Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The **second edition** of **Partial Differential Equations** provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations.

In this book **mathematical** jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.

**Table of Contents:**

Chapter 1/Where PDEs Come From

1.1 What is a Partial Differential Equation? 1

1.2 First-Order Linear Equations 6

1.3 Flows, Vibrations, and Diffusions 10

1.4 Initial and Boundary Conditions 20

1.5 Well-Posed Problems 25

1.6 Types of Second-Order Equations 28

Chapter 2/Waves and Diffusions

2.1 The Wave Equation 33

2.2 Causality and Energy 39

2.3 The Diffusion Equation 42

2.4 Diffusion on the Whole Line 46

2.5 Comparison of Waves and Diffusions 54

Chapter 3/Reflections and Sources

3.1 Diffusion on the Half-Line 57

3.2 Reflections of Waves 61

3.3 Diffusion with a Source 67

3.4 Waves with a Source 71

3.5 Diffusion Revisited 80

Chapter 4/Boundary Problems

4.1 Separation of Variables, The Dirichlet Condition 84

4.2 The Neumann Condition 89

4.3 The Robin Condition 92

Chapter 5/Fourier Series

5.1 The Coefficients 104

5.2 Even, Odd, Periodic, and Complex Functions 113

5.3 Orthogonality and General Fourier Series 118

5.4 Completeness 124

5.5 Completeness and the Gibbs Phenomenon 136

5.6 Inhomogeneous Boundary Conditions 147

Chapter 6/Harmonic Functions

6.1 Laplace’s Equation 152

6.2 Rectangles and Cubes 161

6.3 Poisson’s Formula 165

6.4 Circles, Wedges, and Annuli 172

(The next four chapters may be studied in any order.)

Chapter 7/Green’s Identities and Green’s Functions

7.1 Green’s First Identity 178

7.2 Green’s Second Identity 185

7.3 Green’s Functions 188

7.4 Half-Space and Sphere 191

Chapter 8/Computation of Solutions

8.1 Opportunities and Dangers 199

8.2 Approximations of Diffusions 203

8.3 Approximations of Waves 211

8.4 Approximations of Laplace’s Equation 218

8.5 Finite Element Method 222

Chapter 9/Waves in Space

9.1 Energy and Causality 228

9.2 The Wave Equation in Space-Time 234

9.3 Rays, Singularities, and Sources 242

9.4 The Diffusion and Schro¨ dinger Equations 248

9.5 The Hydrogen Atom 254

Chapter 10/Boundaries in the Plane and in Space

10.1 Fourier’s Method, Revisited 258

10.2 Vibrations of a Drumhead 264

10.3 Solid Vibrations in a Ball 270

10.4 Nodes 278

10.5 Bessel Functions 282

10.6 Legendre Functions 289

10.7 Angular Momentum in Quantum Mechanics 294

Chapter 11/General Eigenvalue Problems

11.1 The Eigenvalues Are Minima of the Potential Energy 299

11.2 Computation of Eigenvalues 304

11.3 Completeness 310

11.4 Symmetric Differential Operators 314

11.5 Completeness and Separation of Variables 318

11.6 Asymptotics of the Eigenvalues 322

Chapter 12/Distributions and Transforms

12.1 Distributions 331

12.2 Green’s Functions, Revisited 338

12.3 Fourier Transforms 343

12.4 Source Functions 349

12.5 Laplace Transform Techniques 353

Chapter 13/PDE Problems from Physics

13.1 Electromagnetism 358

13.2 Fluids and Acoustics 361

13.3 Scattering 366

13.4 Continuous Spectrum 370

13.5 Equations of Elementary Particles 373

Chapter 14/Nonlinear PDEs

14.1 Shock Waves 380

14.2 Solitons 390

14.3 Calculus of Variations 397

14.4 Bifurcation Theory 401

14.5 Water Waves 406

Appendix

A.1 Continuous and Differentiable Functions 414

A.2 Infinite Series of Functions 418

A.3 Differentiation and Integration 420

A.4 Differential Equations 423

A.5 The Gamma Function 425

References 427

Answers and Hints to Selected Exercises 431

Index 446

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