Heat equation

The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

For a function u(x,y,z,t) of three spatial variables (x,y,z) (see cartesian coordinates) and the time variable t, the heat equation is

More generally in any coordinate system:

${\displaystyle {\frac {\partial u}{\partial t}}-\alpha \nabla ^{2}u=0}$

where α is a positive constant, and Δ or ∇² denotes the Laplace operator. In the physical problem of temperature variation, u(x,y,z,t) is the temperature and α is the thermal diffusivity. For the mathematical treatment it is sufficient to consider the case α = 1.

Note that the state equation, given by the first law of thermodynamics (i.e. conservation of energy), is written in the following form (assuming no mass transfer or radiation). This form is more general and particularly useful to recognize which property (e.g. c_p or ${\displaystyle \rho }$) influences which term.

...
Wikipedia