Heat equation

The heat equation predicts that if a hot body is placed in a box of cold water, the temperature of the body will decrease, and eventually (after infinite time, and subject to no external heat sources) the temperature in the box will equalize.

The heat equation is an important partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time. For a function u(x,y,z,t) of three spatial variables (x,y,z) and the time variable t, the heat equation is

$\frac{\partial u}{\partial t} -\alpha\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right)=0$

also written

$\frac{\partial u}{\partial t} - \alpha \nabla^2 u=0$

or sometimes

$\frac{\partial u}{\partial t} - \alpha \Delta u=0$

where $\alpha$ is a constant and $\Delta\$ or $\nabla^2\$ denotes the Laplacian operator. For the mathematical treatment it is sufficient to consider the case α=1.

The heat equation is of fundamental importance in diverse scientific fields. In mathematics, it is the prototypical parabolic partial differential equation. In probability theory, the heat equation is connected with the study of Brownian motion via the Fokker–Planck equation. In financial mathematics it is used to solve the Black–Scholes partial differential equation. The diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes.

General description

Graphical representation of the solution to a 1D heat equation PDE. (View animated version)

Suppose one has a function u which describes the temperature at a given location (x, y, z). This function will change over time as heat spreads throughout space. The heat equation is used to determine the change in the function u over time. The image to the right is animated and describes the way heat changes in time along a metal bar. One of the interesting properties of the heat equation is the maximum principle which says that the maximum value of u is either earlier in time than the region of concern or on the edge of the region of concern. This is essentially saying that temperature comes either from some source or from earlier in time because heat permeates but is not created from nothingness. 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 u has a discontinuity at an initial time t = t₀, the temperature becomes smooth as soon as t > t₀. For example, if a bar of metal has temperature 0 and another has temperature 100 and they are stuck together end to end, then very quickly the temperature at the point of connection is 50 and the graph of the temperature is smoothly running from 0 to 100.

The heat equation is used in probability and describes random walks. It is also applied in financial mathematics for this reason.

It is also important in Riemannian geometry and thus topology: it was adapted by Richard Hamilton when he defined the Ricci flow that was later used by Grigori Perelman to solve the topological Poincaré conjecture.

The physical problem and the equation

Derivation in one dimension

The heat equation is derived from Fourier's law and conservation of energy (Cannon 1984). By Fourier's law, the flow rate of heat energy through a surface is proportional to the negative temperature gradient across the surface,

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

where k is the thermal conductivity and u is the temperature. In one dimension, the gradient is an ordinary spatial derivative, and so Fourier's law is

$\mathbf{q} = -k u_x \,$

where u_x is du/dx. In the absence of work done, a change in internal energy per unit volume in the material, ΔQ, is proportional to the change in temperature, Δu. That is,

$\Delta Q = c_p\rho\Delta u \,$

where c_p is the specific heat capacity and ρ is the mass density of the material. (In this section only, Δ is the ordinary difference operator, not the Laplacian.) Choosing zero energy at absolute zero temperature, this can be rewritten as

$Q = c_p\rho u \,$ .

The increase in internal energy in a small spatial region of the material

$x-\Delta x \le \xi \le x+\Delta x$

over the time period

$t-\Delta t\le \tau \le t+\Delta t$

is given by^[1]

$c_p\rho \int_{x-\Delta x}^{x+\Delta x} [u(\xi,t+\Delta t)-u(\xi,t-\Delta t)]\, d\xi = c_p\rho\int_{t-\Delta t}^{t+\Delta t}\int_{x-\Delta x}^{x+\Delta x} \frac{\partial u}{\partial\tau}\,d\xi d\tau$

where the fundamental theorem of calculus was used. Additionally, with no work done and absent any heat sources or sinks, the change in internal energy in the interval [x-Δx, x+Δx] is accounted for entirely by the flux of heat across the boundaries. By Fourier's law, this is

$k\int_{t-\Delta t}^{t+\Delta t}\left[\frac{\partial u}{\partial x}(x+\Delta x,\tau)-\frac{\partial u}{\partial x}(x-\Delta x,\tau)\right]\,d\tau = k\int_{t-\Delta t}^{t+\Delta t}\int_{x-\Delta x}^{x+\Delta x}\frac{\partial^2u}{\partial\xi^2}\,d\xi d\tau$

again by the fundamental theorem of calculus.^[2] By conservation of energy,

$\int_{t-\Delta t}^{t+\Delta t}\int_{x-\Delta x}^{x+\Delta x} [c_p\rho u_\tau - k u_{\xi\xi}]\, d\xi d\tau = 0.$

