region in IR^2......Page 360
4.2.6 A Poisson Problem over IR^3......Page 372
4.3.1 Solving a Wave Equation Over IR^3 × (0, T)......Page 378
4.3.2 Solving Helmholtz Equation......Page 386
4.4.1 A Nonlinear Population Density Problem......Page 387
4.4.2 Navier– Stokes Example......Page 405
4.5.1 The Problems......Page 426
4.5.2 The Comparisons......Page 427
5.1 Wavelet Formulas......Page 431
5.2.1 Standard Sinc Transformations......Page 432
5.2.2 Sinc Points and Weights......Page 433
5.2.3 Interpolation at Sinc Points......Page 434
5.2.4 Derivative Matrices......Page 435
5.2.5 Quadrature......Page 436
5.2.7 Indefinite Convolution......Page 438
5.2.9 Hilbert Transform Programs......Page 440
5.2.10 Cauchy Transform Programs......Page 441
5.2.11 Analytic Continuation......Page 442
5.2.13 Initial Value Problems......Page 443
5.3.2 Transf. for Poisson Green\'s Functions......Page 444
5.3.4 Transforms of Hyperbolic Green\'s Functions......Page 445
5.3.5 Transforms of Parabolic Green\'s Functions......Page 446
5.3.7 Transforms of Biharmonic Green\'s Functions......Page 447
5.3.8 Example Programs for PDE Solutions......Page 448
Bibliography......Page 451