Heat equation


In physics and mathematics, the heat equation is a partial differential equation that describes how the distribution of some quantity evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is a special case of the diffusion equation.
This equation was first developed and solved by Joseph Fourier in 1822 to describe heat flow. However, it is of fundamental importance in diverse scientific fields. In probability theory, the heat equation is connected with the study of random walks and Brownian motion, via the Fokker–Planck equation. In financial mathematics, it is used to solve the Black–Scholes partial differential equation. In quantum mechanics, it is used for finding spread of wave function in potential free region. A variant was also instrumental in the solution of the longstanding Poincaré conjecture of topology.

Statement of the equation

For a function of three spatial variables and the time variable, the heat equation is
where is a real coefficient called the diffusivity of the medium. Using Newton's notation for derivatives, and the notation of vector calculus, the heat equation can be written in compact form as
Here denotes the Laplace operator, and is the time derivative of. One advantage of this formula is that the operator can usually be defined in purely physical terms, independently of the choice of coordinate system.
This equation describes the flow of heat in a homogeneous and isotropic medium, with being the temperature at the point and time. However, it also describes many other physical phenomena as well.
The value of affects the speed and spatial scale of the process; changing it has the same effect as changing the unit of measure for time, and/or the unit of measure of length. Therefore, in mathematical studies of this equation, one often sets. With this simplification, the heat equation is the prototypical parabolic partial differential equation.

Interpretation

Meaning of the 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 is the temperature, tells whether 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 a material, its temperature increases, in proportion to the amount of heat divided by the amount of material, with a proportionality factor called the specific heat capacity of the material.
Therefore, the equation says that the rate at which the material at a point will heat up is proportional to how much hotter the surrounding material is. The coefficient in the equation takes into account the thermal conductivity, the specific heat, and the density of the material.

Character of the solutions

The heat equation implies that peaks of will be gradually eroded down, while depressions 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, then the value at the center of that neighborhood will not be changing at that time.
A more subtle consequence is the maximum principle, that says that the maximum value of in any region of the medium will not exceed the maximum value that previously occurred in, unless it is on the boundary of. That is, the maximum temperature in a region can increase only if heat comes in from outside. This is a property of parabolic partial differential equations and is not difficult to prove mathematically.
Another interesting property is that even if initially has a sharp jump 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 and , 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 will gradually vary between and.
If a certain amount of heat is suddenly applied to a point 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.

Specific examples

Heat flow in a uniform rod

For heat flow, the heat equation follows from the physical laws of conduction of heat and conservation of energy.
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:
where is the thermal conductivity of the material, is the temperature, and is a vector field that represents the magnitude and direction of the heat flow at the point of space and time.
If the medium is a thin rod of uniform section and material, the position is a single coordinate, the heat flow towards increasing is a scalar field , and the gradient is an ordinary derivative with respect to the. The equation becomes
Let 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,, is proportional to the rate of change of its temperature,. That is,
where is the specific heat capacity and is the density 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, one concludes that the rate at which heat accumulates at a given point is equal to the derivative of the heat flow at that point, negated. That is,
From the above equations it follows that
which is the heat equation in one dimension, with diffusivity coefficient
This quantity is called the thermal diffusivity of the medium.

Accounting for radiative loss

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, where is the temperature of the surroundings, and is a coefficient that depends on physical properties of the material. The rate of change in internal energy becomes
and the equation for the evolution of becomes

Non-uniform isotropic medium

Note that the state equation, given by the first law of thermodynamics, is written in the following form. This form is more general and particularly useful to recognize which property influences which term.
where is the volumetric heat source.

Three-dimensional problem

In the special cases of propagation of heat in an isotropic and medium in a 3-dimensional space, this equation is
where:
The heat equation is a consequence of Fourier's law of 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.
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
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. 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 have solutions with finite heat transmission speed.

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 at a rate given by a known function q varying in space and time. Then the heat per unit volume u satisfies an equation
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

The following solution technique for the heat equation was proposed by Joseph Fourier in his treatise Théorie analytique de la chaleur, published in 1822. Consider the heat equation for one space variable. This could be used to model heat conduction in a rod. The equation is
where u = u is a function of two variables x and t. Here
We assume the initial condition
where the function f is given, and the boundary conditions
Let us attempt to find a solution of that is not identically zero satisfying the boundary conditions but with the following property: u is a product in which the dependence of u on x, t is separated, that is:
This solution technique is called separation of variables. Substituting u back into equation,
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:
and
We will now show that nontrivial solutions for for values of λ ≤ 0 cannot occur:
This solves the heat equation in the special case that the dependence of u has the special form.
In general, the sum of solutions to that satisfy the boundary conditions also satisfies and. We can show that the solution to, and is given by
where

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 eigenfunctions. This leads naturally to one of the basic ideas of the spectral theory of linear self-adjoint operators.
Consider the linear operator Δu = uxx. The infinite sequence of functions
for n ≥ 1 are eigenfunctions of Δ. Indeed,
Moreover, any eigenfunction f of Δ with the boundary conditions f = f = 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 . This means
Finally, the sequence n ∈ N spans a dense linear subspace of L2). 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.
Putting these equations together gives the general equation of heat flow:
Remarks.
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, for an introductory treatment.
In one variable, the Green's function is a solution of the initial value problem
where δ is the Dirac delta function. The solution to this problem is the fundamental solution
One can obtain the general solution of the one variable heat equation with initial condition u = g for −∞ < x < ∞ and 0 < t < ∞ by applying a convolution:
In several spatial variables, the fundamental solution solves the analogous problem
The n-variable fundamental solution is the product of the fundamental solutions in each variable; i.e.,
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 = g, one has
The general problem on a domain Ω in Rn is
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, it may not be possible to write it down explicitly. Other methods for obtaining Green's functions include the method of images, separation of variables, and Laplace transforms.

