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heat equation

The PDE $u_t = ku_{xx}$ (or its higher-dimensional analog) modeling how a quantity spreads out by diffusion over time.

The heat equation $u_t = k\nabla^2u$ (one spatial dimension: $u_t=ku_{xx}$ ) describes how temperature -- or any quantity that diffuses the same way, such as a dissolved concentration -- evolves in time, where $k>0$ is the material's diffusivity.

See also

conservation law Fourier's law

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