Heat equation

The heat equation is a partial differential equation describing how temperature or another diffusing quantity changes over time and spreads through space. Its solutions smooth initial variations as diffusion proceeds.

Fourier's law: A constitutive law stating that heat flux is proportional to the negative temperature gradient. Combined with energy conservation, it yields the heat equation for conductive heat flow.
Partial differential equation: An equation involving an unknown function and its partial derivatives with respect to multiple variables. The heat equation is a canonical example of this broad equation class.
Joseph Fourier: A French mathematician and physicist whose work established Fourier analysis and a mathematical theory of heat. His 1822 treatise derived and applied the equation to heat conduction.
Heat conduction: The transfer of thermal energy through matter by microscopic interactions without bulk material flow. The equation's original application predicts temperature evolution in conducting solids.
Wave equation: A partial differential equation describing propagation of waves at finite speed. Unlike heat flow, wave solutions propagate disturbances rather than instantly smoothing them.
Thermal diffusivity: A material property measuring how quickly temperature differences spread by conduction. It sets the coefficient controlling the rate of smoothing in the equation.
Diffusion: The net spreading of particles, heat, or other quantities from regions of higher concentration toward lower concentration. The heat equation is the standard continuum model for diffusion over time.
The Analytical Theory of Heat: Joseph Fourier's 1822 treatise presenting a mathematical theory of heat conduction and trigonometric series. This book gave the heat equation its foundational physical and mathematical treatment.
Brownian motion: The random motion of particles suspended in a fluid, driven by molecular collisions. Its probability density evolves according to a heat equation under standard assumptions.
Laplace's equation: The elliptic equation stating that a function's Laplacian is zero. It describes steady-state heat distributions, whereas the heat equation includes time evolution.

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