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Heat equation


frame The heat equation predicts that if a hot body is placed in a box of cold water, the temperature of the body will decrease, and eventually (after infinite time, and subject to no external heat sources) the temperature in the box will equalize.
The heat equation is an important [partial differential equation] which 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 ) and the time variable t , the heat equation is

:\frac{\partial u}{\partial t} -\alpha\left(\frac{\partial^2u}{\partial x^2} \frac{\partial^2u}{\partial y^2} \frac{\partial^2u}{\partial z^2}\right)=0

or equivalently

:\frac{\partial u}{\partial t} = \alpha \nabla^2 u

where \alpha is a constant. For the mathematical treatment it is sufficient to consider the case α=1.

The heat equation is of fundamental importance in diverse scientific fields. In [mathematics] , it is the prototypical [parabolic partial differential equation] . In [probability theory] , the heat equation is connected with the study of [Brownian motion] via the [Fokker–Planck equation] . The [diffusion equation] , a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes.

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