the heat equation

by Ali Sayegh

This project explores how a simple conservation law, combined with an empirical observation about heat flow, leads to one of the most important partial differential equations in physics and engineering.

deliverables

1. The Recipe

Before diving into math, identify the three physical ingredients that together produce the heat equation.

Requirements

• Watch the Steve Brunton lecture (linked in Resources) and extract the main ideas he uses to build the equation.
• For each idea, write a one‑sentence plain‑English summary. No equations yet.
• Explain intuitively why you need all ideas. What does each one contribute?

2. The Derivation

Now combine all ingredients into the heat equation.

Requirements

• Using your recipe from Deliverable 1, combine the ingredients into a single PDE for \(u(x,t)\).
• Define any constants that appear, and group them meaningfully.

• State the final result: the 1D heat equation

  \[\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2} + \frac{Q(x,t)}{\rho c}\]

  and define each term, including the thermal diffusivity \(\alpha^2 = \kappa / (\rho c)\).

3. Reflection

Step back and interpret what the equation is actually saying.

Requirements

• Give an intuitive analogy for each term in the heat equation (diffusion term, source term).
• Identify one real‑world application where the heat equation is used and explain why it matters.
• Bonus. What happens in steady state, when \(\partial u / \partial t = 0\)? What equation do you get, and what does it mean physically?

Resources

1. Derivation of the Heat Equation. Steve Brunton. Start here! Extract the recipe!
2. But what is a partial differential equation? 3Blue1Brown. Great visual intuition