Editing Heat equation

{{Short description|Partial differential equation describing the evolution of temperature in a region}}
{{More citations needed|date=August 2024}}
[[Image:Heat eqn.gif|thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. The height and redness indicate the temperature at each point.  The initial state has a uniformly hot hoof-shaped region (red) surrounded by uniformly cold region (yellow). As time passes the heat diffuses into the cold region.]]

In [[mathematics]] and [[physics]] (more specifically [[thermodynamics]]), the '''heat equation''' is a [[parabolic partial differential equation]]. The theory of the heat equation was first developed by [[Joseph Fourier]] in 1822 for the purpose of modeling how a quantity such as [[heat]] diffuses through a given region. Since then, the heat equation and its variants have been found to be fundamental in many parts of both pure and applied mathematics.

== 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.

== Interpretation ==
Informally, the Laplacian operator {{math|∆}} gives the difference between the average value of a function in the neighborhood of a point, and its value at that point.  Thus, if {{mvar|u}} is the temperature, {{math|∆u}} conveys if (and by how much) the material surrounding each point is hotter or colder, on the average, than the material at that point.

By the [[second law of thermodynamics]], heat will flow from hotter bodies to adjacent colder bodies, in proportion to the difference of temperature and of the [[thermal conductivity]] of the material between them. When heat flows into (respectively, out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount ([[mass]]) of material, with a [[proportionality (mathematics)|proportionality factor]] called the [[specific heat capacity]] of the material.

By the combination of these observations, the heat equation says the rate <math>\dot u</math> at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The coefficient {{math|α}} in the equation takes into account the thermal conductivity, specific heat, and [[density]] of the material.

=== Interpretation of the equation ===
The first half of the above physical thinking can be put into a mathematical form. The key is that, for any fixed {{mvar|x}}, one has
: <math>\begin{align}
u_{(x)}(0)&=u(x)\\
u_{(x)}'(0)&=0\\
u_{(x)}''(0)&=\frac{1}{n}\Delta u(x)
\end{align}</math>
where {{math|''u''<sub>(''x'')</sub>(''r'')}} is the single-variable function denoting the ''average value'' of {{mvar|u}} over the surface of the sphere of radius {{mvar|r}} centered at {{mvar|x}}; it can be defined by
: <math>u_{(x)}(r)=\frac{1}{\omega_{n-1}r^{n-1}}\int_{\{y:|x-y|=r\}}u\,d\mathcal{H}^{n-1},</math>
in which {{math|ω<sub>''n'' − 1</sub>}} denotes the surface area of the unit ball in {{mvar|n}}-dimensional Euclidean space. This formalizes the above statement that the value of {{math|∆''u''}} at a point {{mvar|x}} measures the difference between the value of {{math|''u''(''x'')}} and the value of {{mvar|u}} at points nearby to {{mvar|x}}, in the sense that the latter is encoded by the values of {{math|''u''<sub>(''x'')</sub>(''r'')}} for small positive values of {{mvar|r}}.

Following this observation, one may interpret the heat equation as imposing an ''infinitesimal averaging'' of a function. Given a solution of the heat equation, the value of {{math|''u''(''x'', ''t'' + τ)}} for a small positive value of {{math|τ}} may be approximated as {{math|{{sfrac|1|2''n''}}}} times the average value of the function {{math|''u''(⋅, ''t'')}} over a sphere of very small radius centered at {{mvar|x}}.

=== Character of the solutions ===
[[Image:Heatequation exampleB.gif|right|frame|Solution of a 1D heat partial differential equation. The temperature (<math>u</math>) is initially distributed over a one-dimensional, one-unit-long interval (''x''&nbsp;=&nbsp;[0,1]) with insulated endpoints. The distribution approaches equilibrium over time.]]

[[File:Heat Transfer.gif|thumb|The behavior of temperature when the sides of a 1D rod are at fixed temperatures (in this case, 0.8 and 0 with initial Gaussian distribution). The temperature approaches a linear function because that is the stable solution of the equation: wherever temperature has a nonzero second spatial derivative, the time derivative is nonzero as well.]]

The heat equation implies that peaks ([[local maximum|local maxima]]) of <math>u</math> will be gradually eroded down, while depressions ([[local minimum|local minima]]) will be filled in.  The value at some point will remain stable only as long as it is equal to the average value in its immediate surroundings.  In particular, if the values in a neighborhood are very close to a linear function <math>A x + B y + C z + D</math>, then the value at the center of that neighborhood will not be changing at that time (that is, the derivative <math>\dot u</math> will be zero).

A more subtle consequence is the [[maximum principle]], that says that the maximum value of <math>u</math> in any region <math>R</math> of the medium will not exceed the maximum value that previously occurred in <math>R</math>, unless it is on the boundary of <math>R</math>. That is, the maximum temperature in a region <math>R</math> can increase only if heat comes in from outside <math>R</math>. This is a property of [[parabolic partial differential equation]]s and is not difficult to prove mathematically (see below).

Another interesting property is that even if <math>u</math> initially has a sharp jump (discontinuity) of value across some surface inside the medium, the jump is immediately smoothed out by a momentary, infinitesimally short but infinitely large rate of flow of heat through that surface. For example, if two isolated bodies, initially at uniform but different temperatures <math>u_0</math> and  <math>u_1</math>, are made to touch each other, the temperature at the point of contact will immediately assume some intermediate value, and a zone will develop around that point where <math>u</math> will gradually vary between <math>u_0</math> and <math>u_1</math>.

