Heat equation

Informally, the Laplacian operator ∆ gives the difference between the average value of a function in the neighborhood of a point, and its value at that point. Thus, if u is the temperature, ∆ tells whether (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 factor called the specific heat capacity of the material.

By the combination of these observations, the heat equation says that the rate ${\displaystyle {\dot {u}}}$ 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 α in the equation takes into account the thermal conductivity, the specific heat, and the density of the material.

The first half of the above physical thinking can be put into a mathematical form. The key is that, for any fixed x, one has

${\displaystyle {\begin{aligned}u_{(x)}(0)&=u(x)\\u_{(x)}'(0)&=0\\u_{(x)}''(0)&={\frac {1}{n}}\Delta u(x)\end{aligned}}}$

where u_(x)(r) is the single-variable function denoting the average value of u over the surface of the sphere of radius r centered at x; it can be defined by

${\displaystyle u_{(x)}(r)={\frac {1}{\omega _{n-1}r^{n-1}}}\int _{\{y:|x-y|=r\}}u\,d{\mathcal {H}}^{n-1},}$

in which ω_{n − 1} denotes the surface area of the unit ball in n-dimensional Euclidean space. This formalizes the above statement that the value of ∆u at a point x measures the difference between the value of u(x) and the value of u at points nearby to x, in the sense that the latter is encoded by the values of u_(x)(r) for small positive values of 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 u(x, t + τ) for a small positive value of τ may be approximated as 1/2n times the average value of the function u(⋅, t) over a sphere of very small radius centered at x.

Solution of a 1D heat partial differential equation. The temperature (${\displaystyle u}$) is initially distributed over a one-dimensional, one-unit-long interval (x = [0,1]) with insulated endpoints. The distribution approaches equilibrium over time.

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 maxima) of ${\displaystyle u}$ will be gradually eroded down, while depressions (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 ${\displaystyle Ax+By+Cz+D}$, then the value at the center of that neighborhood will not be changing at that time (that is, the derivative ${\displaystyle {\dot {u}}}$ will be zero).

A more subtle consequence is the maximum principle, that says that the maximum value of ${\displaystyle u}$ in any region ${\displaystyle R}$ of the medium will not exceed the maximum value that previously occurred in ${\displaystyle R}$, unless it is on the boundary of ${\displaystyle R}$. That is, the maximum temperature in a region ${\displaystyle R}$ can increase only if heat comes in from outside ${\displaystyle R}$. This is a property of parabolic partial differential equations and is not difficult to prove mathematically (see below).

Another interesting property is that even if ${\displaystyle u}$ 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 ${\displaystyle u_{0}}$ and ${\displaystyle u_{1}}$, 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 ${\displaystyle u}$ will gradually vary between ${\displaystyle u_{0}}$ and ${\displaystyle u_{1}}$.

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 elastic and electromagnetic waves, 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.

For heat flow, the heat equation follows from the physical laws of conduction of heat and conservation of energy (Cannon 1984).

By 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:

${\displaystyle \mathbf {q} =-k\,\nabla u\ }$

where ${\displaystyle k}$ is the thermal conductivity of the material, ${\displaystyle u=u(\mathbf {x} ,t)}$ is the temperature, and ${\displaystyle \mathbf {q} =\mathbf {q} (\mathbf {x} ,t)}$ is a vector field that represents the magnitude and direction of the heat flow at the point ${\displaystyle \mathbf {x} }$ of space and time ${\displaystyle t}$.

If the medium is a thin rod of uniform section and material, the position is a single coordinate ${\displaystyle x}$, the heat flow towards increasing ${\displaystyle x}$ is a scalar field ${\displaystyle q=q(t,x)}$ , and the gradient is an ordinary derivative with respect to the ${\displaystyle x}$. The equation becomes

${\displaystyle q=-k\,{\frac {\partial u}{\partial x}}}$

Let ${\displaystyle Q=Q(x,t)}$ be the internal heat energy per unit volume of the bar at each point and time. In the absence of heat energy generation, from external or internal sources, the rate of change in internal heat energy per unit volume in the material, ${\displaystyle \partial Q/\partial t}$, is proportional to the rate of change of its temperature, ${\displaystyle \partial u/\partial t}$. That is,

