Editing Heat equation (section)

== 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 [[Schrödinger's 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]].