ISBN-10:
1439821585
ISBN-13:
9781439821589
Pub. Date:
12/02/2010
Publisher:
Taylor & Francis
Handbook of Sinc Numerical Methods / Edition 1

Handbook of Sinc Numerical Methods / Edition 1

by Frank Stenger

Hardcover

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Product Details

ISBN-13: 9781439821589
Publisher: Taylor & Francis
Publication date: 12/02/2010
Series: Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series
Pages: 482
Product dimensions: 6.10(w) x 9.30(h) x 1.20(d)

About the Author

Frank Stenger is a professor emeritus at the University of Utah, where he received the distinguished research award. One of the leading contributors to the area of numerical analysis, Dr. Stenger is the main developer of Sinc numerical methods and has authored over 160 papers in various journals.

Table of Contents

Preface xv

1 One Dimensional Sinc Theory 1

1.1 Introduction and Summary 1

1.1.1 Some Introductory Remarks 2

1.1.2 Uses and Misuses of Sinc 5

1.2 Sampling over the Real Line 7

Problems for Section 1.2 11

1.3 More General Sinc Approximation on R 18

1.3.1 Infinite Term Sinc Approximation on R 18

1.3.2 Finite Term Sinc Approximation on R 25

Problems for Section 1.3 31

1.4 Sinc, Wavelets, Trigonometric and Algebraic Polynomials and Quadratures 32

1.4.1 A General Theorem 35

1.4.2 Explicit Special Cases on [0, 2π] 36

1.4.3 Wavelets and Trigonometric Polynomials 40

1.4.4 Error of Approximation 42

1.4.5 Algebraic Interpolation and Quadrature 48

1.4.6 Wavelet Differentiation 60

1.4.7 Wavelet Indefinite Integration 62

1.4.8 Hilbert Transforms 63

1.4.9 Discrete Fourier Transform 65

Problems for Section 1.4 68

1.5 Sinc Methods on Γ 70

1.5.1 Sinc Approximation on a Finite Interval 70

1.5.2 Sinc Spaces for Intervals and Arcs 72

1.5.3 Important Explicit Transformations 79

1.5.4 Interpolation on Γ 82

1.5.5 Sinc Approximation of Derivatives 87

1.5.6 Sinc Collocation 89

1.5.7 Sinc Quadrature 90

1.5.8 Sinc Indefinite Integration 92

1.5.9 Sinc Indefinite Convolution 93

1.5.10 Laplace Transform Inversion 100

1.5.11 More General 1 - d Convolutions 101

1.5.12 Hilbert and Cauchy Transforms 105

1.5.13 Analytic Continuation 113

1.5.14 Initial Value Problems 116

1.5.15 Wiener-Hopf Equations 118

Problems for Section 1.5 120

1.6 Rational Approximation at Sinc Points 125

1.6.1 Rational Approximation in Mα,β,d(φ) 126

1.6.2 Thiele-Like Algorithms 127

Problems for Section 1.6 128

1.7 Polynomial Methods at Sinc Points 129

1.7.1 Sinc Polynomial Approximation on (0, 1) 130

1.7.2 Polynomial Approximation on Γ 133

1.7.3 Approximation of the Derivative on Γ 134

Problems for Section 1.7 137

2 Sinc Convolution-BIE Methods for PDE & IE 139

2.1 Introduction and Summary 139

2.2 Some Properties of Green's Functions 141

2.2.1 Directional Derivatives 141

2.2.2 Integrals along Arcs 142

2.2.3 Surface Integrals 142

2.2.4 Some Green's Identities 143

Problems for Section 2.2 150

2.3 Free-Space Green's Functions for PDE 150

2.3.1 Heat Problems 151

2.3.2 Wave Problems 151

2.3.3 Helmholtz Equations 152

2.3.4 Biharmonic Green's Functions 154

2.4 Laplace Transforms of Green's Functions 155

2.4.1 Transforms for Poisson Problems 158

2.4.2 Transforms for Helmholtz Equations 163

2.4.3 Transforms for Hyperbolic Problems 168

2.4.4 Wave Equation in R3 × (0, T) 169

2.4.5 Transforms for Parabolic Problems 172

2.4.6 Navier-Stokes Equations 173

2.4.7 Transforms for Biharmonic Green's Functions 180

Problems for Section 2.4 184

2.5 Multi-Dimensional Convolution Based on Sinc 187

2.5.1 Rectangular Region in 2 - d 187

2.5.2 Rectangular Region in 3 - d 191

2.5.3 Curvilinear Region in 2 - d 192

2.5.4 Curvilinear Region in 3 - d 199

2.5.5 Boundary Integral Convolutions 207

Problems for Section 2.5 209

