- Chapter 1: A Preview of Applications and Techniques
- Section 1.1: What Is a Partial Differential Equation?
- Section 1.2: Solving and Interpreting a Partial Differential Equation

- Chapter 2: Fourier Series
- Section 2.1: Periodic Functions
- Section 2.2: Fourier Series
- Section 2.3: Fourier Series of Functions with Arbitrary Periods
- Section 2.4: Half-Range Expansions: The Cosine and Sine Series
- Section 2.5: Mean Square Approximation and Parseval's Identity
- Section 2.6: Complex Form of Fourier Series
- Section 2.7: Forced Oscillations
- Section 2.8: Proof of the Fourier Series Representation Theorem
- Section 2.9: Uniform Convergence and Fourier Series
- Section 2.10: Dirichlet Test and Convergence of Fourier Series

- Chapter 3: Partial Differential Equations in Rectangular Coordinates
- Section 3.1: Partial Differential Equations in Physics and Engineering
- Section 3.2: Modeling: Vibrating Strings and the Wave Equation
- Section 3.3: Solution of the One Dimensional Wave Equation: The Method of Separation of Variables
- Section 3.4: D'Alembert's Method
- Section 3.5: The One Dimensional Heat Equation
- Section 3.6: Heat Conduction in Bars: Varying the Boundary Conditions
- Section 3.7: The Two Dimensional Wave and Heat Equations
- Section 3.8: Laplace's Equation in Rectangular Coordinates
- Section 3.9: Poisson's Equation: The Method of Eigenfunction Expansions
- Section 3.10: Neumann and Robin Conditions
- Section 3.11: The Maximum Principle

- Chapter 4: Partial Differential Equations in Polar and Cylindrical Coordinates
- Section 4.1: The Laplacian in Various Coordinate Systems
- Section 4.2: Vibrations of a Circular Membrane: Symmetric Case
- Section 4.3: Vibrations of a Circular Membrane: General Case
- Section 4.4: Laplace's Equation in Circular Regions
- Section 4.5: Laplace's Equation in a Cylinder
- Section 4.6: The Helmholtz and Poisson Equations
- Section 4.7: Bessel's Equation and Bessel Functions
- Section 4.8: Bessel Series Expansions
- Section 4.9: Integral Formulas and Asymptotics for Bessel Functions

- Chapter 5: Partial Differential Equations in Spherical Coordinates
- Section 5.1: Preview of Problems and Methods
- Section 5.2: Dirichlet Problems with Symmetry
- Section 5.3: Spherical Harmonics and the General Dirichlet Problem
- Section 5.4: The Helmholtz Equation with Applications to the Poisson, Heat, and Wave Equations
- Section 5.5: Legendre's Differential Equation
- Section 5.6: Legendre Polynomials and Legendre Series Expansions
- Section 5.7: Associated Legendre Functions and Series Expansions

- Chapter 6: Sturm-Liouville Theory with Engineering Applications
- Section 6.1: Orthogonal Functions
- Section 6.2: Sturm-Liouville Theory
- Section 6.3: The Hanging Chain
- Section 6.4: Fourth Order Sturm-Liouville Theory
- Section 6.5: Elastic Vibrations and Buckling of Beams
- Section 6.6: The Biharmonic Operator
- Section 6.7: Vibrations of Circular Plates

- Chapter 7: The Fourier Transform and Its Applications
- Section 7.1: The Fourier Integral Representation
- Section 7.2: The Fourier Transform
- Section 7.3: The Fourier Transform Method
- Section 7.4: The Heat Equation and Gauss's Kernel
- Section 7.5: A Dirichlet Problem and the Poisson Integral Formula
- Section 7.6: The Fourier Cosine and Sine Transforms
- Section 7.7: Problems Involving Semi-Infinite Intervals
- Section 7.8: Generalized Functions
- Section 7.9: The Nonhomogeneous Heat Equation
- Section 7.10: Duhamel’s Principle

- Chapter 8: The Laplace and Hankel Transforms with Applications
- Section 8.1: The Laplace Transform
- Section 8.2: Further Properties of the Laplace Transform
- Section 8.3: The Laplace Transform Method
- Section 8.4: The Hankel Transform with Applications

- Chapter 9: Finite Difference Numerical Methods
- Section 9.1: The Finite Difference Method for the Heat Equation
- Section 9.2: The Finite Difference Method for the Wave Equation
- Section 9.3: The Finite Difference Method for Laplace's Equation
- Section 9.4: Iteration Methods for Laplace's Equation

- Chapter 10: Sampling and Discrete Fourier Analysis with Applications to Partial Differential Equations
- Section 10.1: The Sampling Theorem
- Section 10.2: Partial Differential Equations and the Sampling Theorem
- Section 10.3: The Discrete and Fast Fourier Transforms
- Section 10.4: The Fourier and Discrete Fourier Transforms

- Chapter 11: An Introduction to Quantum Mechanics
- Section 11.1: Schrodinger's Equation
- Section 11.2: The Hydrogen Atom
- Section 11.3: Heisenberg's Uncertainty Principle
- Section 11.4: Hermite and Laguerre Polynomials

- Chapter 12: Green's Functions and Conformal Mappings
- Section 12.1: Green's Theorem and Identities
- Section 12.2: Harmonic Functions and Green’s Identities
- Section 12.3: Green's Functions
- Section 12.4: Green's Functions for the Disk and the Upper Half-Plane
- Section 12.5: Analytic Functions
- Section 12.6: Solving Dirichlet Problems with Conformal Mappings
- Section 12.7: Green's Functions and Conformal Mappings
- Section 12.8: Neumann Functions and the Solution of Neumann Problems

- Appendix A: Ordinary Differential Equations: Review of Concepts and Methods
- Section A.1: Linear Ordinary Differential Equations
- Section A.2: Linear Ordinary Differential Equations with Constant Coefficients
- Section A.3: Linear Ordinary Differential Equations with Nonconstant Coefficients
- Section A.4: The Power Series Method, Part I
- Section A.5: The Power Series Method, Part II
- Section A.6: The Method of Frobenius

Exercises 1.1.1 - 1.2.2

Exercises 1.2.3 - 1.2.18

Exercises 1.2.19 - 2.1.7

Exercises 2.1.8 - 2.1.23