Editing Heat equation (section)

== Specific examples ==

=== Heat flow in a uniform rod ===
For heat flow, the heat equation follows from the physical laws of [[conduction (heat)|conduction of heat]] and [[conservation of energy]] {{harv|Cannon|1984}}.

By [[Thermal_conduction#Fourier's_law|Fourier's law]] for an isotropic medium, the rate of flow of heat energy per unit area through a surface is proportional to the negative temperature gradient across it:
: <math>\mathbf{q} = - k \, \nabla u </math>
where <math>k</math> is the [[thermal conductivity]] of the material, <math>u=u(\mathbf{x},t)</math> is the temperature, and <math>\mathbf{q} = \mathbf{q}(\mathbf{x},t)</math> is a [[vector (physics)|vector]] field that represents the magnitude and direction of the heat flow at the point <math>\mathbf{x}</math> of space and time <math>t</math>.

If the medium is a thin rod of uniform section and material, the position ''x'' is a single coordinate and the heat flow <math>q = q(t,x)</math> towards <math>x</math> is a [[scalar field]]. The equation becomes
: <math>q = -k \,\frac{\partial u}{\partial x}</math>

Let <math>Q=Q(x,t)</math> be the [[internal energy]] (heat) per unit volume of the bar at each point and time. The rate of change in heat per unit volume in the material, <math>\partial Q/\partial t</math>, is proportional to the rate of change of its temperature, <math>\partial u/\partial t</math>. That is,
: <math>\frac{\partial Q}{\partial t} = c \, \rho \,  \frac{\partial u}{\partial t}</math>
where <math>c</math> is the specific heat capacity (at constant pressure, in case of a gas) and <math>\rho</math> is the density (mass per unit volume) of the material. This derivation assumes that the material has constant mass density and heat capacity through space as well as time.

Applying the law of conservation of energy to a small element of the medium centred at <math>x</math>, one concludes that the rate at which heat changes at a given point <math>x</math> is equal to the derivative of the heat flow at that point (the difference between the heat flows either side of the particle). That is,
: <math>\frac{\partial Q}{\partial t} = - \frac{\partial q}{\partial x}</math>

From the above equations it follows that
: <math>\frac{\partial u}{\partial t} \;=\; - \frac{1}{c \rho} \frac{\partial q}{\partial x}
\;=\; - \frac{1}{c \rho} \frac{\partial}{\partial x} \left(-k \,\frac{\partial u}{\partial x} \right)
\;=\; \frac{k}{c \rho} \frac{\partial^2 u}{\partial x^2}</math>
which is the heat equation in one dimension, with diffusivity coefficient
: <math>\alpha = \frac{k}{c\rho}</math>

This quantity is called the [[thermal diffusivity]] of the medium.

==== Accounting for radiative loss ====
An additional term may be introduced into the equation to account for radiative loss of heat. According to the [[Stefan–Boltzmann law]], this term is <math>\mu \left(u^4 - v^4\right)</math>, where <math>v=v(x,t)</math> is the temperature of the surroundings, and <math>\mu</math> is a coefficient that depends on the [[Stefan–Boltzmann constant|Stefan-Boltzmann constant]],  the [[emissivity]] of the material, and the geometry. The rate of change in internal energy becomes
: <math>\frac{\partial Q}{\partial t} = - \frac{\partial q}{\partial x} - \mu \left(u^4 - v^4\right)</math>
and the equation for the evolution of <math>u</math> becomes
: <math>\frac{\partial u}{\partial t} = \frac{k}{c \rho} \frac{\partial^2 u}{\partial x^2} - \frac{\mu}{c \rho}\left(u^4 - v^4\right).</math>

==== Non-uniform isotropic medium ====
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<sub>p</sub>'' or ''<math>\rho</math>'') influences which term.
: <math>\rho c_p \frac{\partial T}{\partial t} - \nabla \cdot \left( k \nabla T \right) = \dot q_V </math>
where <math>\dot q_V </math> is the volumetric heat source.

