The heat equation is a partial differential equation that describes how heat energy is distributed over time in a given medium. It mathematically expresses the relationship between temperature changes and the spatial distribution of heat, often represented as $$\frac{\partial u}{\partial t} = \alpha \nabla^2 u$$, where $$u$$ is the temperature, $$t$$ is time, and $$\alpha$$ is the thermal diffusivity. This equation is crucial for understanding steady-state conduction in various geometries and is also essential for implementing numerical methods to solve conduction problems.
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