The Intuition behind The Heat Equation

There are three fundamental equations in mathematical physics. The heat, wave, and Laplace’s equations. These three universal equations find numerous applications both in physics and engineering. They are all differential equations containing partial derivatives. But what do these mathematical brilliancies express intuitively? In this article, we are going to dive into the first one, the heat equation.

The Problem of the Iron Rod

Suppose we have an iron rod and we know how heat is distributed across that rod at a specific point in time i.e. we know what the temperature is for every single one of its points. What we are interested in is the following question.

How will the heat distribution change over time?

As we know from high school, heat tends to flow from warmer points to colder ones. So, what we are actually trying to do is to find an equation that describes this process of change. And of course, whenever we want to model a procedure that involves “change” we use partial — far more usually than ordinary — derivatives.

If we imagine that our one-dimensional rod in our problem lies on the x-axis then the differential equation that describes how its heat distribution evolves over time is the following: