Editing Heat equation (section)

=== Three-dimensional problem ===
In the special cases of propagation of heat in an [[isotropic]] and [[wiktionary:Homogeneous|homogeneous]] medium in a 3-[[dimension]]al space, this equation is
: <math> \frac{\partial u}{\partial t} = \alpha \nabla^2 u =
\alpha \left(\frac{\partial^2 u}{\partial x^2} +
\frac{\partial^2 u}{\partial y^2} +
\frac{\partial^2 u}{\partial z^2 }\right) </math>   <math> = \alpha \left( u_{xx} + u_{yy} + u_{zz} \right) </math>
where:
* <math> u = u(x, y, z, t) </math> is temperature as a function of space and time;
* <math> \tfrac{\partial u}{\partial t} </math> is the rate of change of temperature at a point over time;
* <math> u_{xx} </math>, <math> u_{yy} </math>, and <math> u_{zz} </math> are the second spatial [[derivative]]s (''thermal conductions'') of temperature in the <math> x </math>, <math> y </math>, and <math> z </math> directions, respectively;
* <math>\alpha \equiv \tfrac{k}{c_p\rho}</math> is the [[thermal diffusivity]], a material-specific quantity depending on the ''[[thermal conductivity]]'' <math> k </math>, the ''[[specific heat capacity]]'' <math> c_p </math>, and the ''[[mass density]]'' <math> \rho </math>.

The heat equation is a consequence of Fourier's law of conduction (see [[heat conduction]]).

If the medium is not the whole space, in order to solve the heat equation uniquely we also need to specify [[boundary condition]]s for ''u''. To determine uniqueness of solutions in the whole space it is necessary to assume additional conditions, for example an exponential bound on the growth of solutions<ref>{{citation|title=Computational Financial Mathematics using MATHEMATICA: Optimal Trading in Stocks and Options|first=Srdjan|last=Stojanovic|publisher=Springer|year=2003|isbn=9780817641979|pages=112–114|url=https://books.google.com/books?id=ERYzXjt3iYkC&pg=PA112|contribution=3.3.1.3 Uniqueness for heat PDE with exponential growth at infinity}}</ref> or a sign condition (nonnegative solutions are unique by a result of [[David Widder]]).<ref>{{Cite book |last=John |first=Fritz |url=https://books.google.com/books?id=cBib_bsGGLYC&q=fritz+john+partial |title=Partial Differential Equations |date=1991-11-20 |publisher=Springer Science & Business Media |isbn=978-0-387-90609-6 |pages=222 |language=en}}</ref>

Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of [[heat]] from warmer to colder areas of an object. Generally, many different states and starting conditions will tend toward the same stable [[thermodynamic equilibrium|equilibrium]]. As a consequence, to reverse the solution and conclude something about earlier times or initial conditions from the present heat distribution is very inaccurate except over the shortest of time periods.

The heat equation is the prototypical example of a [[parabolic partial differential equation]].

Using the [[Laplace operator]], the heat equation can be simplified, and generalized to similar equations over spaces of arbitrary number of dimensions, as
: <math>u_t = \alpha \nabla^2 u = \alpha \Delta u, </math>
where the Laplace operator, denoted as either Δ or as ∇<sup>2</sup> (the divergence of the gradient), is taken in the spatial variables.

The heat equation governs heat diffusion, as well as other diffusive processes, such as [[particle diffusion]] or the propagation of [[action potential]] in nerve cells. Although they are not diffusive in nature, some quantum mechanics problems are also governed by a mathematical analog of the heat equation (see below). It also can be used to model some phenomena arising in [[finance]], like the [[Black–Scholes]] or the [[Ornstein-Uhlenbeck process]]es. The equation, and various non-linear analogues, has also been used in image analysis.

The heat equation is, technically, in violation of [[special relativity]], because its solutions involve instantaneous propagation of a disturbance. The part of the disturbance outside the forward [[light cone]] can usually be safely neglected, but if it is necessary to develop a reasonable speed for the transmission of heat, a [[Hyperbolic partial differential equation|hyperbolic problem]] should be considered instead – like a partial differential equation involving a second-order time derivative.  Some models of nonlinear heat conduction (which are also parabolic equations) have solutions with finite heat transmission speed.<ref>The [http://mathworld.wolfram.com/PorousMediumEquation.html Mathworld: Porous Medium Equation] and the other related models have solutions with finite wave propagation speed.</ref><ref name="pme">{{Citation | isbn=978-0-19-856903-9|title=The Porous Medium Equation: Mathematical Theory | author=Juan Luis Vazquez|publisher=Oxford University Press, USA|date=2006-12-28}}</ref>