${\displaystyle {\frac {\partial Q}{\partial t}}=c\,\rho \,{\frac {\partial u}{\partial t}}}$

where ${\displaystyle c}$ is the specific heat capacity (at constant pressure, in case of a gas) and ${\displaystyle \rho }$ 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 centered at ${\displaystyle x}$, one concludes that the rate at which heat accumulates at a given point ${\displaystyle x}$ is equal to the derivative of the heat flow at that point, negated. That is,

${\displaystyle {\frac {\partial Q}{\partial t}}=-{\frac {\partial q}{\partial x}}}$

From the above equations it follows that

${\displaystyle {\frac {\partial u}{\partial t}}\;=\;-{\frac {1}{c\,\rho }}{\frac {\partial q}{\partial x}}\;=\;-{\frac {1}{c\,\rho }}{\frac {\partial }{\partial x}}\left(-k\,{\frac {\partial u}{\partial x}}\right)\;=\;{\frac {k}{c\,\rho }}{\frac {\partial ^{2}u}{\partial x^{2}}}}$

which is the heat equation in one dimension, with diffusivity coefficient

${\displaystyle \alpha ={\frac {k}{c\rho }}}$

This quantity is called the thermal diffusivity of the medium.

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 ${\displaystyle \mu (u^{4}-v^{4})}$, where ${\displaystyle v=v(x,t)}$ is the temperature of the surroundings, and ${\displaystyle \mu }$ is a coefficient that depends on physical properties of the material. The rate of change in internal energy becomes

${\displaystyle {\frac {\partial Q}{\partial t}}=-{\frac {\partial q}{\partial x}}-\mu (u^{4}-v^{4})}$

and the equation for the evolution of ${\displaystyle u}$ becomes

${\displaystyle {\frac {\partial u}{\partial t}}={\frac {k}{c\,\rho }}{\frac {\partial ^{2}u}{\partial x^{2}}}-{\frac {\mu }{c\,\rho }}(u^{4}-v^{4})}$.

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_p or ${\displaystyle \rho }$) influences which term.

${\displaystyle \rho c_{p}{\frac {\partial T}{\partial t}}-\nabla \cdot \left(k\nabla T\right)={\dot {q}}_{V}}$

where ${\displaystyle {\dot {q}}_{V}}$ is the volumetric heat source.

In the special cases of propagation of heat in an isotropic and homogeneous medium in a 3-dimensional space, this equation is

${\displaystyle {\partial u \over \partial t}=\alpha \nabla ^{2}u=\alpha \left({\partial ^{2}u \over \partial x^{2}}+{\partial ^{2}u \over \partial y^{2}}+{\partial ^{2}u \over \partial z^{2}}\right)}$   ${\displaystyle =\alpha (u_{xx}+u_{yy}+u_{zz})\quad }$

where:

• ${\displaystyle u=u(x,y,z,t)}$ is temperature as a function of space and time;
• ${\displaystyle {\tfrac {\partial u}{\partial t}}}$ is the rate of change of temperature at a point over time;
• ${\displaystyle u_{xx}}$, ${\displaystyle u_{yy}}$, and ${\displaystyle u_{zz}}$ are the second spatial derivatives (thermal conductions) of temperature in the ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ directions, respectively;
• ${\displaystyle \alpha \equiv {\tfrac {k}{c_{p}\rho }}}$ is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity ${\displaystyle k}$, the specific heat capacity ${\displaystyle c_{p}}$, and the mass density ${\displaystyle \rho }$.

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 conditions for u. To determine uniqueness of solutions in the whole space it is necessary to assume an exponential bound on the growth of solutions.[2]

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 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

${\displaystyle u_{t}=\alpha \nabla ^{2}u=\alpha \Delta u,\quad }$

where the Laplace operator, Δ or ∇², 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 processes. 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 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.[3][4]

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.[5] Then the heat per unit volume u satisfies an equation

${\displaystyle {\frac {1}{\alpha }}{\frac {\partial u}{\partial t}}=\left({\partial ^{2}u \over \partial x^{2}}+{\partial ^{2}u \over \partial y^{2}}+{\partial ^{2}u \over \partial z^{2}}\right)+{\frac {1}{k}}q.}$

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.

