SECOND EDITION

TRANSFORM METHODS FOR SOLVING PARTIAL

DIFFERENTIAL EQUATIONS

Dean G. Duffy

CHAPMAN & HALL/CRC A CRC Press Company

Boca Raton London New York Washington, D.C.

Contents

1 The Fundamentals 1

1.1 Fourier Transforms 1

1.2 Laplace Transforms 7

1.3 Linear Ordinary Differential Equations 13

1.4 Complex Variables 16

1.5 Multivalued Functions, Branch Points, Branch Cuts and Riemann Surfaces 24

1.6 Some Examples of Integration That Involve

Multivalued Functions 26

1.7 Bessel Functions 49

1.8 What Are Transform Methods? 54

2 Methods Involving Single-Valued Laplace Transforms 61

2.1 Inversion of Laplace Transforms by Contour Integration 61

xv

xvi Transform Methods for Solving Partial Differential Equations

2.2 The Heat Equation 78

2.3 The Wave Equation 141

2.4 Laplace's and Poisson's Equations 182

Papers Using Laplace Transforms to Solve Partial

Differential Equations 191

3 Methods Involving Single-Valued Fourier

and Hankel Transforms 221

3.1 Inversion of Fourier Transforms by Contour Integration 221

3.2 The Wave Equation 236

3.3 The Heat Equation 245

3.4 Laplace's Equation 250

3.5 The Solution of Partial Differential Equations by

Hankel Transforms 270

3.6 Numerical Inversion of Hankel Transforms 293

Papers Using Fourier Transforms to Solve Partial

Differential Equations 302 Papers Using Hankel Transforms to Solve Partial

Differential Equations 305

4 Methods Involving Multivalued Laplace Transforms 307

4.1 Inversion of Laplace Transforms by Contour Integration 307

4.2 Numerical Inversion of Laplace Transforms 372

4.3 The Wave Equation 386

4.4 The Heat Equation 394

Papers Using Laplace Transforms to Solve Partial

Differential Equations 416

Contents xvii

5 Methods Involving Multivalued Fourier Transforms 425

5.1 Inversion of Fourier Transforms by Contour Integration 425

5.2 Numerical Inversion of Fourier Transforms 444

5.3 Solution of Partial Differential Equations 454

Papers Using Fourier Transforms to Solve Partial

Differential Equations 468

6 The Joint Transform Method 471

6.1 The Wave Equation 471

6.2 The Heat and Other Partial Differential Equations 496

6.3 Inversion of the Joint Transform by Cagniard's Method 509

6.4 The Modification of Cagniard's Method by De Hoop 523

Papers Using the Joint Transform Technique 544

Papers Using the Cagniard Technique 553

Papers Using the Cagniard-De Hoop Technique 557

7 The Wiener-Hopf Technique 565

7.1 The Wiener-Hopf Technique When the Factorization Contains No Branch Points 571

7.2 The Wiener-Hopf Technique When the Factorization Contains Branch Points 591

Papers Using the Wiener-Hopf Technique 605

Worked Solutions to Some of the Problems 627

Index 705