If a certain amount of heat is suddenly applied to a point in the medium, it will spread out in all directions in the form of a [[diffusion wave]].  Unlike the [[mechanical wave|elastic]] and [[electromagnetic wave]]s, the speed of a diffusion wave drops with time: as it spreads over a larger region, the temperature gradient decreases, and therefore the heat flow decreases too.

== 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.

== Solving the heat equation using Fourier series ==
[[Image:Temp Rod homobc.svg|right|thumb|300px|Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions.]]
The following solution technique for the heat equation was proposed by [[Joseph Fourier]] in his treatise ''Théorie analytique de la chaleur'', published in 1822. Consider the heat equation for one space variable. This could be used to model heat conduction in a rod. The equation is
{{NumBlk|:|<math>\displaystyle u_t = \alpha u_{xx}</math>|{{EqRef|1}}}}
where ''u'' = ''u''(''x'', ''t'') is a function of two variables ''x'' and ''t''. Here 
* ''x'' is the space variable, so ''x'' ∈ [0, ''L''], where ''L'' is the length of the rod.
* ''t'' is the time variable, so ''t'' ≥ 0.

We assume the initial condition
{{NumBlk|:|<math>u(x,0) = f(x) \quad \forall x \in [0,L]</math> |{{EqRef|2}}}}
where the function ''f'' is given, and the boundary conditions
{{NumBlk|:|<math>u(0,t) = 0 = u(L,t) \quad \forall t > 0 </math>.|{{EqRef|3}}}}

Let us attempt to find a solution of {{EqNote|1}} that is not identically zero satisfying the boundary conditions {{EqNote|3}} but with the following property: ''u'' is a product in which the dependence of ''u'' on ''x'', ''t'' is separated, that is:
{{NumBlk|:|<math> u(x,t) = X(x) T(t).</math>|{{EqRef|4}}}}

This solution technique is called [[separation of variables]]. Substituting ''u'' back into equation {{EqNote|1}},
: <math>\frac{T'(t)}{\alpha T(t)} = \frac{X''(x)}{X(x)}.</math>

Since the right hand side depends only on ''x'' and the left hand side only on ''t'', both sides are equal to some constant value −''λ''. Thus:
{{NumBlk|:|<math>T'(t) = - \lambda \alpha T(t)</math>|{{EqRef|5}}}}
and
{{NumBlk|:|<math>X''(x) = - \lambda X(x).</math>|{{EqRef|6}}}}

We will now show that nontrivial solutions for {{EqNote|6}} for values of ''λ'' ≤ 0 cannot occur:
# Suppose that ''λ'' < 0. Then there exist real numbers ''B'', ''C'' such that <math display="block">X(x) = B e^{\sqrt{-\lambda} \, x} + C e^{-\sqrt{-\lambda} \, x}.</math> From {{EqNote|3}} we get  ''X''(0) = 0 = ''X''(''L'') and therefore ''B'' = 0 = ''C'' which implies ''u'' is identically 0.
# Suppose that ''λ'' = 0. Then there exist real numbers ''B'', ''C'' such that ''X''(''x'') = ''Bx'' + ''C''. From equation {{EqNote|3}} we conclude in the same manner as in 1 that ''u'' is identically 0.
# Therefore, it must be the case that ''λ'' > 0. Then there exist real numbers ''A'', ''B'', ''C'' such that <math display="block">T(t) = A e^{-\lambda \alpha t}</math> and <math display="block">X(x) = B \sin\left(\sqrt{\lambda} \, x\right) + C \cos\left(\sqrt{\lambda} \, x\right).</math> From {{EqNote|3}} we get ''C'' = 0 and that for some positive integer ''n'', <math display="block">\sqrt{\lambda} = n \frac{\pi}{L}.</math>

This solves the heat equation in the special case that the dependence of ''u'' has the special form {{EqNote|4}}.

In general, the sum of solutions to {{EqNote|1}} that satisfy the boundary conditions {{EqNote|3}} also satisfies {{EqNote|1}} and {{EqNote|3}}. We can show that the solution to {{EqNote|1}}, {{EqNote|2}} and {{EqNote|3}} is given by
: <math>u(x,t) = \sum_{n = 1}^{\infty} D_n \sin \left(\frac{n\pi x}{L}\right) e^{-\frac{n^2 \pi^2 \alpha t}{L^2}}</math>
where
: <math>D_n = \frac{2}{L} \int_0^L f(x) \sin \left(\frac{n\pi x}{L}\right ) \, dx.</math>

=== Generalizing the solution technique ===
The solution technique used above can be greatly extended to many other types of equations. The idea is that the operator ''u<sub>xx</sub>'' with the zero boundary conditions can be represented in terms of its [[eigenfunction]]s. This leads naturally to one of the basic ideas of the [[spectral theory]] of linear [[self-adjoint operator]]s.