This is true for any rectangle [t−Δt, t+Δt] × [x−Δx, x+Δx]. Consequently, the integrand must vanish identically:

$c_p\rho u_t - k u_{xx} = 0. \,\!$

Which can be rewritten as:

$u_t = \frac{k}{c_p\rho}u_{xx},$

or:

$\frac{\partial u}{\partial t} = \frac{k}{c_p\rho} \left(\frac{\partial^2u}{\partial x^2}\right)$

which is the heat equation. The coefficient k/(c_pρ) is called thermal diffusivity and is often denoted α.

Derivation in three dimensions

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

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

where:

u = u(x, y, z, t) is temperature as a function of space and time;

$\frac{\partial u}{\partial t}$ is the rate of change of temperature at a point over time;

u_xx, u_yy, and u_zz are the second spatial derivatives (thermal conductions) of temperature in the x, y, and z directions, respectively;

$\alpha=k/c_p\rho$ is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity, k, the mass density, $\rho$ , and the specific heat capacity, $c_p$ .

The heat equation is a consequence of Fourier's law of cooling (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, this assumption is consistent with observed experiments.

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

$u_t = \alpha \nabla^2 u = \alpha \Delta u, \quad \,\!$

where the Laplace operator, Δ or $\scriptstyle\nabla^2$ , 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.

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/L) at a rate given by a known function q varying in space and time.^[3] Then the heat per unit volume u satisfies an equation

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

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.

Solving the heat equation using Fourier series

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. Let us consider the heat equation for one space variable. This could be used to model heat conduction in a rod. The equation is

$(1) \ u_t = \alpha u_{xx} \quad$

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

$(2) \ u(x,0) = f(x) \quad \forall \ x \in [0,L] \quad$

where the function f is given and the boundary conditions

$(3) \ u(0,t) = 0 = u(L,t) \quad \forall \ t > 0 \quad$ .

Let us attempt to find a solution of (1) which is not identically zero satisfying the boundary conditions (3) but with the following property: u is a product in which the dependence of u on x, t is separated, that is:

$(4) \ u(x,t) = X(x) T(t). \quad$

This solution technique is called separation of variables. Substituting u back into equation (1),

$\frac{T'(t)}{\alpha T(t)} = \frac{X''(x)}{X(x)}. \quad$

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:

$(5) \ T'(t) = - \lambda \alpha T(t) \quad$

and

$(6) \ X''(x) = - \lambda X(x). \quad$

We will now show that solutions for (6) for values of λ ≤ 0 cannot occur:

Suppose that λ < 0. Then there exist real numbers B, C such that

$X(x) = B e^{\sqrt{-\lambda} \, x} + C e^{-\sqrt{-\lambda} \, x}.$

From (3) we get

$X(0) = 0 = X(L). \quad$

and therefore B = 0 = C which implies u is identically 0.
Suppose that λ = 0. Then there exist real numbers B, C such that

$X(x) = Bx + C. \quad$

From equation (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

$T(t) = A e^{-\lambda \alpha t} \quad$

and

$X(x) = B \sin(\sqrt{\lambda} \, x) + C \cos(\sqrt{\lambda} \, x).$

From (3) we get C = 0 and that for some positive integer n,

$\sqrt{\lambda} = n \frac{\pi}{L}.$

This solves the heat equation in the special case that the dependence of u has the special form (4).

In general, the sum of solutions to (1) which satisfy the boundary conditions (3) also satisfies (1) and (3). We can show that the solution to (1), (2) and (3) is given by

$u(x,t) = \sum_{n = 1}^{+\infty} D_n \left(\sin \frac{n\pi x}{L}\right) e^{-\frac{n^2 \pi^2 \alpha t}{L^2}}$

where

$D_n = \frac{2}{L} \int_0^L f(x) \sin \frac{n\pi x}{L} \, dx.$

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_xx with the zero boundary conditions can be represented in terms of its eigenvectors. This leads naturally to one of the basic ideas of the spectral theory of linear self-adjoint operators.