=== Heat flow in non-homogeneous anisotropic media ===
In general, the study of heat conduction is based on several principles. Heat flow is a form of [[energy]] flow, and as such it is meaningful to speak of the time rate of flow of heat into a region of space.
* The time rate of heat flow into a region ''V'' is given by a time-dependent quantity ''q''<sub>''t''</sub>(''V''). We assume ''q'' has a [[Radon-Nikodym Derivative|density]] ''Q'', so that <math display="block"> q_t(V) = \int_V Q(x,t)\,d x \quad </math>
* Heat flow is a time-dependent vector function '''H'''(''x'') characterized as follows: the time rate of heat flowing through an infinitesimal surface element with area ''dS'' and with unit normal vector '''n''' is <math display="block"> \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS .</math> Thus the rate of heat flow into ''V'' is also given by the surface integral <math display="block">  q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS </math> where '''n'''(''x'') is the outward pointing normal vector at ''x''.
* The [[law of heat conduction|Fourier law]] states that heat energy flow has the following linear dependence on the temperature gradient <math display="block"> \mathbf{H}(x) = -\mathbf{A}(x) \cdot \nabla u (x) </math> where '''A'''(''x'') is a 3&nbsp;×&nbsp;3 real [[matrix (mathematics)|matrix]] that is [[symmetric]] and [[positive-definite matrix|positive definite]].
* By the [[divergence theorem]], the previous surface integral for heat flow into ''V'' can be transformed into the volume integral <math display="block">\begin{align}
q_t(V) &= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS \\
&= \int_{\partial V} \mathbf{A}(x) \cdot \nabla u (x) \cdot \mathbf{n}(x) \, dS \\
&= \int_V \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t) \bigr)\,dx 
\end{align}</math>
* The time rate of temperature change at ''x'' is proportional to the heat flowing into an infinitesimal volume element, where the constant of proportionality is dependent on a constant ''κ'' <math display="block"> \partial_t u(x,t) = \kappa(x) Q(x,t)</math>

Putting these equations together gives the general equation of heat flow:
: <math> \partial_t u(x,t) = \kappa(x) \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t)\bigr) </math>

'''Remarks'''
* The coefficient ''κ''(''x'') is the inverse of [[specific heat]] of the substance at ''x'' × [[density]] of the substance at ''x'': <math>\kappa = 1/(\rho c_p)</math>.
* In the case of an isotropic medium, the matrix '''A''' is a scalar matrix equal to [[thermal conductivity]] ''k''.
* In the anisotropic case where the coefficient matrix '''A''' is not scalar and/or if it depends on ''x'', then an explicit formula for the solution of the heat equation can seldom be written down, though it is usually possible to consider the associated abstract [[Cauchy problem]] and show that it is a [[well-posed problem]] and/or to show some qualitative properties (like preservation of positive initial data, infinite speed of propagation, convergence toward an equilibrium, smoothing properties). This is usually done by [[one-parameter semigroup]]s theory: for instance, if ''A'' is a symmetric matrix, then the [[elliptic operator]] defined by <math display="block">Au(x):=\sum_{i, j} \partial_{x_i} a_{i j}(x) \partial_{x_j} u (x)</math> is [[self-adjoint]] and dissipative, thus by the [[spectral theorem]] it generates a [[one-parameter semigroup]].

=== 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>

=== Internal heat generation ===
The function ''u'' above represents temperature of a body. Alternatively, it is sometimes convenient to change units and represent ''u'' as the heat density of a medium. Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units.

Suppose that a body obeys the heat equation and, in addition, generates its own heat per unit volume (e.g., in watts/litre - W/L) at a rate given by a known function ''q'' varying in space and time.<ref>Note that the units of ''u'' must be selected in a manner compatible with those of ''q''. Thus instead of being for thermodynamic temperature ([[Kelvin]] - K), units of ''u'' should be J/L.</ref> Then the heat per unit volume ''u'' satisfies an equation
: <math>\frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q.</math>

For example, a tungsten light bulb filament generates heat, so it would have a positive nonzero value for ''q'' when turned on. While the light is turned off, the value of ''q'' for the tungsten filament would be zero.