Consider the [[linear operator]] Δ''u'' = ''u<sub>xx</sub>''. The infinite sequence of functions
: <math> e_n(x) = \sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right)</math>
for ''n'' ≥ 1 are eigenfunctions of Δ. Indeed,
: <math> \Delta e_n = -\frac{n^2 \pi^2}{L^2} e_n. </math>

Moreover, any eigenfunction ''f'' of Δ with the boundary conditions ''f''(0) = ''f''(''L'') = 0 is of the form ''e''<sub>''n''</sub> for some ''n'' ≥ 1. The functions ''e''<sub>''n''</sub> for ''n'' ≥ 1 form an [[orthonormal]] sequence with respect to a certain [[inner product]] on the space of real-valued functions on [0, ''L'']. This means
: <math> \langle e_n, e_m \rangle = \int_0^L e_n(x) e^*_m(x) dx = \delta_{mn}</math>

Finally, the sequence {''e''<sub>''n''</sub>}<sub>''n'' ∈ '''N'''</sub> spans a dense linear subspace of ''L''<sup>2</sup>((0, ''L'')). This shows that in effect we have [[diagonal matrix|diagonalized]] the operator Δ.

=== Mean-value property ===
Solutions of the heat equations 
: <math>(\partial_t -\Delta)u=0</math>
satisfy a mean-value property analogous to the [[Harmonic_function#The_mean_value_property|mean-value properties of harmonic functions]], solutions of 
: <math>\Delta u = 0,</math>
though a bit more complicated. Precisely, if ''u'' solves 
: <math>(\partial_t -\Delta)u=0</math>
and 
: <math>(x,t)+E_\lambda\subset\mathrm{dom}(u)</math>
then
: <math>u(x,t)=\frac{\lambda}{4}\int_{E_\lambda}u(x-y,t-s)\frac{|y|^2}{s^2}ds\,dy,</math>
where ''E<sub>λ</sub>'' is a ''heat-ball'', that is a super-level set of the fundamental solution of the heat equation:
: <math>E_\lambda := \{(y,s) : \Phi(y,s) > \lambda\},</math>
: <math>\Phi(x,t) := (4t\pi)^{-\frac{n}{2}}\exp\left(-\frac{|x|^2}{4t}\right).</math>
Notice that 
: <math>\mathrm{diam}(E_\lambda)=o(1)</math>
as ''λ'' → ∞ so the above formula holds for any (''x'', ''t'') in the (open) set dom(''u'') for ''λ'' large enough.<ref>Conversely, any function ''u'' satisfying the above mean-value property on an open domain of '''R'''<sup>''n''</sup> × '''R''' is a solution of the heat equation</ref>

== Fundamental solutions ==
{{see also|Weierstrass transform}}
A [[fundamental solution]] of the heat equation is a solution that corresponds to the initial condition of an initial point source of heat at a known position. These can be used to find a general solution of the heat equation over certain domains (see, for instance, {{harv|Evans|2010}}).

In one variable, the [[Green's function]] is a solution of the initial value problem (by [[Duhamel's principle]], equivalent to the definition of Green's function as one with a delta function as solution to the first equation)
: <math>\begin{cases}
u_t(x,t) - k u_{xx}(x,t) = 0& (x, t) \in \R \times (0, \infty)\\
u(x,0)=\delta(x)&
\end{cases}</math>
where ''<math>\delta</math>'' is the [[Dirac delta function]]. The fundamental solution to this problem is given by the [[heat kernel]]
: <math>\Phi(x,t)=\frac{1}{\sqrt{4\pi kt}}\exp\left(-\frac{x^2}{4kt}\right).</math>

One can obtain the general solution of the one variable heat equation with initial condition ''u''(''x'', 0) = ''g''(''x'') for −∞ < ''x'' < ∞ and 0 < ''t'' < ∞ by applying a [[convolution]]:
: <math>u(x,t) = \int \Phi(x-y,t) g(y) dy.</math>

In several spatial variables, the fundamental solution solves the analogous problem
: <math>\begin{cases}
u_t(\mathbf{x},t) - k \sum_{i=1}^nu_{x_ix_i}(\mathbf{x},t) = 0 & (\mathbf{x}, t) \in \R^n \times (0, \infty)\\
u(\mathbf{x},0)=\delta(\mathbf{x})
\end{cases}</math>

The ''n''-variable fundamental solution is the product of the fundamental solutions in each variable; i.e.,
: <math>\Phi(\mathbf{x},t) = \Phi(x_1,t) \Phi(x_2,t) \cdots \Phi(x_n,t) = \frac{1}{\sqrt{(4\pi k t)^n}} \exp \left (-\frac{\mathbf{x}\cdot\mathbf{x}}{4kt} \right).</math>

The general solution of the heat equation on '''R'''<sup>''n''</sup> is then obtained by a convolution, so that to solve the initial value problem with ''u''('''x''', 0) = ''g''('''x'''), one has
: <math>u(\mathbf{x},t) = \int_{\R^n}\Phi(\mathbf{x}-\mathbf{y},t)g(\mathbf{y})d\mathbf{y}.</math>

The general problem on a domain Ω in '''R'''<sup>''n''</sup> is
: <math> \begin{cases}
u_t(\mathbf{x},t) - k \sum_{i=1}^nu_{x_ix_i}(\mathbf{x},t) = 0& (\mathbf{x}, t) \in \Omega\times (0, \infty)\\
u(\mathbf{x},0)=g(\mathbf{x})&\mathbf{x}\in\Omega
\end{cases}</math>
with either [[Dirichlet problem|Dirichlet]] or [[Neumann problem|Neumann]] boundary data.  A [[Green's function]] always exists, but unless the domain Ω can be readily decomposed into one-variable problems (see below), it may not be possible to write it down explicitly.  Other methods for obtaining Green's functions include the [[method of images]], [[separation of variables]], and [[Laplace transform]]s (Cole, 2011). 