The solution technique used above can be greatly extended to many other types of equations. The idea is that the operator u_xx with the zero boundary conditions can be represented in terms of its eigenfunctions. This leads naturally to one of the basic ideas of the spectral theory of linear self-adjoint operators.

Consider the linear operator Δu = u_xx. The infinite sequence of functions

${\displaystyle e_{n}(x)={\sqrt {\frac {2}{L}}}\sin \left({\frac {n\pi x}{L}}\right)}$

for n ≥ 1 are eigenfunctions of Δ. Indeed,

${\displaystyle \Delta e_{n}=-{\frac {n^{2}\pi ^{2}}{L^{2}}}e_{n}.}$

Moreover, any eigenfunction f of Δ with the boundary conditions f(0) = f(L) = 0 is of the form e_n for some n ≥ 1. The functions e_n 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

${\displaystyle \langle e_{n},e_{m}\rangle =\int _{0}^{L}e_{n}(x)e_{m}^{*}(x)dx=\delta _{mn}}$

Finally, the sequence {e_n}_{n ∈ N} spans a dense linear subspace of L²((0, L)). This shows that in effect we have diagonalized the operator Δ.

A variety of elementary Green's function solutions in one-dimension are recorded here; many others are available elsewhere.[6] In some of these, the spatial domain is (−∞,∞). In others, it is the semi-infinite interval (0,∞) with either Neumann or Dirichlet boundary conditions. One further variation is that some of these solve the inhomogeneous equation

${\displaystyle u_{t}=ku_{xx}+f.}$

where f is some given function of x and t.

Initial value problem on (−∞,∞)

${\displaystyle {\begin{cases}u_{t}=ku_{xx}&(x,t)\in \mathbf {R} \times (0,\infty )\\u(x,0)=g(x)&IC\end{cases}}}$

${\displaystyle u(x,t)={\frac {1}{\sqrt {4\pi kt}}}\int _{-\infty }^{\infty }\exp \left(-{\frac {(x-y)^{2}}{4kt}}\right)g(y)\,dy}$

Fundamental solution of the one-dimensional heat equation. Red: time course of ${\displaystyle \Phi (x,t)}$. Blue: time courses of ${\displaystyle \Phi (x_{0},t)}$ for two selected points x₀ = 0.2 and x₀ = 1. Note the different rise times/delays and amplitudes.
Interactive version.

Comment. This solution is the convolution with respect to the variable x of the fundamental solution

${\displaystyle \Phi (x,t):={\frac {1}{\sqrt {4\pi kt}}}\exp \left(-{\frac {x^{2}}{4kt}}\right),}$

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

${\displaystyle \left(\partial _{t}-k\partial _{x}^{2}\right)(\Phi *g)=\left[\left(\partial _{t}-k\partial _{x}^{2}\right)\Phi \right]*g=0.}$

Moreover,

${\displaystyle \Phi (x,t)={\frac {1}{\sqrt {t}}}\,\Phi \left({\frac {x}{\sqrt {t}}},1\right)}$

${\displaystyle \int _{-\infty }^{\infty }\Phi (x,t)\,dx=1,}$

so that, by general facts about 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 Φ(⋅, t) ∗ g converges uniformly to g as t → 0, meaning that u(x, t) is continuous on R × [0, ∞) with u(x, 0) = g(x).

