- 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

Section 3.1 | Section 3.2 | Section 3.3 | Section 3.4 | Section 3.5 | ||
---|---|---|---|---|---|---|

Exercise 1 | Exercise 1 | Exercise 9 | Exercise 1 | Exercise 1 | Exercise 9 | Exercise 1 |

Exercise 2 | Exercise 2 | Exercise 10 | Exercise 2 | Exercise 2 | Exercise 10 | Exercise 2 |

Exercise 3 | Exercise 3 | Exercise 11 | Exercise 3 | Exercise 3 | Exercise 11 | |

Exercise 4 | Exercise 4 | Exercise 4 | Exercise 12 | |||

Exercise 5 | Exercise 5 | Exercise 5 | Exercise 13 | |||

Exercise 6 | Exercise 6 | Exercise 14 | ||||

Exercise 7 | Exercise 7 | Exercise 15 | ||||

Exercise 8 | Wave simulation | Exercise 8 |

Section 4.1 | Section 4.2 | Section 4.3 | ||
---|---|---|---|---|

Exercise 1 | Exercise 1 | Exercise 1 | Exercise 8 | Exercise 15 |

Exercise 2 | Exercise 2 | Exercise 2 | Exercise 9 | Exercise 16 |

Exercise 3 | Exercise 3 | Exercise 3 | Exercise 10 | Exercise 17 |

Exercise 4 | Exercise 4 | Exercise 4 | Exercise 11 | Exercise 18 |

Exercise 5 | Exercise 5 | Exercise 12 | Exercise 19 | |

Exercise 6 | Exercise 6 | Exercise 13 | ||

Exercise 7 | Exercise 14 |

Section 6.1 | Section 6.2 | Section 6.3 | Section 6.4 | ||
---|---|---|---|---|---|

Exercise 1 | Exercise 8 | Exercise 1 | Exercise 1 | Exercise 1 | Exercise 8 |

Exercise 2 | Exercise 9 | Exercise 2 | Exercise 2 | Exercise 2 | Exercise 9 |

Exercise 3 | Exercise 10 | Exercise 3 | Exercise 3 | Exercise 3 | Exercise 10 |

Exercise 4 | Exercise 11 | Exercise 4 | Exercise 4 | Exercise 4 | Exercise 11 |

Exercise 5 | Exercise 12 | Exercise 5 | Exercise 5 | Exercise 12 | |

Exercise 6 | Exercise 6 | Exercise 6 | Exercise 13 | ||

Exercise 7 | Exercise 7 | Exercise 7 | Exercise 14 |

Section 8.1 | Section 8.2 | Section 8.3 | Section 8.4 | Section 8.5 |
---|---|---|---|---|

Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Solution |

Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Solution |

Exercise 3 | Solution | Exercise 3 | Solution | Exercise 3 | Solution | Exercise 3 | Solution | Exercise 3 | Solution |

Exercise 4 | Solution | Exercise 4 | Solution | Exercise 4 | Solution | Exercise 4 | Solution | |

Exercise 5 | Solution | Exercise 5 | Solution | Exercise 5 | Solution | Exercise 5 | Solution | |

Exercise 6 | Solution | Exercise 6 | Solution | Exercise 6 | Solution | ||

Exercise 7 | Solution | Exercise 7 | Solution | Exercise 7 | Solution | ||

Exercise 8 | Solution | Exercise 8 | Solution | Exercise 8 | Solution | ||

Exercise 9 | Solution | Exercise 9 | Solution | Exercise 9 | Solution | ||

Exercise 10 | Solution | Exercise 10 | Solution | Exercise 10 | Solution | ||

Exercise 11 | Solution | Exercise 11 | Solution | Exercise 11 | Solution | ||

Exercise 12 | Solution | Exercise 12 | Solution | |||

Exercise 13 | Solution | ||||

Exercise 14 | Solution | ||||

Exercise 15 | Solution |

Section 9.1 | Section 9.2 | Section 9.3 | Section 9.4 | Section 9.5 | |
---|---|---|---|---|---|

Exercise 1 | Exercise 1 | Exercise 11 | Exercise 1 | Exercise 1 | Exercise 1 |

Exercise 2 | Exercise 2 | Exercise 12 | Exercise 2 | Exercise 2 | Exercise 2 |

Exercise 3 | Exercise 3 | Exercise 13 | Exercise 3 | Exercise 3 | |

Exercise 4 | Exercise 4 | Exercise 14 | Exercise 4 | Exercise 4 | |

Exercise 5 | Exercise 5 | Exercise 15 | Exercise 5 | Exercise 5 | |

Exercise 6 | Exercise 6 | Exercise 16 | Exercise 6 | Exercise 6 | |

Exercise 7 | Exercise 7 | Exercise 17 | Exercise 7 | Exercise 7 | |

Exercise 8 | Exercise 8 | Exercise 18 | Exercise 8 | ||

Exercise 9 | Exercise 19 | Exercise 9 | |||

Exercise 10 | Exercise 20 |

Section 10.1 | Section 10.2 | Section 10.3 | Section 10.4 | Section 10.5 | Section 10.6 | Section 10.7 |
---|---|---|---|---|---|---|

Exercise 1 | Exercise 1 | Exercise 1 | Exercise 1 | Exercise 1 | Exercise 1 | Exercise 1 |

Exercise 2 | Exercise 2 | Exercise 2 | Exercise 2 | Exercise 2 | Exercise 2 | Exercise 2 |