=== Some Green's function solutions in 1D ===
A variety of elementary Green's function solutions in one-dimension are recorded here; many others are available elsewhere.<ref>The [https://www.engr.unl.edu/~glibrary/home/index.html Green's Function Library] contains a variety of fundamental solutions to the heat equation.</ref>  In some of these, the spatial domain is (−∞,∞).  In others, it is the semi-infinite interval (0,∞) with either [[Neumann problem|Neumann]] or [[Dirichlet problem|Dirichlet]] boundary conditions.  One further variation is that some of these solve the inhomogeneous equation
: <math>u_{t}=ku_{xx}+f.</math>
where ''f'' is some given function of ''x'' and ''t''.

==== Homogeneous heat equation ====
; Initial value problem on (−∞,∞) :
: <math>\begin{cases}
u_{t}=ku_{xx} & (x, t) \in \R \times (0, \infty) \\ 
u(x,0)=g(x) & \text{Initial condition}
\end{cases} </math>
: <math>u(x,t) = \frac{1}{\sqrt{4\pi kt}} \int_{-\infty}^{\infty} \exp\left(-\frac{(x-y)^2}{4kt}\right)g(y)\,dy </math>

[[Image:Fundamental solution to the heat equation.gif|right|thumb|upright=2|Fundamental solution of the one-dimensional heat equation. Red: time course of <math>\Phi(x,t)</math>. Blue: time courses of <math>\Phi(x_0,t)</math> for two selected points x<sub>0</sub> = 0.2 and x<sub>0</sub> = 1. Note the different rise times/delays and amplitudes.<br/> [https://www.geogebra.org/m/SV6PruXx Interactive version.]]]

''Comment''. This solution is the [[convolution]] with respect to the variable ''x'' of the fundamental solution
: <math>\Phi(x,t) := \frac{1}{\sqrt{4\pi kt}} \exp\left(-\frac{x^2}{4kt}\right),</math>
and the function ''g''(''x''). (The [[Green's function number]] of the fundamental solution is X00.)

Therefore, according to the general properties of the convolution with respect to differentiation, ''u'' = ''g'' ∗ Φ is a solution of the same heat equation, for
: <math>\left (\partial_t-k\partial_x^2 \right )(\Phi*g)=\left [\left (\partial_t-k\partial_x^2 \right )\Phi \right ]*g=0.</math>

Moreover,
: <math>\Phi(x,t)=\frac{1}{\sqrt{t}}\,\Phi\left(\frac{x}{\sqrt{t}},1\right)</math>
: <math>\int_{-\infty}^{\infty}\Phi(x,t)\,dx=1,</math>
so that, by general facts about [[mollifier|approximation to the identity]], Φ(⋅, ''t'') ∗ ''g'' → ''g'' as ''t'' → 0 in various senses, according to the specific ''g''. For instance, if ''g'' is assumed bounded and continuous on '''R''' then {{nowrap|Φ(⋅, ''t'') ∗ ''g''}} converges uniformly to ''g'' as ''t'' → 0, meaning that ''u''(''x'', ''t'') is continuous on {{nowrap|'''R''' × [0, ∞)}} with {{nowrap|1=''u''(''x'', 0) = ''g''(''x'').}}

; Initial value problem on (0,∞) with homogeneous Dirichlet boundary conditions :
: <math>\begin{cases}
u_{t}=ku_{xx} & (x, t) \in [0, \infty) \times (0, \infty) \\ 
u(x,0)=g(x) & \text{IC} \\
u(0,t)=0 & \text{BC}
\end{cases} </math>
: <math>u(x,t)=\frac{1}{\sqrt{4\pi kt}} \int_{0}^{\infty} \left[\exp\left(-\frac{(x-y)^2}{4kt}\right)-\exp\left(-\frac{(x+y)^2}{4kt}\right)\right] g(y)\,dy </math>

''Comment.'' This solution is obtained from the preceding formula as applied to the data ''g''(''x'') suitably extended to '''R''', so as to be an [[odd function]], that is, letting ''g''(−''x'') := −''g''(''x'') for all ''x''. Correspondingly, the solution of the initial value problem on (−∞,∞) is an odd function with respect to the variable ''x'' for all values of ''t'', and in particular it satisfies the homogeneous Dirichlet boundary conditions ''u''(0, ''t'') = 0.
The [[Green's function number]] of this solution is X10.

; Initial value problem on (0,∞) with homogeneous Neumann boundary conditions :
: <math>\begin{cases}
u_{t}=ku_{xx} & (x, t) \in [0, \infty) \times (0, \infty) \\
u(x,0)=g(x) & \text{IC} \\
u_{x}(0,t)=0 & \text{BC}
\end{cases} </math>
: <math>u(x,t)=\frac{1}{\sqrt{4\pi kt}} \int_{0}^{\infty} \left[\exp\left(-\frac{(x-y)^2}{4kt}\right)+\exp\left(-\frac{(x+y)^2}{4kt}\right)\right]g(y)\,dy </math>

''Comment.'' This solution is obtained from the first solution formula as applied to the data ''g''(''x'') suitably extended to '''R''' so as to be an [[even function]], that is, letting ''g''(−''x'') := ''g''(''x'') for all ''x''. Correspondingly, the solution of the initial value problem on '''R''' is an even  function with respect to the variable ''x'' for all values of ''t'' > 0, and in particular, being smooth, it satisfies the homogeneous Neumann boundary conditions ''u<sub>x</sub>''(0, ''t'') = 0.  The [[Green's function number]] of this solution is X20.