Consider the linear operator Δ u = u_{x x}. The infinite sequence of functions

$e_n(x) = \sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L}$

for n ≥ 1 are eigenvectors of Δ. Indeed

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

Moreover, any eigenvector 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

$\langle e_n, e_m \rangle = \int_0^L e_n(x) e_m(x) dx = \left\{ \begin{matrix} 0 & n \neq m \\ 1 & m = n\end{matrix}\right..$

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.

Heat conduction 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_t(V). We assume q has a density, so that

$q_t(V) = \int_V Q(x,t)\,d x \quad$

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 d S and with unit normal vector n is

$\mathbf{H}(x) \cdot \mathbf{n}(x) \, dS$

Thus the rate of heat flow into V is also given by the surface integral

$q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS$

where n(x) is the outward pointing normal vector at x.

The Fourier law states that heat energy flow has the following linear dependence on the temperature gradient

$\mathbf{H}(x) = -\mathbf{A}(x) \cdot \nabla u (x)$

where A(x) is a 3 × 3 real matrix that is symmetric and positive definite.

By Green's theorem, the previous surface integral for heat flow into V can be transformed into the volume integral

$q_t(V) = - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS$

$= \int_{\partial V} \mathbf{A}(x) \cdot \nabla u (x) \cdot \mathbf{n}(x) \, dS$

$= \int_V \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t) \bigr)\,dx$

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

$\partial_t u(x,t) = \kappa(x) Q(x,t)\,$

Putting these equations together gives the general equation of heat flow:

$\partial_t u(x,t) = \kappa(x) \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t)\bigr)$

Remarks.

The coefficient κ(x) is the inverse of specific heat of the substance at x × density of the substance at x.

In the case of an isotropic medium, the matrix A is a scalar matrix equal to thermal conductivity.

In the anisotropic case where the coefficient matrix A is not scalar (i.e., 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 semigroups theory: for instance, if A is a symmetric matrix, then the elliptic operator defined by

$Au(x):=\sum_{i, j} \partial_{x_i} a_{i j}(x) \partial_{x_j} u (x)$

is self-adjoint and dissipative, thus by the spectral theorem it generates a one-parameter semigroup.

Fundamental solutions

A fundamental solution, also called a heat kernel, is a solution of the heat equation corresponding 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, (Evans 1998) for an introductory treatment.

In one variable, the Green's function is a solution of the initial value problem

$\begin{cases} u_t(x,t) - k u_{xx}(x,t) = 0& -\infty<x<\infty,\quad 0<t<\infty\\ u(x,t=0)=\delta(x)& \end{cases}$

where δ is the Dirac delta function. The solution to this problem is the fundamental solution

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

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:

$u(x,t) = \int \Phi(x-y,t) g(y) dy.$

In several spatial variables, the fundamental solution solves the analogous problem

$\begin{cases} u_t(\mathbf{x},t) - k \sum_{i=1}^nu_{x_ix_i}(\mathbf{x},t) = 0\\ u(\mathbf{x},t=0)=\delta(\mathbf{x}) \end{cases}$

in -∞<x_i<∞, i=1,...,n, and 0<t<∞. The n-variable fundamental solution is the product of the fundamental solutions in each variable; i.e.,

$\Phi(\mathbf{x},t) = \Phi(x_1,t)\Phi(x_2,t)\dots\Phi(x_n,t)=\frac{1}{(4\pi k t)^{n/2}}e^{-\mathbf{x}\cdot\mathbf{x}/4kt}.$

The general solution of the heat equation on Rⁿ is then obtained by a convolution, so that to solve the initial value problem with u(x,t=0)=g(x), one has

$u(\mathbf{x},t) = \int_{\mathbb{R}^n}\Phi(\mathbf{x}-\mathbf{y},t)g(\mathbf{y})d\mathbf{y}.$

The general problem on a domain Ω in Rⁿ is

$\begin{cases} u_t(\mathbf{x},t) - k \sum_{i=1}^nu_{x_ix_i}(\mathbf{x},t) = 0& \mathbf{x}\in\Omega\quad 0<t<\infty\\ u(\mathbf{x},t=0)=g(\mathbf{x})&\mathbf{x}\in\Omega \end{cases}$

with either Dirichlet or 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. The method of images provides one additional technique for obtaining Green's functions for non-trivial domains.

