Handbook of Sinc Numerical Methods

by Stenger; Frank

• ISBN13: 9781439821589
• ISBN10: 1439821585
• Edition: CD
• Format: Hardcover
• Copyright: 2010-12-02
• Publisher: CRC Press

Summary

This handbook is essential for solving numerical problems in mathematics, computer science, and engineering. The methods presented are similar to finite elements but more adept at solving analytic problems with singularities over irregularly shaped yet analytically described regions. The author makes sinc methods accessible to potential users by limiting details as to how or why these methods work. From calculus to partial differential and integral equations, the book can be used to approximate almost every type of operation. It includes more than 470 MATLAB ® programs, along with a CD-ROM containing these programs for ease of use.

Table of Contents

Preface	p. xv
One Dimensional Sinc Theory	p. 1
Introduction and Summary	p. 1
Some Introductory Remarks	p. 2
Uses and Misuses of Sinc	p. 5
Sampling over the Real Line	p. 7
Problems for Section 1.2	p. 11
More General Sinc Approximation on R	p. 18
Infinite Term Sinc Approximation on R	p. 18
Finite Term Sinc Approximation on R	p. 25
Problems for Section 1.3	p. 31
Sinc, Wavelets, Trigonometric and Algebraic Polynomials and Quadratures	p. 32
A General Theorem	p. 35
Explicit Special Cases on [0, 2¿]	p. 36
Wavelets and Trigonometric Polynomials	p. 40
Error of Approximation	p. 42
Algebraic Interpolation and Quadrature	p. 48
Wavelet Differentiation	p. 60
Wavelet Indefinite Integration	p. 62
Hilbert Transforms	p. 63
Discrete Fourier Transform	p. 65
Problems for Section 1.4	p. 68
Sinc Methods on ¿	p. 70
Sinc Approximation on a Finite Interval	p. 70
Sinc Spaces for Intervals and Arcs	p. 72
Important Explicit Transformations	p. 79
Interpolation on ¿	p. 82
Sinc Approximation of Derivatives	p. 87
Sinc Collocation	p. 89
Sinc Quadrature	p. 90
Sinc Indefinite Integration	p. 92
Sinc Indefinite Convolution	p. 93
Laplace Transform Inversion	p. 100
More General 1 - d Convolutions	p. 101
Hilbert and Cauchy Transforms	p. 105
Analytic Continuation	p. 113
Initial Value Problems	p. 116
Wiener-Hopf Equations	p. 118
Problems for Section 1.5	p. 120
Rational Approximation at Sinc Points	p. 125
Rational Approximation in M¿,ß,d(¿)	p. 126
Thiele-Like Algorithms	p. 127
Problems for Section 1.6	p. 128
Polynomial Methods at Sinc Points	p. 129
Sinc Polynomial Approximation on (0, 1)	p. 130
Polynomial Approximation on ¿	p. 133
Approximation of the Derivative on ¿	p. 134
Problems for Section 1.7	p. 137
Sinc Convolution-BIE Methods for PDE & IE	p. 139
Introduction and Summary	p. 139
Some Properties of Green's Functions	p. 141
Directional Derivatives	p. 141
Integrals along Arcs	p. 142
Surface Integrals	p. 142
Some Green's Identities	p. 143
Problems for Section 2.2	p. 150
Free-Space Green's Functions for PDE	p. 150
Heat Problems	p. 151
Wave Problems	p. 151
Helmholtz Equations	p. 152
Biharmonic Green's Functions	p. 154
Laplace Transforms of Green's Functions	p. 155
Transforms for Poisson Problems	p. 158
Transforms for Helmholtz Equations	p. 163
Transforms for Hyperbolic Problems	p. 168
Wave Equation in R3 × (0, T)	p. 169
Transforms for Parabolic Problems	p. 172
