Handbook of Sinc Numerical Methods

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  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2010-12-02
  • Publisher: CRC Press

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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

Prefacep. xv
One Dimensional Sinc Theoryp. 1
Introduction and Summaryp. 1
Some Introductory Remarksp. 2
Uses and Misuses of Sincp. 5
Sampling over the Real Linep. 7
Problems for Section 1.2p. 11
More General Sinc Approximation on Rp. 18
Infinite Term Sinc Approximation on Rp. 18
Finite Term Sinc Approximation on Rp. 25
Problems for Section 1.3p. 31
Sinc, Wavelets, Trigonometric and Algebraic Polynomials and Quadraturesp. 32
A General Theoremp. 35
Explicit Special Cases on [0, 2]p. 36
Wavelets and Trigonometric Polynomialsp. 40
Error of Approximationp. 42
Algebraic Interpolation and Quadraturep. 48
Wavelet Differentiationp. 60
Wavelet Indefinite Integrationp. 62
Hilbert Transformsp. 63
Discrete Fourier Transformp. 65
Problems for Section 1.4p. 68
Sinc Methods on p. 70
Sinc Approximation on a Finite Intervalp. 70
Sinc Spaces for Intervals and Arcsp. 72
Important Explicit Transformationsp. 79
Interpolation on p. 82
Sinc Approximation of Derivativesp. 87
Sinc Collocationp. 89
Sinc Quadraturep. 90
Sinc Indefinite Integrationp. 92
Sinc Indefinite Convolutionp. 93
Laplace Transform Inversionp. 100
More General 1 - d Convolutionsp. 101
Hilbert and Cauchy Transformsp. 105
Analytic Continuationp. 113
Initial Value Problemsp. 116
Wiener-Hopf Equationsp. 118
Problems for Section 1.5p. 120
Rational Approximation at Sinc Pointsp. 125
Rational Approximation in M,,d()p. 126
Thiele-Like Algorithmsp. 127
Problems for Section 1.6p. 128
Polynomial Methods at Sinc Pointsp. 129
Sinc Polynomial Approximation on (0, 1)p. 130
Polynomial Approximation on p. 133
Approximation of the Derivative on p. 134
Problems for Section 1.7p. 137
Sinc Convolution-BIE Methods for PDE & IEp. 139
Introduction and Summaryp. 139
Some Properties of Green's Functionsp. 141
Directional Derivativesp. 141
Integrals along Arcsp. 142
Surface Integralsp. 142
Some Green's Identitiesp. 143
Problems for Section 2.2p. 150
Free-Space Green's Functions for PDEp. 150
Heat Problemsp. 151
Wave Problemsp. 151
Helmholtz Equationsp. 152
Biharmonic Green's Functionsp. 154
Laplace Transforms of Green's Functionsp. 155
Transforms for Poisson Problemsp. 158
Transforms for Helmholtz Equationsp. 163
Transforms for Hyperbolic Problemsp. 168
Wave Equation in R3 (0, T)p. 169
Transforms for Parabolic Problemsp. 172
Navier-Stokes Equationsp. 173
Transforms for Biharmonic Green's Functionsp. 180
Problems for Section 2.4p. 184
Multi-Dimensional Convolution Based on Sincp. 187
Rectangular Region in 2 - dp. 187
Rectangular Region in 3 - dp. 191
Curvilinear Region in 2 - dp. 192
Curvilinear Region in 3 - dp. 199
Boundary Integral Convolutionsp. 207
Problems for Section 2.5p. 209
Theory of Separation of Variablesp. 209
Regions and Function Spacesp. 210
Analyticity and Separation of Variablesp. 222
Problems for Section 2.6p. 242
Explicit 1-d Programs Solutions via Sinc-Packp. 243
Introduction and Summaryp. 243
Sinc Interpolationp. 245
Sinc Points Programsp. 246
Sinc Basis Programsp. 248
Interpolation and Approximationp. 251
Singular, Unbounded Functionsp. 257
Problems for Section 3.2p. 257
Approximation of Derivativesp. 258
Problems for Section 3.3p. 261
Sinc Quadraturep. 262
Problems for Section 3.4p. 265
Sinc Indefinite Integrationp. 266
Problems for Section 3.5p. 268
Sinc Indefinite Convolutionp. 270
Problems for Section 3.6p. 274
Laplace Transform Inversionp. 275
Problems for Section 3.7p. 278
Hilbert and Cauchy Transformsp. 280
Problems for Section 3.8p. 283
Sinc Solution of ODEp. 284
Nonlinear ODE-IVP on (0, T) via Picardp. 285
Linear ODE-IVP on (0, T) via Picardp. 286
Linear ODE-IVP on (0, T) via Direct Solutionp. 289
Second-Order Equationsp. 292
Wiener-Hopf Equationsp. 298
Wavelet Examplesp. 300
Wavelet Approximationsp. 302
Wavelet Sol'n of a Nonlinear ODE via Picardp. 309
Explicit Program Solutions of PDE via Sinc-Packp. 315
Introduction and Summaryp. 315
Elliptic PDEp. 315
Harmonic Sinc Approximationp. 315
A Poisson-Dirichlet Problem over R2p. 320
A Poisson-Dirichlet Problem over a Squarep. 323
Neumann to a Dirichlet Problem on Lemniscatep. 332
A Poisson Problem over a Curvilinear Region in R2p. 338
A Poisson Problem over R3p. 350
Hyperbolic PDEp. 356
Solving a Wave Equation Over R3 (0, T)p. 356
Solving Helmholtz Equationp. 364
Parabolic PDEp. 365
A Nonlinear Population Density Problemp. 365
Navier-Stokes Examplep. 383
Performance Comparisonsp. 404
The Problemsp. 404
The Comparisonsp. 405
Directory of Programsp. 409
Wavelet Formulasp. 409
One Dimensional Sinc Programsp. 410
Standard Sinc Transformationsp. 410
Sinc Points and Weightsp. 411
Interpolation at Sinc Pointsp. 412
Derivative Matricesp. 413
Quadraturep. 414
Indefinite Integrationp. 416
Indefinite Convolutionp. 416
Laplace Transform Inversionp. 418
Hilbert Transform Programsp. 418
Cauchy Transform Programsp. 419
Analytic Continuationp. 420
Cauchy Transformsp. 421
Initial Value Problemsp. 421
Wiener-Hopf Equationsp. 422
Multi-Dimensional Laplace Transform Programsp. 422
Q - Function Programp. 422
Transf. for Poisson Green's Functionsp. 422
Transforms of Helmholtz Green's Functionp. 423
Transforms of Hyperbolic Green's Functionsp. 423
Transforms of Parabolic Green's Functionsp. 424
Transf. of Navier-Stokes Green's Functionsp. 425
Transforms of Biharmonic Green's Functionsp. 425
Example Programs for PDE Solutionsp. 426
Bibliographyp. 429
Indexp. 461
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