Editing Heat equation (section)

== Definition ==

Given an open subset {{mvar|U}} of {{math|'''R'''<sup>''n''</sup>}} and a subinterval {{mvar|I}} of {{math|'''R'''}}, one says that a function {{math|''u'' : ''U'' × ''I'' → '''R'''}} is a solution of the '''heat equation''' if
: <math>\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x_1^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2},</math>
where {{math|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>, ''t'')}} denotes a general point of the domain.{{sfn|Evans|2010|p=44}} It is typical to refer to {{mvar|t}} as time and {{math|''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}} as spatial variables, even in abstract contexts where these phrases fail to have their intuitive meaning. The collection of spatial variables is often referred to simply as {{mvar|x}}. For any given value of {{mvar|t}}, the right-hand side of the equation is the [[Laplace operator|Laplacian]] of the function {{math|''u''(⋅, ''t'') : ''U'' → '''R'''}}. As such, the heat equation is often written more compactly as
{{Equation box 1
|equation=<math>\frac{\partial u}{\partial t}=\Delta u</math>
|indent=:
|cellpadding
|border
|border colour = #50C878
|background colour=#ECFCF4}}

In physics and engineering contexts, especially in the context of diffusion through a medium, it is more common to fix a [[Cartesian coordinate system]] and then to consider the specific case of a [[function (mathematics)|function]]  {{math|''u''(''x'', ''y'', ''z'', ''t'')}} of three spatial variables {{math|(''x'', ''y'', ''z'')}} and [[time]] variable {{mvar|t}}. One then says that {{mvar|u}} is a solution of the heat equation if
:<math>\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)</math>
in which {{math|''α''}} is a positive [[coefficient]] called the [[thermal diffusivity]] of the medium. In addition to other physical phenomena, this equation describes the flow of heat in a homogeneous and isotropic medium, with {{math|''u''(''x'', ''y'', ''z'', ''t'')}} being the temperature at the point {{math|(''x'', ''y'', ''z'')}} and time {{mvar|t}}. If the medium is not homogeneous and isotropic, then {{math|α}} would not be a fixed coefficient, and would instead depend on {{math|(''x'', ''y'', ''z'')}}; the equation would also have a slightly different form. In the physics and engineering literature, it is common to use {{math|∇<sup>2</sup>}} to denote the Laplacian, rather than {{math|∆}}.

In mathematics as well as in physics and engineering, it is common to use [[Newton's notation]] for time derivatives, so that <math>\dot u</math> is used to denote {{math|{{sfrac|''∂u''|''∂t''}}}}, so the equation can be written

{{Equation box 1
|equation=<math>\dot u=\Delta u</math>
|indent=:
|cellpadding
|border
|border colour = #50C878
|background colour=#ECFCF4}}

Note also that the ability to use either {{math|∆}} or {{math|∇<sup>2</sup>}} to denote the Laplacian, without explicit reference to the spatial variables, is a reflection of the fact that the Laplacian is independent of the choice of coordinate system. In mathematical terms, one would say that the Laplacian is translationally and rotationally invariant. In fact, it is (loosely speaking) the simplest differential operator which has these symmetries. This can be taken as a significant (and purely mathematical) justification of the use of the Laplacian and of the heat equation in modeling any physical phenomena which are homogeneous and isotropic, of which heat diffusion is a principal example.

=== Diffusivity constant ===

The diffusivity constant {{math|''α''}} is often not present in mathematical studies of the heat equation, while its value can be very important in engineering. This is not a major difference, for the following reason. Let {{mvar|u}} be a function with
:<math>\frac{\partial u}{\partial t}=\alpha\Delta u.</math>
Define a new function <math>v(t,x)=u(t/\alpha,x) </math>. Then, according to the [[chain rule]], one has
{{NumBlk|:|<math>
 \frac{\partial}{\partial t} v(t,x)
= \frac{\partial}{\partial t} u(t/\alpha,x)
= \alpha^{-1}\frac{\partial u}{\partial t}(t/\alpha,x)
= \Delta u(t/\alpha,x)
= \Delta v(t,x)
</math>|{{EquationRef|⁎}}}}
Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of {{math|α}} and solutions of the heat equation with {{math|1=''α'' = 1}}. As such, for the sake of mathematical analysis, it is often sufficient to only consider the case {{math|1=''α'' = 1}}. 

Since <math>\alpha>0</math> there is another option to define a <math>v</math> satisfying <math display="inline">\frac{\partial}{\partial t} v = \Delta v </math> as in ({{EquationNote|⁎}}) above by setting <math>v(t,x) = u(t, \alpha^{1/2} x) </math>. Note that the two possible means of defining the new function <math>v</math> discussed here amount, in physical terms, to changing the unit of measure of time or the unit of measure of length.

=== Nonhomogeneous heat equation ===
The nonhomogeneous heat equation is
: <math>\frac{\partial u}{\partial t} = \Delta u + f</math>
for a given function <math>f = f(x,t)</math> which is allowed to depend on both {{mvar|x}} and {{mvar|t}}.{{sfn|Evans|2010|p=44}} The inhomogeneous heat equation models thermal problems in which a heat source modeled by {{mvar|f}} is switched on. For example, it can be used to model the temperature throughout a room with a heater switched on. If <math>S \subset U</math> is the region of the room where the heater is and the heater is constantly generating {{mvar|q}} units of heat per unit of volume, then {{mvar|f}} would be given by <math>f(x,t) = q 1_S(x)</math>.

=== Steady-state equation ===
A solution to the heat equation <math>\partial u/\partial t = \Delta u</math> is said to be a steady-state solution if it does not vary with respect to time:
: <math>0 = \frac{\partial u}{\partial t} = \Delta u.</math>
Flowing {{math|''u''}} via. the heat equation causes it to become closer and closer as time increases to a steady-state solution. For very large time, {{mvar|''u''}} is closely approximated by a steady-state solution. A steady state solution of the heat equation is equivalently a solution of [[Laplace's equation]].

Similarly, a solution to the nonhomogeneous heat equation <math>\partial u/\partial t = \Delta u + f</math> is said to be a steady-state solution if it does not vary with respect to time:
: <math>0 = \frac{\partial u}{\partial t} = \Delta u + f.</math>
This is equivalently a solution of [[Poisson's equation]].

In the steady-state case, a nonzero spatial thermal gradient <math>\nabla u</math> may (or may not) be present, but if it is, it does not change in time. The steady-state equation describes the end result in all thermal problems in which a source is switched on (for example, an engine started in an automobile), and enough time has passed for all permanent temperature gradients to establish themselves in space, after which these spatial gradients no longer change in time (as again, with an automobile in which the engine has been running for long enough). The other (trivial) solution is for all spatial temperature gradients to disappear as well, in which case the temperature become uniform in space, as well. The steady-state equations are simpler and can help to understand better the physics of the materials without focusing on the dynamics of heat transport. It is widely used for simple engineering problems assuming there is equilibrium of the temperature fields and heat transport, with time.