; Problem on (0,∞) with homogeneous initial conditions and non-homogeneous Dirichlet boundary conditions :
: <math>\begin{cases}
u_{t}=ku_{xx} & (x, t) \in [0, \infty) \times (0, \infty) \\
u(x,0)=0 & \text{IC} \\
u(0,t)=h(t) & \text{BC}
\end{cases} </math>
: <math>u(x,t)=\int_{0}^{t} \frac{x}{\sqrt{4\pi k(t-s)^3}} \exp\left(-\frac{x^2}{4k(t-s)}\right)h(s)\,ds, \qquad\forall x>0</math>

''Comment''. This solution is the [[convolution]] with respect to the variable ''t'' of
: <math>\psi(x,t):=-2k \partial_x \Phi(x,t) = \frac{x}{\sqrt{4\pi kt^3}} \exp\left(-\frac{x^2}{4kt}\right)</math>
and the function ''h''(''t''). Since Φ(''x'', ''t'') is the fundamental solution of
: <math>\partial_t-k\partial^2_x,</math>
the function ''ψ''(''x'', ''t'') is also a solution of the same heat equation, and so is ''u'' := ''ψ'' ∗ ''h'', thanks to general properties of the convolution with respect to differentiation. Moreover,
: <math>\psi(x,t)=\frac{1}{x^2}\,\psi\left(1,\frac{t}{x^2}\right)</math>
: <math>\int_0^{\infty}\psi(x,t)\,dt=1,</math>
so that, by general facts about [[mollifier|approximation to the identity]], ''ψ''(''x'', ⋅) ∗ ''h'' → ''h'' as ''x'' → 0 in various senses, according to the specific ''h''. For instance, if ''h'' is assumed continuous on '''R''' with support in [0, ∞) then ''ψ''(''x'', ⋅) ∗ ''h'' converges uniformly on compacta to ''h'' as ''x'' → 0, meaning that ''u''(''x'', ''t'') is continuous on {{nowrap|[0, ∞) × [0, ∞)}} with {{nowrap|1=''u''(0, ''t'') = ''h''(''t'').}}
[[File:2D Nonhomogeneous heat equation .gif|thumb|Depicted is a numerical solution of the non-homogeneous heat equation. The equation has been solved with 0 initial and boundary conditions and a source term representing a stove top burner.]]

==== Inhomogeneous heat equation  ====
; Problem on (-∞,∞) homogeneous initial conditions :
: 
: 

''Comment''. This solution is the convolution in '''R'''<sup>2</sup>, that is with respect to both the variables ''x'' and ''t'', of the fundamental solution
: <math>\Phi(x,t) := \frac{1}{\sqrt{4\pi kt}} \exp\left(-\frac{x^2}{4 kt}\right)</math>
and the function ''f''(''x'', ''t''), both meant as defined on the whole '''R'''<sup>2</sup> and identically 0 for all ''t'' → 0. One verifies that
: <math>\left (\partial_t-k \partial_x^2 \right )(\Phi*f)=f,</math>
which expressed in the language of distributions becomes
: <math>\left (\partial_t-k \partial_x^2 \right )\Phi=\delta,</math>
where the distribution δ is the [[Dirac's delta function]], that is the evaluation at 0.

; Problem on (0,∞) with homogeneous Dirichlet boundary conditions and initial conditions:
: <math>\begin{cases}
u_{t}=ku_{xx}+f(x,t) & (x, t) \in [0, \infty) \times (0, \infty)  \\
u(x,0)=0 & \text{IC} \\
u(0,t)=0 & \text{BC}
\end{cases} </math>
: <math>u(x,t)=\int_{0}^{t}\int_{0}^{\infty} \frac{1}{\sqrt{4\pi k(t-s)}} \left(\exp\left(-\frac{(x-y)^2}{4k(t-s)}\right)-\exp\left(-\frac{(x+y)^2}{4k(t-s)}\right)\right)
f(y,s)\,dy\,ds </math>

''Comment''. This solution is obtained from the preceding formula as applied to the data ''f''(''x'', ''t'') suitably extended to '''R''' × [0,∞), so as to be an odd function of the variable ''x'', that is, letting ''f''(−''x'', ''t'') := −''f''(''x'', ''t'') for all ''x'' and ''t''. Correspondingly, the solution of the inhomogeneous problem on (−∞,∞) is an odd function with respect to the variable ''x'' for all values of ''t'', and in particular it satisfies the homogeneous Dirichlet boundary conditions ''u''(0, ''t'') = 0.

