An ordinary differential equation governs a function of one variable. But the real world unfolds across space and time at once: heat seeping through a plate, a guitar string trembling, the electric potential filling a room. Their unknown is a function of several variables, and the law relating it to its partial derivatives is a partial differential equation (PDE).

PDEs are the working language of physics and engineering — and, by reputation, hard. This course makes them approachable the same way the rest of the site does: one small idea at a time, each with a picture you can play with.

The shape of the journey

We start with the simplest PDEs of all and work up to the three great equations of physics and the methods — exact and numerical — that solve them. You should already have met the teaser introduction to PDEs and the Fourier series toolkit, which the separation method depends on.

• Orientation. Order, linearity, and the superposition principle that underlies every method here.
• First-order PDEs. The transport equation and the method of characteristics — PDEs as ODEs in disguise.
• Classification. Elliptic, parabolic, hyperbolic; boundary and initial conditions; well-posedness.
• The heat equation. Separation of variables, Fourier modes, diffusion.
• The wave equation. d'Alembert's travelling waves and a string's standing harmonics.
• Laplace & Poisson. Steady states, harmonic functions, the maximum principle.
• Transforms & numerics. The heat kernel on the line; finite differences and their stability.

The path

Orientation.

1. Order and Linearity of PDEs
2. The Superposition Principle

First-order PDEs.

1. The Transport Equation
2. The Method of Characteristics
3. Quasilinear Equations and Shocks

Classification & conditions.

1. Elliptic, Parabolic, Hyperbolic
2. Boundary and Initial Conditions
3. Well-Posedness

The heat equation.

1. Separation of Variables
2. Deriving the Heat Equation
3. The Heated Rod
4. Steady State and Transient

The wave equation.

1. Deriving the Wave Equation
2. d'Alembert's Solution
3. Standing Waves and Harmonics

Laplace & Poisson.

1. Laplace's Equation
2. Laplace on a Rectangle
3. Laplace on a Disk
4. The Maximum Principle
5. Poisson's Equation

Transforms & numerics.

1. The Heat Kernel
2. Finite Differences
3. Numerical Stability

Begin → Order and Linearity of PDEs