Some Green's function solutions in 1D

A variety of elementary Green's function solutions in one-dimension are recorded here; many others are available elsewhere. In some of these, the spatial domain is. In others, it is the semi-infinite interval with either Neumann or Dirichlet boundary conditions. One further variation is that some of these solve the inhomogeneous equation
where f is some given function of x and t.

Homogeneous heat equation

;Initial value problem on
]
Comment. This solution is the convolution with respect to the variable x of the fundamental solution
and the function g.
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
Moreover,
so that, by general facts about approximation to the identity, Φ ∗ 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 ∗ g converges uniformly to g as t → 0, meaning that u with u = g.
;Initial value problem on with homogeneous Dirichlet boundary conditions
Comment. This solution is obtained from the preceding formula as applied to the data g suitably extended to R, so as to be an odd function, that is, letting g := −g 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.
The Green's function number of this solution is X10.
;Initial value problem on with homogeneous Neumann boundary conditions
Comment. This solution is obtained from the first solution formula as applied to the data g suitably extended to R so as to be an even function, that is, letting g := g 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 ux = 0. The Green's function number of this solution is X20.
;Problem on with homogeneous initial conditions and non-homogeneous Dirichlet boundary conditions
Comment. This solution is the convolution with respect to the variable t of
and the function h. Since Φ is the fundamental solution of
the function ψ 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,
so that, by general facts about approximation to the identity, ψ then ψ ∗ h converges uniformly on compacta to h as x → 0, meaning that u × 0, ∞) with u = h''.

Inhomogeneous heat equation

;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
and the function f, both meant as defined on the whole R2 and identically 0 for all t → 0. One verifies that
which expressed in the language of distributions becomes
where the distribution δ is the [Dirac's delta function, that is the evaluation at 0.
;Problem on with homogeneous Dirichlet boundary conditions and initial conditions
Comment. This solution is obtained from the preceding formula as applied to the data f, so as to be an odd function of the variable x, that is, letting f := −f 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.
;Problem on with homogeneous Neumann boundary conditions and initial conditions
Comment. This solution is obtained from the first formula as applied to the data f, so as to be an even function of the variable x, that is, letting f := f 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 ux = 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
let u = w + v where w and v solve the problems
Similarly, to solve
let u = w + v + r where w, v, and r solve the problems

Mean-value property for the heat equation

Solutions of the heat equations
satisfy a mean-value property analogous to the mean-value properties of harmonic functions, solutions of
though a bit more complicated. Precisely, if u solves
and
then
where Eλ is a "heat-ball", that is a super-level set of the fundamental solution of the heat equation:
Notice that
as λ → ∞ so the above formula holds for any in the set dom for λ large enough. This can be shown by an argument similar to the analogous one for harmonic functions.

Steady-state heat equation

The steady-state heat equation is by definition not dependent on time. In other words, it is assumed conditions exist such that:
This condition depends on the time constant and the amount of time passed since boundary conditions have been imposed. Thus, the condition is fulfilled in situations in which the time equilibrium constant is fast enough that the more complex time-dependent heat equation can be approximated by the steady-state case. Equivalently, the steady-state condition exists for all cases in which enough time has passed that the thermal field u no longer evolves in time.
In the steady-state case, a spatial thermal gradient may exist, but if it does, it does not change in time. This equation therefore describes the end result in all thermal problems in which a source is switched on, and enough time has passed for all permanent temperature gradients to establish themselves in space, after which these spatial gradients no longer change in time. The other solution is for all spatial temperature gradients to disappear as well, in which case the temperature become uniform in space, as well.
The equation is much simpler and can help to understand better the physics of the materials without focusing on the dynamic of the heat transport process. It is widely used for simple engineering problems assuming there is equilibrium of the temperature fields and heat transport, with time.
Steady-state condition:
The steady-state heat equation for a volume that contains a heat source, is the Poisson's equation:
where u is the temperature, k is the thermal conductivity and q the heat-flux density of the source.
In electrostatics, this is equivalent to the case where the space under consideration contains an electrical charge.
The steady-state heat equation without a heat source within the volume is the equation in electrostatics for a volume of free space that does not contain a charge. It is described by Laplace's equation:

Applications

Particle diffusion

One can model particle diffusion by an equation involving either:
In either case, one uses the heat equation
or
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, then one gets the nonlinear diffusion equation.

Brownian motion

Let the stochastic process be the solution of the stochastic differential equation
where is the Wiener process. Then the probability density function of is given at any time by
which is the solution of the initial value problem
where is the Dirac delta function.

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:
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:
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:
with
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.

Thermal diffusivity in polymers

A direct practical application of the heat equation, in conjunction with Fourier theory, in spherical coordinates, is the prediction of thermal transfer profiles and the measurement of the thermal diffusivity in polymers. 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
where is the initial temperature of the sphere and the temperature at the surface of the sphere, of radius. This equation has also found applications in protein energy transfer and thermal modeling in biophysics.

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 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.
The heat equation is also widely used in image analysis 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. This method can be extended to many of the models with no closed form solution, see for instance.
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.