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Physical Sciences
Thermodynamics
1822
Intermediate

Heat Equation

∂u∂t=α∇2u\frac{\partial u}{\partial t} = \alpha \nabla^2 u

Temperature changes proportionally to how curved the temperature distribution is—heat flows from hot to cold.

By Joseph Fourier

Physical Sciences
Heat Equation
1822 · Joseph Fourier
Human Reviewed
84%

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Five doors into the universe behind this equation. Choose your path.

Story PortalWho discovered this?Discover how Joseph Fourier and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — heat exchangers.Math PortalWhat does it derive from?Trace the mathematical lineage from diffusion equation.Future PortalWhere is it going?Fusion reactor thermal management
Launch collection — full story, visual, sound & share card
Why it matters: Enabled thermal engineering, climate modeling, and financial mathematics (Brownian motion).

Discoverers: Joseph Fourier (1822)

What does it mean?

Temperature changes proportionally to how curved the temperature distribution is—heat flows from hot to cold.

Why should I care?

Enabled thermal engineering, climate modeling, and financial mathematics (Brownian motion).

Equation Compass

North — Prerequisites

West — History

East — Applications

South — Derivations

Variables & Units

SymbolNameUnitMeaning
uuTemperature—Temperature field
ααDiffusivitym²/sThermal diffusivity
ttTime—Time

Worked Example

Rod ends at 0°C and 100°C reach steady linear temperature profile.

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Sources & further reading

Search WikipediaSearch Wolfram MathWorldOpen in Wolfram|AlphaWatch explainers on YouTube

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Equation Universe

Heat Equation

∂u∂t=α∇2u\frac{\partial u}{\partial t} = \alpha \nabla^2 u

Real-world impact

Life sciences

Mathematical models drive medicine and biotech.

Photo: Unsplash — laboratory

Temperature changes proportionally to how curved the temperature distribution is—heat flows from hot to cold.

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