Exercise 3 | Exercise 3 | Exercise 3 | Exercise 3 | Exercise 3 | Exercise 3 | Exercise 3 |

Exercise 4 | Exercise 4 | Exercise 4 | Exercise 4 | Exercise 4 | Exercise 4 | Exercise 4 |

Exercise 5 | Exercise 5 | Exercise 5 | Exercise 5 | Exercise 5 | Exercise 5 | |

Exercise 6 | Exercise 6 | Exercise 6 | Exercise 6 | Exercise 6 | ||

Exercise 7 | Exercise 7 | Exercise 7 | Exercise 7 | Exercise 7 | ||

Exercise 8 | Exercise 8 | Exercise 8 | ||||

Exercise 9 | Exercise 9 | |||||

Exercise 10 | Exercise 10 | |||||

Exercise 11 | Exercise 11 | |||||

Exercise 12 | Exercise 12 | |||||

Exercise 13 | Exercise 13 | |||||

Exercise 14 | ||||||

Exercise 15 | ||||||

Exercise 16 | ||||||

Exercise 17 | ||||||

Exercise 18 | ||||||

Exercise 19 |

Section 11.1 | Section 11.2 | Section 11.3 | Section 11.4 | Section 11.5 | Section 11.6 |
---|---|---|---|---|---|

Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Solution |

Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Solution |

Exercise 3 | Solution | Exercise 3 | Solution | Exercise 3 | Solution | Exercise 3 | Solution | Exercise 3 | Solution | Exercise 3 | Solution |

Exercise 4 | Solution | Exercise 4 | Solution | Exercise 4 | Solution | Exercise 4 | Solution | ||

Exercise 5 | Solution | Exercise 5 | Solution | Exercise 5 | Solution | Exercise 5 | Solution | ||

Exercise 6 | Solution | Exercise 6 | Solution | Exercise 6 | Solution | |||

Exercise 7 | Solution | Exercise 7 | Solution | Exercise 7 | Solution | |||

Exercise 8 | Solution | Exercise 8 | Solution | ||||

Exercise 9 | Solution | Exercise 9 | Solution |

Section 12.1 | Section 12.2 | Section 12.3 | Section 12.4 | Section 12.5 |
---|---|---|---|---|

Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Exercise 1 | Exercise 1 |

Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Exercise 2 | Exercise 2 |

Exercise 3 | Solution | Exercise 3 | Solution | Exercise 3 | Exercise 3 | Exercise 3 |

Exercise 4 | Solution | Exercise 4 | Solution | Exercise 4 | Exercise 4 | Exercise 4 |

Exercise 5 | Solution | Exercise 5 | Solution | Exercise 5 | Solution | Exercise 5 | Exercise 5 |

Exercise 6 | Solution | Exercise 6 | Solution | Exercise 6 | Solution | Exercise 6 | Exercise 6 |

Exercise 7 | Solution | Exercise 7 | Solution | Exercise 7 | Solution | Exercise 7 | |

Exercise 8 | Solution | Exercise 8 | Solution | Exercise 8 | Solution | ||

Exercise 9 | Solution | Exercise 9 | Solution | Exercise 9 | ||

Exercise 10 | Solution | Exercise 10 | Solution | |||

Exercise 11 | Solution | Exercise 11 | Solution | |||

Exercise 12 | Solution | Exercise 12 | Solution | |||

Exercise 13 | Solution |

Section 13.1 | Section 13.2 | Section 13.3 | Section 13.4 | Section 13.5 |
---|---|---|---|---|

Exercise 1 | Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Solution |

Exercise 2 | Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Solution |

Exercise 3 | Exercise 3 | Solution | Exercise 3 | Solution | ||

Exercise 4 | Exercise 4 | Solution | Exercise 4 | Solution | ||

Exercise 5 | Exercise 5 | Solution | Exercise 5 | Solution | ||

Exercise 6 | Solution | Exercise 6 | Solution | Exercise 6 | Solution | ||

Exercise 7 | Exercise 7 | Solution | Exercise 7 | Solution | ||

Exercise 8 | Exercise 8 | Solution | |||

Exercise 9 | Solution | ||||

Exercise 10 | Solution | ||||

Exercise 11 | Solution |

Section 14.1 | Section 14.2 | Section 14.3 | Section 14.4 | Section 14.5 |
---|---|---|---|---|

Exercise 1 | Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Solution | Exercise 1 | Solution |

Exercise 2 | Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Solution | Exercise 2 | Solution |

Exercise 3 | Exercise 3 | Solution | Exercise 3 | Solution | Exercise 3 | Solution | Exercise 3 | Solution |

Exercise 4 | Exercise 4 | Solution | Exercise 4 | Solution | Exercise 4 | Solution | Exercise 4 | Solution |

Exercise 5 | Exercise 5 | Solution | Exercise 5 | Solution | Exercise 5 | Solution | |

Exercise 6 | Exercise 6 | Solution | Exercise 6 | Solution | Exercise 6 | Solution | |

Exercise 7 | Exercise 7 | Solution | Exercise 7 | Solution | Exercise 7 | Solution | |

Exercise 8 | Solution | Exercise 8 | Solution | Exercise 8 | Solution | ||

Exercise 9 | Exercise 9 | Solution | Exercise 9 | Solution | ||

Exercise 10 | Solution | Exercise 10 | Solution | Exercise 10 | Solution | ||

Exercise 11 | Solution | Exercise 11 | Solution | Exercise 11 | Solution | ||

Exercise 12 | Exercise 12 | Solution | |||

Exercise 13 | Solution |