- Chapter 1: Where PDEs Come From
- Section 1.1: What is a Partial Differential Equation?
- Section 1.2: First-Order Linear Equations
- Section 1.3: Flows, Vibrations, and Diffusions
- Section 1.4: Initial and Boundary Conditions
- Section 1.5: Well-Posed Problems
- Section 1.6: Types of Second-Order Equations
- Chapter 2: Waves and Diffusions
- Section 2.1: The Wave Equation
- Section 2.2: Causality and Energy
- Section 2.3: The Diffusion Equation
- Section 2.4: Diffusion on the Whole Line
- Section 2.5: Comparison of Waves and Diffusions
- Chapter 3: Reflections and Sources
- Section 3.1: Diffusion on the Half-Line
- Section 3.2: Reflections of Waves
- Section 3.3: Diffusion with a Source
- Section 3.4: Waves with a Source
- Section 3.5: Diffusion Revisited
- Chapter 4: Boundary Problems
- Section 4.1: Separation of Variables, The Dirichlet Condition
- Section 4.2: The Neumann Condition
- Section 4.3: The Robin Condition
- Chapter 5: Fourier Series
- Section 5.1: The Coefficients
- Section 5.2: Even, Odd, Periodic, and Complex Functions
- Section 5.3: Orthogonality and General Fourier Series
- Section 5.4: Completeness
- Section 5.5: Completeness and the Gibbs Phenomenon
- Section 5.6: Inhomogeneous Boundary Conditions
- Chapter 6: Harmonic Functions
- Section 6.1: Laplace's Equation
- Section 6.2: Rectangles and Cubes
- Section 6.3: Poisson's Formula
- Section 6.4: Circles, Wedges, and Annuli
- Chapter 7: Green's Identities and Green's Functions
- Section 7.1: Green's First Identity
- Section 7.2: Green's Second Identity
- Section 7.3: Green's Functions
- Section 7.4: Half-Space and Sphere
- Chapter 8: Computation of Solutions
- Section 8.1: Opportunities and Dangers
- Section 8.2: Approximations of Diffusions
- Section 8.3: Approximations of Waves
- Section 8.4: Approximations of Laplace's Equation
- Section 8.5: Finite Element Method
- Chapter 9: Waves in Space
- Section 9.1: Energy and Causality
- Section 9.2: The Wave Equation in Space-Time
- Section 9.3: Rays, Singularities, and Sources
- Section 9.4: The Diffusion and Schrodinger Equation
- Section 9.5: The Hydrogen Atom
- Chapter 10: Boundaries in the Plane and in Space
- Section 10.1: Fourier's Method, Revisited
- Section 10.2: Vibrations of a Drumhead
- Section 10.3: Solid Vibrations in a Ball
- Section 10.4: Nodes
- Section 10.5: Bessel Functions
- Section 10.6: Legendre Functions
- Section 10.7: Angular Momentum in Quantum Mechanics
- Chapter 11: General Eigenvalue Problems
- Section 11.1: The Eigenvalues Are Minima of the Potential Energy
- Section 11.2: Computation of Eigenvalues
- Section 11.3: Completeness
- Section 11.4: Symmetric Differential Operators
- Section 11.5: Completeness and Separation of Variables
- Section 11.6: Asymptotics of the Eigenvalues
- Chapter 12: Distributions and Transforms
- Section 12.1: Distributions
- Section 12.2: Green's Functions, Revisited
- Section 12.3: Fourier Transforms
- Section 12.4: Source Functions
- Section 12.5: Laplace Transform Techniques
- Chapter 13: PDE Problems from Physics
- Section 13.1: Electromagnetism
- Section 13.2: Fluids and Acoustics
- Section 13.3: Scattering
- Section 13.4: Continuous Spectrum
- Section 13.5: Equations of Elementary Particles
- Chapter 14: Nonlinear PDEs
- Section 14.1: Shock Waves
- Section 14.2: Solitons
- Section 14.3: Calculus of Variations
- Section 14.4: Bifurcation Theory
- Section 14.5: Water Waves
Exercises 1.1.1 - 1.2.4
Exercises 1.2.5 - 1.3.7
Exercises 1.3.8 - 1.5.5
Exercises 1.5.6 - 2.1.9
Exercises 2.1.10 - 2.3.8
Exercises 2.4.1 - 2.4.16
Exercises 2.4.17 - 3.2.8
Exercises 3.2.9 - 3.4.9
Exercises 3.4.10 - 4.2.2
Exercises 4.2.3 - 4.3.14
Exercises 4.3.15 - 5.1.11
Exercises 5.2.1 - 5.2.16
Exercises 5.2.17 - 5.3.15
Exercises 5.4.1 - 5.4.16
Exercises 5.4.17 - 5.5.12
Exercises 5.5.13 - 6.1.1
Exercises 6.1.2 - 6.2.5
Exercises 6.2.6 - 6.4.10
Exercises 6.4.11 - 7.2.2
Exercises 7.2.3 - 7.4.12
Exercises 7.4.13 - 8.1.3
Exercises 8.2.1 - 8.3.1
Exercises 8.3.2 - 8.4.6
Exercises 8.4.7 - 9.1.5
Exercises 9.1.6 - 9.2.13
Exercises 9.2.14 - 9.3.9
Exercises 9.4.1 - 10.1.7
Exercises 10.2.1 - 10.3.9
Exercises 10.3.10 - 10.5.8
Exercises 10.5.9 - 10.6.5
Exercises 10.6.6 - 11.2.3
Exercises 11.2.4 - 11.4.7
Exercises 11.5.1 - 12.1.4
Exercises 12.1.5 - 12.2.8
Exercises 12.2.9 - 12.4.2
Exercises 12.4.3 - 13.1.5
Exercises 13.1.6 - 13.5.2
Exercises 13.5.3 - 14.1.7
Exercises 14.1.8 - 14.2.11
Exercises 14.2.12 - 14.4.3
Exercises 14.4.4 - 14.5.7
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