; Problem on (0,∞) with homogeneous Neumann boundary conditions and initial conditions :
: <math>\begin{cases}
u_{t} = ku_{xx}+f(x,t) & (x, t) \in [0, \infty) \times (0, \infty)  \\
u(x,0)=0 & \text{IC} \\
u_x(0,t)=0 & \text{BC}
\end{cases} </math>
: <math>u(x,t)=\int_{0}^{t}\int_{0}^{\infty} \frac{1}{\sqrt{4\pi k(t-s)}} \left(\exp\left(-\frac{(x-y)^2}{4k(t-s)}\right)+\exp\left(-\frac{(x+y)^2}{4k(t-s)}\right)\right)
f(y,s)\,dy\,ds </math>

''Comment''. This solution is obtained from the first formula as applied to the data ''f''(''x'', ''t'') suitably extended to '''R''' × [0,∞), so as to be an even function of the variable ''x'', that is, letting ''f''(−''x'', ''t'') := ''f''(''x'', ''t'') for all ''x'' and ''t''. Correspondingly, the solution of the inhomogeneous problem on (−∞,∞) is an even function with respect to the variable ''x'' for all values of ''t'', and in particular, being a smooth function, it satisfies the homogeneous Neumann boundary conditions ''u<sub>x</sub>''(0, ''t'') = 0.

==== Examples ====
Since the heat equation is linear, solutions of other combinations of boundary conditions, inhomogeneous term, and initial conditions can be found by taking an appropriate [[linear combination]] of the above Green's function solutions.

For example, to solve
: <math>\begin{cases}
u_{t}=ku_{xx}+f & (x, t) \in \R \times (0, \infty) \\
u(x,0)=g(x) & \text{IC}
\end{cases} </math>
let ''u'' = ''w'' + ''v'' where ''w'' and ''v'' solve the problems
: <math>\begin{cases} 
v_{t}=kv_{xx}+f, \, w_{t}=kw_{xx} \, & (x, t) \in \R \times (0, \infty) \\ 
v(x,0)=0,\, w(x,0)=g(x) \, & \text{IC}
\end{cases} </math>

Similarly, to solve
: <math>\begin{cases}
u_{t}=ku_{xx}+f & (x, t) \in [0, \infty) \times (0, \infty) \\
u(x,0)=g(x) & \text{IC} \\
u(0,t)=h(t) & \text{BC}
\end{cases} </math>
let ''u'' = ''w'' + ''v'' + ''r'' where ''w'', ''v'', and ''r'' solve the problems
: <math>\begin{cases} 
v_{t}=kv_{xx}+f, \, w_{t}=kw_{xx}, \, r_{t}=kr_{xx} & (x, t) \in [0, \infty) \times (0, \infty)  \\ 
v(x,0)=0, \; w(x,0)=g(x), \; r(x,0)=0 & \text{IC} \\ 
v(0,t)=0, \; w(0,t)=0, \; r(0,t)=h(t) & \text{BC}
\end{cases}</math>

== Applications ==

As the prototypical [[parabolic partial differential equation]], the heat equation is among the most widely studied topics in [[pure mathematics]], and its analysis is regarded as fundamental to the broader field of [[partial differential equation]]s. The heat equation can also be considered on [[Riemannian manifold]]s, leading to many geometric applications. Following work of [[Subbaramiah Minakshisundaram]] and [[Åke Pleijel]], the heat equation is closely related with [[spectral geometry]]. A seminal [[harmonic map|nonlinear variant of the heat equation]] was introduced to [[differential geometry]] by [[James Eells]] and [[Joseph H. Sampson|Joseph Sampson]] in 1964, inspiring the introduction of the [[Ricci flow]] by [[Richard S. Hamilton|Richard Hamilton]] in 1982 and culminating in the proof of the [[Poincaré conjecture]] by [[Grigori Perelman]] in 2003. Certain solutions of the heat equation known as [[heat kernel]]s provide subtle information about the region on which they are defined, as exemplified through their application to the [[Atiyah–Singer index theorem]].<ref>Berline, Nicole; Getzler, Ezra; Vergne, Michèle. Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften, 298. Springer-Verlag, Berlin, 1992. viii+369 pp. {{ISBN|3-540-53340-0}}</ref>

The heat equation, along with variants thereof, is also important in many fields of science and [[applied mathematics]]. In [[probability theory]], the heat equation is connected with the study of [[random walk]]s and [[Brownian motion]] via the [[Fokker–Planck equation]]. The [[Black–Scholes equation]] of [[financial mathematics]] is a small variant of the heat equation, and the [[Schrödinger equation]] of [[quantum mechanics]] can be regarded as a heat equation in [[imaginary time]]. In [[image analysis]], the heat equation is sometimes used to resolve pixelation and to [[Edge detection|identify edges]]. Following [[Robert D. Richtmyer|Robert Richtmyer]] and [[John von Neumann]]'s introduction of artificial viscosity methods, solutions of heat equations have been useful in the mathematical formulation of [[Shock (fluid dynamics)|hydrodynamical shocks]]. Solutions of the heat equation have also been given much attention in the [[numerical analysis]] literature, beginning in the 1950s with work of Jim Douglas, D.W. Peaceman, and Henry Rachford Jr.

=== Particle diffusion ===
{{main|Diffusion equation}}

One can model particle [[diffusion]] by an equation involving either: 
* the volumetric [[concentration]] of particles, denoted ''c'', in the case of [[collective diffusion]] of a large number of particles, or
* the [[probability density function]] associated with the position of a single particle, denoted ''P''.

In either case, one uses the heat equation
: <math>c_t = D \Delta c, </math>
or
: <math>P_t = D \Delta P. </math>

Both ''c'' and ''P'' are functions of position and time. ''D'' is the diffusion coefficient that controls the speed of the diffusive process, and is typically expressed in meters squared over second.  If the diffusion coefficient ''D'' is not constant, but depends on the concentration ''c'' (or ''P'' in the second case), then one gets the [[diffusion equation|nonlinear diffusion equation]].