2.6 Theory of Separation of Variables 209

2.6.1 Regions and Function Spaces 210

2.6.2 Analyticity and Separation of Variables 222

Problems for Section 2.6 242

3 Explicit 1-d Programs Solutions via Sinc-Pack 243

3.1 Introduction and Summary 243

3.2 Sinc Interpolation 245

3.2.1 Sinc Points Programs 246

3.2.2 Sinc Basis Programs 248

3.2.3 Interpolation and Approximation 251

3.2.4 Singular, Unbounded Functions 257

Problems for Section 3.2 257

3.3 Approximation of Derivatives 258

Problems for Section 3.3 261

3.4 Sinc Quadrature 262

Problems for Section 3.4 265

3.5 Sinc Indefinite Integration 266

Problems for Section 3.5 268

3.6 Sinc Indefinite Convolution 270

Problems for Section 3.6 274

3.7 Laplace Transform Inversion 275

Problems for Section 3.7 278

3.8 Hilbert and Cauchy Transforms 280

Problems for Section 3.8 283

3.9 Sinc Solution of ODE 284

3.9.1 Nonlinear ODE-IVP on (0, T) via Picard 285

3.9.2 Linear ODE-IVP on (0, T) via Picard 286

3.9.3 Linear ODE-IVP on (0, T) via Direct Solution 289

3.9.4 Second-Order Equations 292

3.9.5 Wiener-Hopf Equations 298

3.10 Wavelet Examples 300

3.10.1 Wavelet Approximations 302

3.10.2 Wavelet Sol'n of a Nonlinear ODE via Picard 309

4 Explicit Program Solutions of PDE via Sinc-Pack 315

4.1 Introduction and Summary 315

4.2 Elliptic PDE 315

4.2.1 Harmonic Sinc Approximation 315

4.2.2 A Poisson-Dirichlet Problem over R2 320

4.2.3 A Poisson-Dirichlet Problem over a Square 323

4.2.4 Neumann to a Dirichlet Problem on Lemniscate 332

4.2.5 A Poisson Problem over a Curvilinear Region in R2 338

4.2.6 A Poisson Problem over R3 350

4.3 Hyperbolic PDE 356

4.3.1 Solving a Wave Equation Over R3 × (0, T) 356

4.3.2 Solving Helmholtz Equation 364

4.4 Parabolic PDE 365

4.4.1 A Nonlinear Population Density Problem 365

4.4.2 Navier-Stokes Example 383

4.5 Performance Comparisons 404

4.5.1 The Problems 404

4.5.2 The Comparisons 405

5 Directory of Programs 409

5.1 Wavelet Formulas 409

5.2 One Dimensional Sinc Programs 410

5.2.1 Standard Sinc Transformations 410

5.2.2 Sinc Points and Weights 411

5.2.3 Interpolation at Sinc Points 412

5.2.4 Derivative Matrices 413

5.2.5 Quadrature 414

5.2.6 Indefinite Integration 416

5.2.7 Indefinite Convolution 416

5.2.8 Laplace Transform Inversion 418

5.2.9 Hilbert Transform Programs 418

5.2.10 Cauchy Transform Programs 419

5.2.11 Analytic Continuation 420

5.2.12 Cauchy Transforms 421

5.2.13 Initial Value Problems 421

5.2.14 Wiener-Hopf Equations 422

5.3 Multi-Dimensional Laplace Transform Programs 422

5.3.1 Q - Function Program 422

5.3.2 Transf. for Poisson Green's Functions 422

5.3.3 Transforms of Helmholtz Green's Function 423

5.3.4 Transforms of Hyperbolic Green's Functions 423

5.3.5 Transforms of Parabolic Green's Functions 424

5.3.6 Transf. of Navier-Stokes Green's Functions 425

5.3.7 Transforms of Biharmonic Green's Functions 425

5.3.8 Example Programs for PDE Solutions 426

Bibliography 429

Index 461

What People are Saying About This

From the Publisher

The author, a well-known expert in this area, has published many papers dealing with various aspects of sinc methods. A key result is that sinc methods can converge very fast under certain assumptions on the given problem. … practical aspects are covered in great detail. In particular, there is an accompanying CD-ROM that contains about 450 MATLAB programs where sinc methods are implemented to solve various problems. The book provides a good description of these programs, so a user with a certain equation to solve can easily find an appropriate sinc algorithm. … it should be useful reading for practitioners who have heard about sinc methods and want to use them.
—Kai Diethelm, Computing Reviews, September 2011

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