The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time.

The physical problem and the equation

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

$\{\backslash partial\; u\backslash over\; \backslash partial\; t\}\; =$

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

where:

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

• u_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

• k is a material-specific quantity depending on the thermal conductivity , the density and the heat capacity.

The heat equation is a consequence of Fourier's law of cooling (see heat conduction).

To solve the heat equation, we also need to specify boundary conditions for u.

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. 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 generalized to

$u\_t\; =\; k\; \backslash Delta\; u,\; \backslash quad$

where the Laplace operator is taken in the spatial variables.

The heat equation governs heat diffusion, as well as other diffusive processes, such as particle diffusion. 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 processes in finance.

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 an equation with a second-order time derivative must be used. In one dimension, the telegrapher's equation can be adapted.

Solving the heat equation using Fourier series

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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

$(1)\; \backslash \; u\_t\; =\; k\; u\_\{xx\}\; \backslash quad$

where u = u(t, x) is a function of two variables t and x. Here

• x is the space variable, so x ∈ ^*, where L is the length of the rod.
• t is the time variable, so t ≥ 0.

We assume the initial condition

$(2)\; \backslash \; u(0,x)\; =\; f(x)\; \backslash quad\; \backslash forall\; x\; \backslash in*\backslash quad$

where the function f is given and the boundary conditions

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

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:

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

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

$\backslash frac\{T\text{'}(t)\}\{kT(t)\}\; =\; \backslash frac\{X\text{'}\text{'}(x)\}\{X(x)\}.\; \backslash quad$

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:

$(5)\; \backslash \; T\text{'}(t)\; =\; -\; \backslash lambda\; kT(t)\; \backslash quad$

and

$(6)\; \backslash \; X\text{'}\text{'}(x)\; =\; -\; \backslash lambda\; X(x).\; \backslash quad$

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

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

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

   From (3) we get

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

   and therefore B = 0 = C which implies u is identically 0.

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

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

   From equation (3) we conclude in the same manner as in 1 that u is identically 0.

3. Therefore, it must be the case that λ > 0. Then there exist real numbers A, B, C such that

   $T(t)\; =\; A\; e^\{-\backslash lambda\; k\; t\}\; \backslash quad$

   and

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

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

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

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

$u(x,t)\; =\; \backslash sum\_\{n\; =\; 1\}^\{+\backslash infty\}\; D\_n\; \backslash left(\backslash sin\; \backslash frac\{n\backslash pi\; x\}\{L\}\backslash right)\; e^\{-\backslash frac\{n^2\; \backslash pi^2\; kt\}\{L^2\}\}$

where

$D\_n\; =\; \backslash frac\{2\}\{L\}\; \backslash int\_0^L\; f(x)\; \backslash sin\; \backslash frac\{n\backslash pi\; x\}\{L\}\; \backslash ,\; dx.$

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

$e\_n(x)\; =\; \backslash sqrt\{\backslash frac\{2\}\{L\}\}\backslash sin\; \backslash frac\{n\backslash pi\; x\}\{L\}$

for n ≥ 1 are eigenvectors of Δ. Indeed

$\backslash Delta\; e\_n\; =\; -\backslash frac\{n^2\; \backslash pi^2\}\{L^2\}\; e\_n.$

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 L. This means

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

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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

$q\_t(V)\; =\; \backslash int\_V\; Q(t,x)\backslash ,d\; x\; \backslash quad$

• 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

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

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

$q\_t(V)=\; -\; \backslash int\_\{\backslash partial\; V\}\; \backslash mathbf\{H\}(x)\; \backslash cdot\; \backslash mathbf\{n\}(x)\; \backslash ,\; dS$

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

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

where A(x) is a 3 × 3 real matrix, which in fact is symmetric and non-negative.

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

$q\_t(V)\; =\; -\; \backslash int\_\{\backslash partial\; V\}\; \backslash mathbf\{H\}(x)\; \backslash cdot\; \backslash mathbf\{n\}(x)\; \backslash ,\; dS$

$=\; \backslash int\_\{\backslash partial\; V\}\; \backslash mathbf\{A\}(x)\; \backslash cdot\; \backslash nabla\; u\; (x)\; \backslash cdot\; \backslash mathbf\{n\}(x)\; \backslash ,\; dS$

$=\; \backslash int\_V\; \backslash sum\_\{i,\; j\}\; \backslash partial\_\{x\_i\}\; a\_\{i\; j\}(x)\; \backslash partial\_\{x\_j\}\; u\; (t,x)\backslash ,dx$

• 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

$\backslash partial\_t\; u(t,x)\; =\; \backslash kappa(x)\; Q(t,x)\backslash ,\; dx$

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

$\backslash partial\_t\; u(t,x)\; =\; \backslash kappa(x)\; \backslash sum\_\{i,\; j\}\; \backslash partial\_\{x\_i\}\; a\_\{i\; j\}(x)\; \backslash partial\_\{x\_j\}\; u\; (t,x)$

Remarks.