=== Brownian motion ===
Let the [[stochastic process]] <math>X</math> be the solution to the [[stochastic differential equation]]
: <math>\begin{cases}
\mathrm{d}X_t = \sqrt{2k}\; \mathrm{d}B_t \\
X_0=0
\end{cases}</math>
where <math>B</math> is the [[Wiener process]] (standard Brownian motion). The [[probability density function]] of <math>X</math> is given at any time <math>t</math> by
: <math>\frac{1}{\sqrt{4\pi kt}}\exp\left(-\frac{x^2}{4kt}\right)</math>
which is the solution to the initial value problem
: <math>\begin{cases}
u_t(x,t)-ku_{xx}(x,t)=0, & (x,t)\in\R\times(0,+\infty)\\
u(x,0)=\delta(x)
\end{cases}</math>
where <math>\delta</math> is the [[Dirac delta function]].

=== Schrödinger equation for a free particle ===
{{main|Schrödinger equation}}

With a simple division, the [[Schrödinger equation]] for a single particle of [[mass]] ''m'' in the absence of any applied force field can be rewritten in the following way:
: <math>\psi_t = \frac{i \hbar}{2m} \Delta \psi</math>,
where ''i'' is the [[imaginary unit]], ''ħ'' is the [[reduced Planck constant]], and ''ψ'' is the [[wave function]] of the particle.

This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation:
: <math>\begin{align}
c(\mathbf R,t) &\to \psi(\mathbf R,t) \\
D &\to \frac{i \hbar}{2m}
\end{align}</math>

Applying this transformation to the expressions of the Green functions determined in the case of particle diffusion yields the Green functions of the [[Schrödinger equation]], which in turn can be used to obtain the [[wave function]] at any time through an integral on the [[wave function]] at ''t'' = 0:
: <math>\psi(\mathbf R, t) = \int \psi\left(\mathbf R^0,t=0\right) G\left(\mathbf R - \mathbf R^0,t\right) dR_x^0 \, dR_y^0 \, dR_z^0,</math>
with
: <math>G(\mathbf R,t) = \left( \frac{m}{2 \pi i \hbar t} \right)^{3/2} e^{-\frac {\mathbf R^2 m}{2 i \hbar t}}.</math>

Remark: this analogy between quantum mechanics and diffusion is a purely formal one. Physically, the evolution of the [[wave function]] satisfying the [[Schrödinger equation]] might have an origin other than diffusion{{citation needed|date=January 2023}}.

=== Thermal diffusivity in polymers ===
A direct practical application of the heat equation, in conjunction with [[Fourier theory]], in spherical coordinates, is the prediction of thermal transfer profiles and the measurement of the [[thermal diffusivity]] in [[polymers]] (Unsworth and [[F. J. Duarte|Duarte]]). This dual theoretical-experimental method is applicable to rubber, various other polymeric materials of practical interest, and microfluids. These authors derived an expression for the temperature at the center of a sphere {{mvar|T<sub>C</sub>}}
: <math>\frac{T_C - T_S}{T_0 - T_S} =2 \sum_{n = 1}^{\infty} (-1)^{n+1} \exp\left({-\frac{n^2 \pi^2 \alpha t}{L^2}}\right)</math>
where {{math|''T''<sub>0</sub>}} is the initial temperature of the sphere and {{mvar|T<sub>S</sub>}} the temperature at the surface of the sphere, of radius {{mvar|L}}.  This equation has also found applications in protein energy transfer and thermal modeling in biophysics.

=== Financial Mathematics ===

The heat equation arises in a [[List of natural phenomena|number of phenomena]] and is often used in [[financial mathematics]] in the [[Mathematical model|modeling]] of [[Option (finance)|options]]. The [[Black–Scholes]] option pricing model's [[differential equation]] can be transformed into the heat equation allowing relatively easy solutions from a familiar body of mathematics. Many of the extensions to the simple option models do not have closed form solutions and thus must be solved numerically to obtain a modeled option price. The equation describing pressure diffusion in a porous medium is identical in form with the heat equation. [[Diffusion]] problems dealing with [[Dirichlet boundary conditions|Dirichlet]], [[Neumann boundary conditions|Neumann]] and [[Robin boundary condition]]s have closed form analytic solutions {{harv|Thambynayagam|2011}}.

=== Image Analysis ===
The heat equation is also widely used in image analysis {{harv|Perona|Malik|1990}} and in [[machine learning]] as the driving theory behind [[Scale space|scale-space]] or [[graph Laplacian]] methods. The heat equation can be efficiently solved numerically using the implicit [[Crank–Nicolson method]] of {{harv|Crank|Nicolson|1947}}. This method can be extended to many of the models with no closed form solution, see for instance {{harv|Wilmott|Howison|Dewynne|1995}}.

=== Riemannian geometry ===
An abstract form of heat equation on [[manifold]]s provides a major approach to the [[Atiyah–Singer index theorem]], and has led to much further work on heat equations in [[Riemannian geometry]].

