# Heat equation

The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. In the special case of heat propagation in an isotropic and homogeneous medium in the 3-dimensional space, this equation is

$u_{t}=k(u_{{xx}}+u_{{yy}}+u_{{zz}})\quad$

where:

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

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.

The heat equation is the prototypical example of a parabolic partial differential equation.

Using the Laplace operator, the heat equation can be generalized to

$u_{t}=k\Delta u,\quad$

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.

## Solving the heat equation using Fourier series

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

$(1)\ u_{t}=ku_{{xx}}\quad$

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

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

where the function f is given and the boundary conditions

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

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:

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

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

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

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:

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

and

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

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

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

$X(x)=Be^{{{\sqrt {-\lambda k}}x}}+Ce^{{-{\sqrt {-\lambda k}}x}}$.

From (3) we get

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

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

2. Suppose that λ=0. Then there exists 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.

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

$T(t)=Ae^{{-\lambda kt}}\quad$

and

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

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

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

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

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

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 uxx 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 = ux 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 en for some n ≥ 1. The functions en 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 {en}nN 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 qt(V). We assume q has a density, so that
$q_{t}(V)=\int _{V}Q(t,x)\,dx\quad$
• 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
${\mathbf {H}}(x)=-{\mathbf {A}}(x)\cdot [\operatorname {grad}(u)](x)$
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

$q_{t}(V)=-\int _{{\partial V}}{\mathbf {H}}(x)\cdot {\mathbf {n}}(x)\,dS$
$=\int _{{\partial V}}{\mathbf {A}}(x)\cdot [\operatorname {grad}(u)](x)\cdot {\mathbf {n}}(x)\,dS$
$=\int _{V}\sum _{{i,j}}\partial _{{x_{i}}}a_{{ij}}(x)\partial _{{x_{j}}}u(t,x)\,dx$
• 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
$\partial _{t}u(t,x)=\kappa (x)Q(t,x)\,dx$

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

$\partial _{t}u(t,x)=\kappa (x)\sum _{{i,j}}\partial _{{x_{i}}}a_{{ij}}(x)\partial _{{x_{j}}}u(t,x)$

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

### Particle diffusion equation

One can model particle diffusion by an equation involving either:

In either case, one uses the heat equation

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

or

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

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.

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

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

$P({\vec R},t)=G({\vec R},t)={\frac {1}{(4\Pi Dt)^{{3/2}}}}e^{{-{\frac {{\vec R}^{2}}{4Dt}}}}$

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:

$P({\vec R},t)={\frac {1}{(4\Pi Dt)^{{3/2}}}}e^{{-{\frac {R_{x}^{2}+R_{y}^{2}+R_{z}^{2}}{4Dt}}}}=P(R_{x},t)P(R_{y},t)P(R_{z},t)$

The random variables $R_{x}$, $R_{y}$ and $R_{z}$ are distributed according to a normal distribution of mean 0 and of variance $2\,D\,t$. In 3D, the random vector ${\vec R}$ is distributed according to a normal distribution of mean ${\vec 0}$ and of variance $6\,D\,t$.

At t=0, the expression of $P({\vec R},t)$ above is singular. The probability density function corresponding to the initial condition of a particle located in a known position ${\vec R}={\vec 0}$ is the Dirac delta function, noted $\delta ({\vec R})$ (the generalisation to 3D of the Dirac delta function is simply $\delta ({\vec R})=\delta (R_{x})\delta (R_{y})\delta (R_{z})$). 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. 17, 549, 1905. [1]

### 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 ${\vec R}={\vec 0}$, but instead a large number of particles, distributed according to a spatial concentration profile $c({\vec R},t=0)$. 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:

$c({\vec R},t=0)=\int c({\vec R}^{0},t=0)\delta ({\vec R}-{\vec R}^{0})dR_{x}^{0}\,dR_{y}^{0}\,dR_{z}^{0}$

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

$c({\vec R},t)=\int c({\vec R}^{0},t=0)G({\vec R}-{\vec R}^{0},t)dR_{x}^{0}\,dR_{y}^{0}\,dR_{z}^{0}$, where $G({\vec R}-{\vec R}^{0},t)$ 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.

## Schrödinger equation for a free particle

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

$c({\vec R},t)$$\psi ({\vec R},t)$
$D~$${\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 date 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
$G({\vec R},t)={\frac {m^{{3/2}}}{(2i\Pi \hbar t)^{{3/2}}}}e^{{-{\frac {{\vec R}^{2}m}{2i\hbar t}}}}$

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 $c({\vec R},t)$ is a solution of the diffusion equation, then $c({\vec R},-t)$ 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 $\psi ({\vec R},t)$ is a solution of the Schrödinger equation, then the complex conjugate of $\psi ({\vec R},-t)$ 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 $i\hbar /(2m)$ 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

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.