Some Green's function solutions in 1D

A variety of elementary Green's function solutions in one-dimension are recorded here. In some of these, the spatial domain is the entire real line (-∞,∞). 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

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

where f is some given function of x and t.

Homogeneous heat equation

Initial value problem on (-∞,∞)

$\begin{cases} u_{t}=ku_{xx} & -\infty<x<\infty,\,0<t<\infty \\ u(x,0)=g(x) & IC \end{cases}$

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

Comment. This solution is the convolution with respect to the variable $x$ of the fundamental solution $\scriptstyle\Phi(x,t):=\frac{1}{\sqrt{4kt}}\exp\left(-\frac{x^2}{4kt}\right)$ and the function $g(x)$ . Therefore, according to the general properties of the convolution with respect to differentiation, $\scriptstyle u=g*\Phi$ is a solution of the same heat equation, for $\scriptstyle(\partial_t-k\,\partial_x^2)(\Phi*g)=[(\partial_t-k\,\partial_x^2)\Phi]*g)=0.$ Moreover, $\scriptstyle\Phi(x,t)=\frac{1}{\sqrt{t}}\,\Phi(\frac{x}{\sqrt{t}})$ and $\scriptstyle\int_{-\infty}^{+\infty}\Phi(x,t)\,dx=1,$ so that, by general facts about approximation to the identity, $\scriptstyle \Phi(\cdot,t)*g\to g$ as $\scriptstyle t\to0$ in various senses, according to the specific $g.$ For instance, if $g$ is assumed bounded and continuous on $\scriptstyle\R,$ then $\scriptstyle\Phi(\cdot,t)*g$ converges uniformly to $g$ as $\scriptstyle t\to0$ , meaning that $u(x,t)$ is continuous on $\scriptstyle \R\times[0,\infty)$ with $u(x,0)=g(x).$

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

$\begin{cases} u_{t}=ku_{xx} & \, 0\le x<\infty, \, 0<t<\infty \\ u(x,0)=g(x) & IC \\ u(0,t)=0 & BC \end{cases}$

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

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 $\scriptstyle(-\infty,+\infty)$ 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 $u(0,\,t)=0.$

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

$\begin{cases} u_{t}=ku_{xx} & \, 0\le x<\infty, \, 0<t<\infty \\ u(x,0)=g(x) & IC \\ u_{x}(0,t)=0 & BC \end{cases}$

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

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 $\scriptstyle(-\infty,+\infty)$ 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 $u_x(0,\,t)=0.$

Problem on (0,∞) with homogeneous initial conditions and non-homogeneous Dirichlet boundary conditions

$\begin{cases} u_{t}=ku_{xx} & 0\le x<\infty,\,0<t<\infty \\ u(x,0)=0 & IC \\ u(0,t)=h(t) & BC \end{cases}$

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

Comment. This solution is the convolution with respect to the variable $t$ of $\scriptstyle\psi(x,t):=-2k\,\partial_x \Phi(x,t)=\frac{x}{\sqrt{4kt^3}}\exp\left(-\frac{x^2}{4kt}\right)$ and the function $h(t)$ . Since $\Phi(x,t)$ is the fundamental solution of $\scriptstyle\partial_t-k\,\partial^2_x,$ the function $\psi(x,t)$ is also a solution of the same heat equation, and so is $u:=\psi*h$ , thanks to general properties of the convolution with respect to differentiation. Moreover, $\scriptstyle\psi(x,t)=\frac{1}{x^2}\,\psi(1,\frac{t}{x^2})$ and $\scriptstyle\int_0^{+\infty}\psi(x,t)\,dt=1,$ so that, by general facts about approximation to the identity, $\scriptstyle \psi(x,\cdot)*h \to h$ as $\scriptstyle x\to0$ in various senses, according to the specific $h.$ For instance, if $h$ is assumed continuous on $\scriptstyle\R$ with support in $\scriptstyle[0,\infty),$ then $\scriptstyle\psi(x,\cdot)*h$ converges uniformly on compacta to $h$ as $\scriptstyle x\to0$ , meaning that $u(x,t)$ is continuous on $\scriptstyle [0,\infty)\times[0,\infty)$ with $u(0,t)=h(t).$