• The constant 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.

Particle diffusion

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Particle 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

$c\_t\; =\; D\; \backslash Delta\; c,\; \backslash quad$

or

$P\_t\; =\; D\; \backslash Delta\; P.\; \backslash quad$

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.

The random trajectory of a single particle subject to the particle diffusion equation is a brownian motion.

If a particle is placed in $\backslash vec\; R\; =\; \backslash vec\; 0$ at time $t\; =\; 0$, then the probability density function associated to the vector $\backslash vec\; R$ will be the following:

$P(\backslash vec\; R,t)\; =\; G(\backslash vec\; R,t)\; =\; \backslash frac\{1\}\{(4\; \backslash pi\; D\; t)^\{3/2\}\}\; e^\{-\backslash frac\; \{\backslash vec\; R^2\}\{4\; D\; t\}\}$

It is related to the probability density functions associated to each of its components $R\_x$, $R\_y$ and $R\_z$ in the following way:

$P(\backslash vec\; R,t)\; =\; \backslash frac\{1\}\{(4\; \backslash pi\; D\; t)^\{3/2\}\}\; e^\{-\backslash frac\; \{R\_x^2+R\_y^2+R\_z^2\}\{4\; D\; t\}\}\; =\; P(R\_x,t)P(R\_y,t)P(R\_z,t)$

The random variables $R\_x$, $R\_y$ and $R\_z$ are distributed according to a normal distribution of mean 0 and of variance $2\backslash ,D\backslash ,t$. In 3D, the random vector $\backslash vec\; R$ is distributed according to a normal distribution of mean $\backslash vec\; 0$ and of variance $6\backslash ,\; D\backslash ,t$.

At t=0, the expression of $P(\backslash vec\; R,t)$ above is singular. The probability density function corresponding to the initial condition of a particle located in a known position $\backslash vec\; R\; =\; \backslash vec\; 0$ is the Dirac delta function, noted $\backslash delta\; (\backslash vec\; R)$ (the generalisation to 3D of the Dirac delta function is simply $\backslash delta\; (\backslash vec\; R)\; =\; \backslash delta\; (R\_x)\; \backslash delta\; (R\_y)\; \backslash delta\; (R\_z)$). The solution of the diffusion equation associated to this initial condition is also called a Green function.

Historical origin of the diffusion equation

The particle diffusion equation was originally derived by Albert Einstein in 1905. Einstein used it in order to model brownian motion. The reference of the major article he published on this subject is the following:

• Einstein, A. "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen." Ann. Phys. 322 (1905) 549-560. ^*

Solving the diffusion equation through Green functions

Green functions are the solutions of the diffusion equation corresponding to the initial condition of a particle of known position. For another initial condition, the solution to the diffusion equation can be expressed as a decomposition on a set of Green Functions.

Say, for example, that at t=0 we have not only a particle located in a known position $\backslash vec\; R\; =\; \backslash vec\; 0$, but instead a large number of particles, distributed according to a spatial concentration profile $c(\backslash vec\; R,\; t=0)$. Solving the diffusion equation will tell us how this profile will evolve with time.

As any function, the initial concentration profile can be decomposed as an integral sum on Dirac delta functions:

$c(\backslash vec\; R,\; t=0)\; =\; \backslash int\; c(\backslash vec\; R^0,t=0)\; \backslash delta(\backslash vec\; R\; -\; \backslash vec\; R^0)\; dR\_x^0\backslash ,dR\_y^0\backslash ,dR\_z^0$

At subsequent instants, given the linearity of the diffusion equation, the concentration profile becomes:

$c(\backslash vec\; R,\; t)\; =\; \backslash int\; c(\backslash vec\; R^0,t=0)\; G(\backslash vec\; R\; -\; \backslash vec\; R^0,t)\; dR\_x^0\backslash ,dR\_y^0\backslash ,dR\_z^0$, where $G(\backslash vec\; R\; -\; \backslash vec\; R^0,t)$ is the Green function defined above.

Although it is more easily understood in the case of particle diffusion , where an initial condition corresponding to a Dirac delta function can be intuitively described as a particle being located in a known position, such a decomposition of a solution into Green functions can be generalized to the case of any diffusive process, like heat transfer, or momentum diffusion, which is the phenomenon at the origin of viscosity in liquids.