Initial value problem on (0,∞) with homogeneous Dirichlet boundary conditions

${\displaystyle {\begin{cases}u_{t}=ku_{xx}&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=g(x)&IC\\u(0,t)=0&BC\end{cases}}}$

${\displaystyle u(x,t)={\frac {1}{\sqrt {4\pi kt}}}\int _{0}^{\infty }\left[\exp \left(-{\frac {(x-y)^{2}}{4kt}}\right)-\exp \left(-{\frac {(x+y)^{2}}{4kt}}\right)\right]g(y)\,dy}$

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

${\displaystyle {\begin{cases}u_{t}=ku_{xx}&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=g(x)&IC\\u_{x}(0,t)=0&BC\end{cases}}}$

${\displaystyle u(x,t)={\frac {1}{\sqrt {4\pi kt}}}\int _{0}^{\infty }\left[\exp \left(-{\frac {(x-y)^{2}}{4kt}}\right)+\exp \left(-{\frac {(x+y)^{2}}{4kt}}\right)\right]g(y)\,dy}$

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_x(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

${\displaystyle {\begin{cases}u_{t}=ku_{xx}&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=0&IC\\u(0,t)=h(t)&BC\end{cases}}}$

${\displaystyle u(x,t)=\int _{0}^{t}{\frac {x}{\sqrt {4\pi k(t-s)^{3}}}}\exp \left(-{\frac {x^{2}}{4k(t-s)}}\right)h(s)\,ds,\qquad \forall x>0}$

Comment. This solution is the convolution with respect to the variable t of

${\displaystyle \psi (x,t):=-2k\partial _{x}\Phi (x,t)={\frac {x}{\sqrt {4\pi kt^{3}}}}\exp \left(-{\frac {x^{2}}{4kt}}\right)}$

and the function h(t). Since Φ(x, t) is the fundamental solution of

${\displaystyle \partial _{t}-k\partial _{x}^{2},}$

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,

${\displaystyle \psi (x,t)={\frac {1}{x^{2}}}\,\psi \left(1,{\frac {t}{x^{2}}}\right)}$

${\displaystyle \int _{0}^{\infty }\psi (x,t)\,dt=1,}$

so that, by general facts about 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 [0, ∞) × [0, ∞) with u(0, t) = h(t).

Depicted is a numerical solution of the nonhomogeneous heat equation. The equation has been solved with 0 initial and boundary conditions and a source term representing a stove top burner.

Problem on (-∞,∞) homogeneous initial conditions

Comment. This solution is the convolution in R², that is with respect to both the variables x and t, of the fundamental solution

${\displaystyle \Phi (x,t):={\frac {1}{\sqrt {4\pi kt}}}\exp \left(-{\frac {x^{2}}{4kt}}\right)}$

and the function f(x, t), both meant as defined on the whole R² and identically 0 for all t → 0. One verifies that

${\displaystyle \left(\partial _{t}-k\partial _{x}^{2}\right)(\Phi *f)=f,}$

which expressed in the language of distributions becomes

${\displaystyle \left(\partial _{t}-k\partial _{x}^{2}\right)\Phi =\delta ,}$

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

${\displaystyle {\begin{cases}u_{t}=ku_{xx}+f(x,t)&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=0&IC\\u(0,t)=0&BC\end{cases}}}$

${\displaystyle u(x,t)=\int _{0}^{t}\int _{0}^{\infty }{\frac {1}{\sqrt {4\pi k(t-s)}}}\left(\exp \left(-{\frac {(x-y)^{2}}{4k(t-s)}}\right)-\exp \left(-{\frac {(x+y)^{2}}{4k(t-s)}}\right)\right)f(y,s)\,dy\,ds}$

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

${\displaystyle {\begin{cases}u_{t}=ku_{xx}+f(x,t)&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=0&IC\\u_{x}(0,t)=0&BC\end{cases}}}$

${\displaystyle u(x,t)=\int _{0}^{t}\int _{0}^{\infty }{\frac {1}{\sqrt {4\pi k(t-s)}}}\left(\exp \left(-{\frac {(x-y)^{2}}{4k(t-s)}}\right)+\exp \left(-{\frac {(x+y)^{2}}{4k(t-s)}}\right)\right)f(y,s)\,dy\,ds}$

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_x(0, t) = 0.