Inhomogeneous heat equation

Problem on (-∞,∞) homogeneous initial conditions

$\begin{cases} u_{t}=ku_{xx}+f(x,t) & -\infty<x<\infty,\,0<t<\infty \\ u(x,0)=0 & IC \end{cases}$

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

Comment. This solution is the convolution in $\scriptstyle\R^2$ , that is with respect to both the variables $x$ and $t,$ of the fundamental solution $\scriptstyle\Phi(x,t):=\frac{1}{\sqrt{4kt}}\exp\left(-\frac{x^2}{4kt}\right)$ and the function $\scriptstyle f(x,\,t),$ both meant as defined on the whole $\scriptstyle\R^2,$ and identically 0 for all $\scriptstyle t\leq0$ . One verifies that $\scriptstyle(\partial_t-k\,\partial_x^2)(\Phi*f)=f,$ which is expressed in the language of distributions as $\scriptstyle(\partial_t-k\,\partial_x^2)\Phi=\delta,$ where the distribution $\delta$ is the Dirac's delta function, that is the evaluation at 0.

Problem on (0,∞) with homogeneous Dirichlet boundary conditions and initial conditions

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

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

Comment. This solution is obtained from the preceding formula as applied to the data $\scriptstyle f(x,\,t)$ suitably extended to $\scriptstyle\R\times[0,\infty)$ , so as to be an odd function of the variable $x,$ that is, letting $\scriptstyle f(-x,\,t):=-f(x,\,t)$ for all $x$ and $t.$ Correspondingly, the solution of the inhomogeneous problem on $\scriptstyle(-\infty,+\infty)$ 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 $\scriptstyle u(0,\,t)=0.$

Problem on (0,∞) with homogeneous Neumann boundary conditions and initial conditions

$\begin{cases} u_{t}=ku_{xx}+f(x,t) & 0\le x<\infty,\,0<t<\infty \\ u(x,0)=0 & IC \\ u_x(0,t)=0 & BC \end{cases}$

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

Comment. This solution is obtained from the first formula as applied to the data $\scriptstyle f(x,\,t)$ suitably extended to $\scriptstyle\R\times[0,\infty)$ , so as to be an even function of the variable $x,$ that is, letting $\scriptstyle f(-x,\,t):=f(x,\,t)$ for all $x$ and $t.$ Correspondingly, the solution of the inhomogeneous problem on $\scriptstyle(-\infty,+\infty)$ 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 $\scriptstyle u_x(0,\,t)=0.$

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

$\begin{cases} u_{t}=ku_{xx}+f & -\infty<x<\infty,\,0<t<\infty \\ u(x,0)=g(x) & IC\end{cases}$

let

$\quad{u=w+v}$

where u and v solve the problems

$\begin{cases} v_{t}=kv_{xx}+f, \, w_{t}=kw_{xx} \, & -\infty<x<\infty,\,0<t<\infty \\ v(x,0)=0,\, w(x,0)=g(x) \, & IC\end{cases}$

Similarly, to solve

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

let

$\quad{u=w+v+r}$

where w, v, and r solve the problems

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

Theta functions

Solutions of the one-dimensional heat equation on a finite spatial interval (0,1) involve the Jacobi theta function, defined by^[4]

$\theta(x,t) = \sum_{m=-\infty}^\infty \Phi(x+2m, t),$

where

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

is the fundamental solution.

By the method of images, the theta function gives Green's functions for the initial boundary value problems on the interval (0,1). For instance, the solution to the following problem

$u_t=u_{xx},\quad 0<x<1,\quad 0<t$

$u(x,0)=f(x),\quad 0<x<1$

$u(0,t)=0,\quad 0<t$

$u(1,t)=0,\quad 0<t$

is given by

$u(x,t) = \int_0^1\left[\theta(x-\xi,t) - \theta(x+\xi,t)\right]f(\xi)\,d\xi.$

Mean-value property for the heat equation

Solutions of the heat equations $(\partial_t -\Delta)u=0$ satisfy a mean-value property analogous to the mean-value properties of harmonic functions (solutions of $\Delta u=0$ ), though a bit more complicated. Precisely, if $u$ solves $(\partial_t -\Delta)u=0$ and $(x,t)+E_\lambda\subset\mathrm{dom}(u)$ then

