It is a curious historical fact that modern quantum mechanics began with two quite different mathematical formulations: the differential equation of Schroedinger and the matrix algebra of Heisenberg. - Richard Feynman

In this section of the course we will learn how to solve another type of differential equation - the partial differential equation. These are also of vital importance to a physicist as they underly quantum mechanics (via the Schroedinger equation) and electromagnetism (via Maxwell’s equations). They are also used to model heat diffusion and wave propagation, amongst other processes.

It is usually impossible to write down explicit formulas for solutions of partial differential equations, and so there is a vast amount of research dedicated to solving these equations using computers. In this lesson we will start to explore some of these numerical approaches, with a focus on understanding the underlying mathematical methods used rather than importing pre-made functions.

Before you begin

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Lesson outline

Topic Objective Quick test
Introduction to PDEs Which physical systems can be described using a PDE?
What are the del and Laplacian operators?
When to I need boundary conditions and initial conditions?		 🍫
Laplace’s equation for electrostatics How do I use the relaxation method to solve Laplace’s equation? :evergreen_tree:
Heat diffusion How do I combine the relaxation and Euler methods to solve the heat diffusion equation? :chestnut:

Course resources

External resources

youtube: https://youtu.be/ly4S0oi3Yz8

youtube: https://youtu.be/ToIXSwZ1pJU