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17Calculus Partial Differential Equations - Heat Equation

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Heat Equation

Heat Equation - Additional Resources

Videos

MIT OCW - Heat Equation

video by MIT OCW

Christopher Lum - Solving the 1D Heat Equation

video by Christopher Lum

Related Topics and Links

Heat Equation - Wikipedia

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 go-to guy for learning calculus and differential equations.   As usual, he does a great job of clearly explaining the derivation of the heat equation.

Dr Chris Tisdell - Heat equation derivation

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.

Michel vanBiezen - The Diffusion Equation Playlist

video by Michel vanBiezen

Other derivations can be found in the additional resources panel above.

Really UNDERSTAND Differential Equations

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