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

${\displaystyle {\begin{cases}u_{t}=ku_{xx}+f&(x,t)\in \mathbf {R} \times (0,\infty )\\u(x,0)=g(x)&IC\end{cases}}}$

let u = w + v where w and v solve the problems

${\displaystyle {\begin{cases}v_{t}=kv_{xx}+f,\,w_{t}=kw_{xx}\,&(x,t)\in \mathbf {R} \times (0,\infty )\\v(x,0)=0,\,w(x,0)=g(x)\,&IC\end{cases}}}$

Similarly, to solve

${\displaystyle {\begin{cases}u_{t}=ku_{xx}+f&(x,t)\in [0,\infty )\times (0,\infty )\\u(x,0)=g(x)&IC\\u(0,t)=h(t)&BC\end{cases}}}$

let u = w + v + r where w, v, and r solve the problems

${\displaystyle {\begin{cases}v_{t}=kv_{xx}+f,\,w_{t}=kw_{xx},\,r_{t}=kr_{xx}&(x,t)\in [0,\infty )\times (0,\infty )\\v(x,0)=0,\;w(x,0)=g(x),\;r(x,0)=0&IC\\v(0,t)=0,\;w(0,t)=0,\;r(0,t)=h(t)&BC\end{cases}}}$

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

${\displaystyle c_{t}=D\Delta c,\quad }$

or

${\displaystyle P_{t}=D\Delta P.\quad }$

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 nonlinear diffusion equation.

Let the stochastic process ${\displaystyle X}$ be the solution of the stochastic differential equation

${\displaystyle \left\{{\begin{array}{ll}{\text{d}}X_{t}={\sqrt {2k}}\;{\text{d}}B_{t}\\X_{0}=0\end{array}}\right.}$

where ${\displaystyle B}$ is the Wiener process (standard Brownian motion). Then the probability density function of ${\displaystyle X}$ is given at any time ${\displaystyle t}$ by

${\displaystyle {\frac {1}{\sqrt {4\pi kt}}}\exp \left(-{\frac {x^{2}}{4kt}}\right)}$

which is the solution of the initial value problem

${\displaystyle \left\{{\begin{array}{ll}u_{t}(x,t)-ku_{xx}(x,t)=0,\quad (x,t)\in \mathbb {R} \times (0,+\infty )\\u(x,0)=\delta (x)\end{array}}\right.}$

where ${\displaystyle \delta }$ is the Dirac delta function.

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:

${\displaystyle \psi _{t}={\frac {i\hbar }{2m}}\Delta \psi }$,

where i is the imaginary unit, ħ is the reduced Planck's 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:

${\displaystyle {\begin{aligned}c(\mathbf {R} ,t)&\to \psi (\mathbf {R} ,t)\\D&\to {\frac {i\hbar }{2m}}\end{aligned}}}$

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:

${\displaystyle \psi (\mathbf {R} ,t)=\int \psi (\mathbf {R} ^{0},t=0)G(\mathbf {R} -\mathbf {R} ^{0},t)dR_{x}^{0}\,dR_{y}^{0}\,dR_{z}^{0},}$

with

${\displaystyle G(\mathbf {R} ,t)={\bigg (}{\frac {m}{2\pi i\hbar t}}{\bigg )}^{3/2}e^{-{\frac {\mathbf {R} ^{2}m}{2i\hbar t}}}.}$

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.

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 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 T_C

${\displaystyle {\frac {T_{C}-T_{S}}{T_{0}-T_{S}}}=2\sum _{n=1}^{\infty }(-1)^{n+1}\exp \left({-{\frac {n^{2}\pi ^{2}\alpha t}{L^{2}}}}\right)}$

where T₀ is the initial temperature of the sphere and T_S the temperature at the surface of the sphere, of radius L. This equation has also found applications in protein energy transfer and thermal modeling in biophysics.

The heat equation arises in the modeling of a number of phenomena and is often used in financial mathematics in the modeling of options. The famous 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, Neumann and Robin boundary conditions have closed form analytic solutions (Thambynayagam 2011). The heat equation is also widely used in image analysis (Perona & Malik 1990) and in machine-learning as the driving theory behind scale-space or graph Laplacian methods. The heat equation can be efficiently solved numerically using the implicit Crank–Nicolson method of (Crank & Nicolson 1947). This method can be extended to many of the models with no closed form solution, see for instance (Wilmott, Howison & Dewynne 1995).

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