Differential Equations › Section I · Partial differential equations › Topic I.1
The heat equation
Temperature along a metal bar changes from place to place and from moment to moment, so it depends on two variables at once. The heat equation, built from conservation of energy and Fourier's law, is the partial differential equation that governs it.
In this topic
- Equations with more than one variable
- The setting: a thin, insulated bar
- Conservation of energy
- Fourier's law of conduction
- Worked example: heat flux in an aluminium rod
- The heat equation
- Reading the equation
- Worked example: how fast a hot spot cools
- Initial and boundary conditions
- More boundary conditions and higher dimensions
- Common mistakes to avoid
- Key takeaways
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