What are PDEs?

PDEs · Axiom Academy

Discover how partial differential equations describe systems that vary across both space and time! Understanding the Difference: Ordinary vs. Partial When a quantity depends on multiple variables, we need different types of derivatives. Let's visualize the difference! Exploring a Function of Two Variables Let's explore a function that depends on both x and y. Move the sliders to see how the function value and its partial derivatives change! PDEs Describe Real-World Phenomena Partial differential equations connect these partial derivatives to model how quantities evolve in space and time. Here are some famous examples! See a PDE in Action: Heat Spreading Watch how heat spreads according to the heat equation. Click to add hot spots and see how the temperature evolves! Most real-world systems depend on multiple variables. Temperature varies with position and time. Wave height depends on location and time. Concentrations spread through space as time passes. When we have multiple independent variables, we need partial differential equations to describe how the system evolves. PDEs don't just describe one phenomenon. The same PDE can model heat flow in metals, diffusion of chemicals, spread of populations, option pricing in finance, and image processing in computer vision. Understanding PDEs gives you a universal language for describing change across space and time.

This is the written version of the interactive lesson above. See the full PDEs course.