$u(x,t)=\frac{\lambda}{4}\int_{E_\lambda}u(x-y,t-s)\frac{|y|^2}{s^2}ds\,dy,$

where $E_\lambda$ is a "heat-ball", that is a super-level set of the fundamental solution of the heat equation:

$E_\lambda:=\{(y,s)\,:\,\Phi(y,s)>\lambda\},$

$\Phi(x,t):=(4t\pi)^{-\frac{n}{2}}\exp\Big(-\frac{|x|^2}{4t}\Big).$

Notice that $\mathrm{diam}(E_\lambda)=o(1)$ as $\lambda\to\infty,$ so the above formula holds for any $(x,t)$ in the (open) set $\mathrm{dom}(u)$ for $\lambda$ large enough. Conversely, any function u satisfying the above mean-value property on an open domain of $\R^n\times\R$ is a solution of the heat equation. This can be shown by an argument similar to the analogous one for harmonic functions.

Applications

Particle diffusion

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

$c_t = D \Delta c, \quad$

$P_t = D \Delta P. \quad$

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.

Brownian motion

The random trajectory of a single particle subject to the particle diffusion equation (or heat equation) is a Brownian motion. If a particle is placed at $\vec R = \vec 0$ at time $t = 0$ , then the probability density function associated with the position vector of the particle $\vec R$ will be the following:

$P(\vec R,t) = G(\vec R,t) = \frac{1}{(2 \pi)^{3/2} (\det(D) t)^{1/2}} \exp\Big\{-\frac{\vec{R}^{T} D^{-1} \vec{R}}{2t}\Big\}$

which is a (multivariate) normal distribution evolving in time.

Schrödinger equation for a free particle

Main article: 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:

$\psi_t = \frac{i \hbar}{2m} \Delta \psi$ , where i is the unit imaginary number, and $\hbar$ is Planck's constant divided by $2\pi$ , and $\psi$ is the wavefunction of the particle.

This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation:

$c(\vec R,t) \to \psi(\vec R,t)$

$D \to \frac{i \hbar}{2m}.$

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 wavefunction at any time through an integral on the wavefunction at t=0:

$\psi(\vec R, t) = \int \psi(\vec R^0,t=0) G(\vec R - \vec R^0,t) dR_x^0\,dR_y^0\,dR_z^0$ , with

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

Remark: this analogy between quantum mechanics and diffusion is a purely formal one. Physically, the evolution of the wavefunction satisfying Schrödinger's equation might have an origin other than diffusion.

Further applications

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 heat equation can be efficiently solved numerically using the 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.

Notes

↑ Here we are assuming that the material has constant mass density and heat capacity through space as well as time, although generalizations are given below.
↑ In higher dimensions, the divergence theorem is used instead.
↑ Note that the units of u must be selected in a manner compatible with those of q. Thus instead of being temperature (K), units of u should be J/L.
↑ As is the convention when dealing with the heat equation, this differs from the usual definition of the theta function in that it is missing a factor of i on the time variable. See, for instance, Cannon (1984), Chapter 6.

References

Cannon, John (1984), The One-Dimensional Heat Equation, Encyclopedia of mathematics and its applications, Addison-Wesley, ISBN 0-521-30243-9

Crank, J.; Nicolson, P.; Hartree, D. R. (1947), "A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-Conduction Type", Proceedings of the Cambridge Philosophical Society 43: 50–67, doi:10.1017/S0305004100023197

Einstein, Albert (1905), "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen", Ann. Phys. Leipzig 17 322: 549–560, doi:10.1002/andp.19053220806

Evans, L.C. (1998), Partial Differential Equations, American Mathematical Society, ISBN 0-8218-0772-2

John, Fritz (1991), Partial Differential Equations (4th ed.), Springer, ISBN 978-0387906096

Wilmott, P.; Howison, S.; Dewynne, J. (1995), The Mathematics of Financial Derivatives:A Student Introduction, Cambridge University Press

Carslaw, H. S.; Jaeger, I. (1973), Conduction of Heat in Solids (2nd ed.), Oxford University Press

External links

Derivation of the heat equation
Linear heat equations: Particular solutions and boundary value problems - from EqWorld