Heat Equation
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The Heat Equation
There are several forms of the heat equation that you might see. In general, here is what it looks like. We have \(u\) as a function of \(x\) and \(t\). The symbol \(k\) represents a constant, which depends on the context. The 1D heat equation, sometimes called the diffusion equation is \[ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} \] Written more compactly, you may see \[ \frac{\partial u}{\partial t} = \nabla^2 u \] where \( \nabla^2 \) is the Laplacian. This more compact notation can apply to functions of higher dimensions.
As indicated by the name, the heat equation describes how heat flows over time.
Deriving The Heat Equation
There are lots of videos and websites showing the derivation of the heat equation.
However, not all of them are clear and done well.
Here are a few that we found to be the most helpful.
As with most differential equations topics, different instructors use different notation.
So it would be best to go through several of these derivations.
1. Dr Chris Tisdell
This is our goto guy for learning calculus and differential equations. As usual, he does a great job of clearly explaining the derivation of the heat equation.
video by Dr Chris Tisdell 

2. Paul's Online Notes
Paul's Online Notes shows a nice derivation of the heat equation.
3. Michel van Biezen
Here is a playlist containing five videos deriving the heat equation.
video by Michel vanBiezen 

Other derivations can be found in the additional resources panel above.
Really UNDERSTAND Differential Equations
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