Chapter 1 / Where PDEs Come From

1.1 What is a Partial Differential Equation?

1.2 First-Order Linear Equations

1.3 Flows, Vibrations, and Diffusions

1.4 Initial and Boundary Conditions

1.5 Well-Posed Problems

1.6 Types of Second-Order Equations

Chapter 2 / Waves and Diffusions

2.1 The Wave Equation

2.2 Causality and Energy

2.3 The Diffusion Equation

2.4 Diffusion on the Whole Line

2.5 Comparison of Waves and Diffusions

Chapter 3 / Reflections and Sources

3.1 Diffusion on the Half-Line

3.2 Reflections of Waves

3.3 Diffusion with a Source

3.4 Waves with a Source

3.5 Diffusion Revisited

Chapter 4 / Boundary Problems

4.1 Separation of Variables, The Dirichlet Condition

4.2 The Neumann Condition

4.3 The Robin Condition

Chapter 5 / Fourier Series

5.1 The Coefficients

5.2 Even, Odd, Periodic, and Complex Functions

5.3 Orthogonality and General Fourier Series

5.4 Completeness

5.5 Completeness and the Gibbs Phenomenon

5.6 Inhomogeneous Boundary Conditions

Chapter 6 / Harmonic Functions

6.1 Laplace's Equation

6.2 Rectangles and Cubes

6.3 Poisson's Formula

6.4 Circles, Wedges, and Annuli

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

7.1 Green's First Identity

7.2 Green's Second Identity

7.3 Green's Functions

7.4 Half-Space and Sphere

Chapter 8 / Computation of Solutions

8.1 Opportunities and Dangers

8.2 Approximations of Diffusions

8.3 Approximations of Waves

8.4 Approximations of Laplace's Equation

8.5 Finite Element Method

Chapter 9 / Waves in Space

9.1 Energy and Causality

9.2 The Wave Equation in Space-Time

9.3 Rays, Singularities, and Sources

9.4 The Diffusion and Schršdinger Equations

9.5 The Hydrogen Atom

Chapter 10 / Boundaries in the Plane and in Space

10.1 Fourier's Method, Revisited

10.2 Vibrations of a Drumhead

10.3 Solid Vibrations in a Ball

10.4 Nodes

10.5 Bessel Functions

10.6 Legendre Functions

10.7 Angular Momentum in Quantum Mechanics

Chapter 11 / General Eigenvalue Problems

11.1 The Eigenvalues Are Minima of the Potential Energy

11.2 Computation of Eigenvalues

11.3 Completeness

11.4 Symmetric Differential Operators

11.5 Completeness and Separation of Variables

11.6 Asymptotics of the Eigenvalues

Chapter 12 / Distributions and Transforms

12.1 Distributions

12.2 Green's Functions, Revisited

12.3 Fourier Transforms

12.4 Source Functions

12.5 Laplace Transform Techniques

Chapter 13 / PDE Problems from Physics

13.1 Electromagnetism

13.2 Fluids and Acoustics

13.3 Scattering

13.4 Continuous Spectrum

13.5 Equations of Elementary Particles

Chapter 14 / Nonlinear PDEs

14.1 Shock Waves

14.2 Solitons

14.3 Calculus of Variations

14.4 Bifurcation Theory

14.5 Water Waves

Appendix

A.1 Continuous and Differentiable Functions

A.2 Infinite Series of Functions

A.3 Differentiation and Integration

A.4 Differential Equations

A.5 The Gamma Function

References

Answers and Hints to Selected Exercises

Index