List of Green Function Solutions in 1D

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

$u(x,t)=\backslash frac\{1\}\{\backslash sqrt\{4\backslash pi\; kt\}\}\; \backslash int\_\{0\}^\{\backslash infty\}$

\left(\exp\left(-\frac{(x-y)^2}{4kt}\right)-\exp\left(-\frac{(x+y)^2}{4kt}\right)\right) g(y)\,dy

$u(x,t)=\backslash frac\{1\}\{\backslash sqrt\{4\backslash pi\; kt\}\}\; \backslash int\_\{0\}^\{\backslash infty\}$

\left(\exp\left(-\frac{(x-y)^2}{4kt}\right)+\exp\left(-\frac{(x+y)^2}{4kt}\right)\right) g(y)\,dy

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

$u(x,t)=\backslash int\_\{0\}^\{t\}\backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{1\}\{\backslash sqrt\{4\backslash pi\; k(t-s)\}\}$

\left(\exp\left(-\frac{(x-y)^2}{4k(t-s)}\right)-\exp\left(-\frac{(x+y)^2}{4k(t-s)}\right)\right) f(s)\,dyds

$u(x,t)=\backslash int\_\{0\}^\{t\}\; \backslash frac\{x\}\{\backslash sqrt\{4\backslash pi\; k(t-s)^3\}\}$

\exp\left(-\frac{x^2}{4k(t-s)}\right)h(s)\,ds

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

\\ v(x,0)=0,\, w(x,0)=g(x) \, & IC\end{cases}

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

\\ 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}

Schrödinger equation for a free particle

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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:

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

This equation is a mathematical analogue of the particle diffusion equation, which one obtains through the following transformation:

$c(\backslash vec\; R,t)$→$\backslash psi(\backslash vec\; R,t)$

$D~$→$\backslash frac\{i\; \backslash hbar\}\{2m\}$

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 date through an integral on the wavefunction at t=0:

$\backslash psi(\backslash vec\; R,\; t)\; =\; \backslash int\; \backslash psi(\backslash vec\; R^0,t=0)\; G(\backslash vec\; R\; -\; \backslash vec\; R^0,t)\; dR\_x^0\backslash ,dR\_y^0\backslash ,dR\_z^0$, with

$G(\backslash vec\; R,t)\; =\; \backslash frac\{m^\{3/2\}\}\{(2\; i\; \backslash Pi\; \backslash hbar\; t)^\{3/2\}\}\; e^\{-\backslash frac\; \{\backslash vec\; R^2\; m\}\{2\; i\; \backslash hbar\; t\}\}$

Remark: this analogy between quantum mechanics and diffusion is a purely mathematical one. In physics, the evolution of the wavefunction according to Schrödinger equation is not a diffusive process.

Diffusion (of particles, heat, momentum...) describes the return to global thermodynamic equilibrium of an inhomogeneous system, and as such is a time-irreversible phenomenon, associated to an increase in the entropy of the universe: in the case of particle diffusion, if $c(\backslash vec\; R,t)$ is a solution of the diffusion equation, then $c(\backslash vec\; R,-t)$ isn't. Intuitively we know that particle diffusion tends to resorb spatial concentration inhomogeneities, and never amplify them.

As a generalization of classical mechanics, quantum mechanics involves only time-reversible phenomena: if $\backslash psi(\backslash vec\; R,t)$ is a solution of the Schrödinger equation, then the complex conjugate of $\backslash psi(\backslash vec\; R,-t)$ is also a solution. Note that the complex conjugate of a wavefunction has the exact same physical meaning as the wavefunction itself: the two react exactly in the same way to any series of quantum measurements.

It is the imaginary nature of the equivalent diffusion coefficient $i\; \backslash hbar/(2m)$ that makes up for this difference in behavior between quantum and diffusive systems.

On a related note, it is interesting to notice that the imaginary exponentials that appear in the Green functions associated to the Schrödinger equation create interferences between the various components of the decomposition of the wavefunction. This is a symptom of the wavelike properties of quantum particles.

Applications

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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 and this method can be extended to many of the models with no closed form solution. (Wilmott, 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.

See also

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• Heat
• Partial differential equation

References

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• Einstein, A. "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen." Ann. Phys. 17, 549, 1905. ^*

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

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

External links

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

Heat | Partial differential equations | Diffusion

Wärmeleitungsgleichung | Équation de la chaleur | Warmtevergelijking | Уравнение диффузии | Difuzijska enačba | Värmeledningsekvationen