== See also ==

* [[Caloric polynomial]]
* [[Curve-shortening flow]]
* [[Diffusion equation]]
* [[Parabolic partial differential equation]]
* [[Relativistic heat conduction]]
* [[Schrödinger equation]]
* [[Weierstrass transform]]

== Notes ==
{{reflist|30em}}

== References ==
* {{citation| last = Cannon| first = John Rozier| title = The one–dimensional heat equation|location=Reading, MA|year=1984|isbn=0-201-13522-1|publisher=Addison-Wesley Publishing Company, Advanced Book Program|series=Encyclopedia of Mathematics and its Applications|volume=23|mr = 0747979| zbl = 0567.35001}}
* {{citation|last1=Crank|first1=J.|last2=Nicolson|first2=P.|title=A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-Conduction Type|journal=Proceedings of the Cambridge Philosophical Society|year=1947|volume=43|issue=1|pages=50–67|doi=10.1017/S0305004100023197|bibcode = 1947PCPS...43...50C |s2cid=16676040 }}
* {{citation|last=Evans|first=Lawrence C.|series=Graduate Studies in Mathematics|volume=19|title=Partial Differential Equations|edition=2nd|publisher=American Mathematical Society|location=Providence, RI|year=2010|isbn=978-0-8218-4974-3|url=https://books.google.com/books?id=Xnu0o_EJrCQC&q=%22heat%20equation%22}}
* {{citation|last1=Perona|first1=P|last2=Malik|first2=J.|author2-link=Jitendra Malik|year=1990|title=Scale-Space and Edge Detection Using Anisotropic Diffusion|journal=IEEE Transactions on Pattern Analysis and Machine Intelligence|volume=12|number=7|pages=629–639|doi=10.1109/34.56205|s2cid=14502908|url=https://authors.library.caltech.edu/6498/1/PERieeetpami90.pdf}}
* {{citation|last=Thambynayagam|first=R. K. M.|year=2011|title=The Diffusion Handbook: Applied Solutions for Engineers|publisher=McGraw-Hill Professional|isbn=978-0-07-175184-1}}
* {{citation|last1=Wilmott|first1=Paul|last2=Howison|first2=Sam|last3=Dewynne|first3=Jeff|year=1995|title=The mathematics of financial derivatives. A student introduction|publisher=Cambridge University Press|location=Cambridge|isbn=0-521-49699-3}}

== Further reading ==
*{{citation |last1=Carslaw |first1=H.S. |author1-link=Horatio Scott Carslaw |last2=Jaeger |first2=J.C. |year=1988 |title=Conduction of heat in solids |edition=2nd |publisher=The Clarendon Press, Oxford University Press |location=New York |isbn=978-0-19-853368-9 |series=Oxford Science Publications}}
* {{citation|last1=Cole|first1=Kevin D.|last2=Beck|first2=James V.|last3=Haji-Sheikh|first3=A.|last4=Litkouhi|first4=Bahan|year=2011|title=Heat conduction using Green's functions|edition=2nd|publisher=CRC Press|location=Boca Raton, FL|series=Series in Computational and Physical Processes in Mechanics and Thermal Sciences|isbn=978-1-43-981354-6 }}
* {{citation|last=Einstein|first=Albert|author-link= Albert Einstein|title=Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen|journal=Annalen der Physik|pages=549–560|year=1905|doi=10.1002/andp.19053220806|volume=322|bibcode = 1905AnP...322..549E|issue=8 |doi-access=free|url=http://sedici.unlp.edu.ar/bitstream/handle/10915/2785/Documento_completo__.pdf?sequence=1}}
* {{citation|last=Friedman|first=Avner|title=Partial differential equations of parabolic type|publisher=Prentice-Hall |year=1964|location=Englewood Cliffs, N.J.}}
* {{citation|last1=Unsworth|first1=J.|last2=Duarte|first2=F. J.|author2-link=F. J. Duarte|title=Heat diffusion in a solid sphere and Fourier Theory|journal=Am. J. Phys.|pages=891–893|doi=10.1119/1.11601|volume=47|bibcode = 1979AmJPh..47..981U|issue=11| year=1979 }}
* {{citation |last1=Jili |first1=Latif M. |author1-link= |year=2009 |title=Heat Conduction |edition=3rd |publisher=Springer-Verlag |location=Berlin-Heidelberg |isbn=978-3-642-01266-2 |series=Springer}}
* {{citation|last=Widder|first=D.V.|title=The heat equation|series=Pure and Applied Mathematics|volume=67|publisher=Academic Press [Harcourt Brace Jovanovich, Publishers]|location=New York-London|year=1975}}

== External links ==
{{wikiversity|Heat equation}}
{{Commons category|Heat equation}}
* [http://www.mathphysics.com/pde/HEderiv.html Derivation of the heat equation]
* [http://eqworld.ipmnet.ru/en/solutions/lpde/heat-toc.pdf Linear heat equations]: Particular solutions and boundary value problems - from EqWorld
* {{cite web |title=The Heat Equation |work=[[PBS Infinite Series]] |date=November 17, 2017 |url=https://www.youtube.com/watch?v=NHucpzbD600  |archive-url=https://ghostarchive.org/varchive/youtube/20211211/NHucpzbD600| archive-date=2021-12-11 |url-status=live|via=[[YouTube]] }}{{cbignore}}

[[Category:Diffusion|Heat equation]]
[[Category:Heat conduction|Equation]]
[[Category:Parabolic partial differential equations]]
[[Category:Heat transfer]]