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Heat equation

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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