Navier-Stokes Equations	p. 173
Transforms for Biharmonic Green's Functions	p. 180
Problems for Section 2.4	p. 184
Multi-Dimensional Convolution Based on Sinc	p. 187
Rectangular Region in 2 - d	p. 187
Rectangular Region in 3 - d	p. 191
Curvilinear Region in 2 - d	p. 192
Curvilinear Region in 3 - d	p. 199
Boundary Integral Convolutions	p. 207
Problems for Section 2.5	p. 209
Theory of Separation of Variables	p. 209
Regions and Function Spaces	p. 210
Analyticity and Separation of Variables	p. 222
Problems for Section 2.6	p. 242
Explicit 1-d Programs Solutions via Sinc-Pack	p. 243
Introduction and Summary	p. 243
Sinc Interpolation	p. 245
Sinc Points Programs	p. 246
Sinc Basis Programs	p. 248
Interpolation and Approximation	p. 251
Singular, Unbounded Functions	p. 257
Problems for Section 3.2	p. 257
Approximation of Derivatives	p. 258
Problems for Section 3.3	p. 261
Sinc Quadrature	p. 262
Problems for Section 3.4	p. 265
Sinc Indefinite Integration	p. 266
Problems for Section 3.5	p. 268
Sinc Indefinite Convolution	p. 270
Problems for Section 3.6	p. 274
Laplace Transform Inversion	p. 275
Problems for Section 3.7	p. 278
Hilbert and Cauchy Transforms	p. 280
Problems for Section 3.8	p. 283
Sinc Solution of ODE	p. 284
Nonlinear ODE-IVP on (0, T) via Picard	p. 285
Linear ODE-IVP on (0, T) via Picard	p. 286
Linear ODE-IVP on (0, T) via Direct Solution	p. 289
Second-Order Equations	p. 292
Wiener-Hopf Equations	p. 298
Wavelet Examples	p. 300
Wavelet Approximations	p. 302
Wavelet Sol'n of a Nonlinear ODE via Picard	p. 309
Explicit Program Solutions of PDE via Sinc-Pack	p. 315
Introduction and Summary	p. 315
Elliptic PDE	p. 315
Harmonic Sinc Approximation	p. 315
A Poisson-Dirichlet Problem over R2	p. 320
A Poisson-Dirichlet Problem over a Square	p. 323
Neumann to a Dirichlet Problem on Lemniscate	p. 332
A Poisson Problem over a Curvilinear Region in R2	p. 338
A Poisson Problem over R3	p. 350
Hyperbolic PDE	p. 356
Solving a Wave Equation Over R3 × (0, T)	p. 356
Solving Helmholtz Equation	p. 364
Parabolic PDE	p. 365
A Nonlinear Population Density Problem	p. 365
Navier-Stokes Example	p. 383
Performance Comparisons	p. 404
The Problems	p. 404
The Comparisons	p. 405
Directory of Programs	p. 409
Wavelet Formulas	p. 409
One Dimensional Sinc Programs	p. 410
Standard Sinc Transformations	p. 410
Sinc Points and Weights	p. 411
Interpolation at Sinc Points	p. 412
Derivative Matrices	p. 413
Quadrature	p. 414
Indefinite Integration	p. 416
Indefinite Convolution	p. 416
Laplace Transform Inversion	p. 418
Hilbert Transform Programs	p. 418
Cauchy Transform Programs	p. 419
Analytic Continuation	p. 420
Cauchy Transforms	p. 421
Initial Value Problems	p. 421
Wiener-Hopf Equations	p. 422
Multi-Dimensional Laplace Transform Programs	p. 422
Q - Function Program	p. 422
Transf. for Poisson Green's Functions	p. 422
Transforms of Helmholtz Green's Function	p. 423
Transforms of Hyperbolic Green's Functions	p. 423
Transforms of Parabolic Green's Functions	p. 424
Transf. of Navier-Stokes Green's Functions	p. 425
Transforms of Biharmonic Green's Functions	p. 425
Example Programs for PDE Solutions	p. 426
Bibliography	p. 429
Index	p. 461
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