Heat equation

In mathematics and physics, the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. Since then, the heat equation and its variants have been found to be fundamental in many parts of both pure and applied mathematics.

Definition

Given an open subset of and a subinterval of, one says that a function is a solution of the heat equation if
where denotes a general point of the domain. It is typical to refer to as time and as spatial variables, even in abstract contexts where these phrases fail to have their intuitive meaning. The collection of spatial variables is often referred to simply as. For any given value of, the right-hand side of the equation is the Laplacian of the function. As such, the heat equation is often written more compactly as
In physics and engineering contexts, especially in the context of diffusion through a medium, it is more common to fix a Cartesian coordinate system and then to consider the specific case of a function of three spatial variables and time variable. One then says that is a solution of the heat equation if
in which is a positive coefficient called the thermal diffusivity of the medium. In addition to other physical phenomena, this equation describes the flow of heat in a homogeneous and isotropic medium, with being the temperature at the point and time. If the medium is not homogeneous and isotropic, then would not be a fixed coefficient, and would instead depend on ; the equation would also have a slightly different form. In the physics and engineering literature, it is common to use to denote the Laplacian, rather than.
In mathematics as well as in physics and engineering, it is common to use Newton's notation for time derivatives, so that is used to denote, so the equation can be written
Note also that the ability to use either or to denote the Laplacian, without explicit reference to the spatial variables, is a reflection of the fact that the Laplacian is independent of the choice of coordinate system. In mathematical terms, one would say that the Laplacian is translationally and rotationally invariant. In fact, it is the simplest differential operator which has these symmetries. This can be taken as a significant justification of the use of the Laplacian and of the heat equation in modeling any physical phenomena which are homogeneous and isotropic, of which heat diffusion is a principal example.

Diffusivity constant

The diffusivity constant is often not present in mathematical studies of the heat equation, while its value can be very important in engineering. This is not a major difference, for the following reason. Let be a function with
Define a new function. Then, according to the chain rule, one has
Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of and solutions of the heat equation with. As such, for the sake of mathematical analysis, it is often sufficient to only consider the case.
Since there is another option to define a satisfying as in above by setting. Note that the two possible means of defining the new function discussed here amount, in physical terms, to changing the unit of measure of time or the unit of measure of length.

Nonhomogeneous heat equation

The nonhomogeneous heat equation is
for a given function which is allowed to depend on both and. The inhomogeneous heat equation models thermal problems in which a heat source modeled by is switched on. For example, it can be used to model the temperature throughout a room with a heater switched on. If is the region of the room where the heater is and the heater is constantly generating units of heat per unit of volume, then would be given by.

Steady-state equation

A solution to the heat equation is said to be a steady-state solution if it does not vary with respect to time:
Flowing via. the heat equation causes it to become closer and closer as time increases to a steady-state solution. For very large time, is closely approximated by a steady-state solution. A steady state solution of the heat equation is equivalently a solution of Laplace's equation.
Similarly, a solution to the nonhomogeneous heat equation is said to be a steady-state solution if it does not vary with respect to time:
This is equivalently a solution of Poisson's equation.
In the steady-state case, a nonzero spatial thermal gradient may be present, but if it is, it does not change in time. The steady-state equation 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 steady-state equations are simpler and can help to understand better the physics of the materials without focusing on the dynamics of heat transport. It is widely used for simple engineering problems assuming there is equilibrium of the temperature fields and heat transport, with time.

Interpretation

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, conveys if 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.
By the combination of these observations, the heat equation says 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, specific heat, and density of the material.

Interpretation of the equation

The first half of the above physical thinking can be put into a mathematical form. The key is that, for any fixed, one has
where is the single-variable function denoting the average value of over the surface of the sphere of radius centered at ; it can be defined by
in which denotes the surface area of the unit ball in -dimensional Euclidean space. This formalizes the above statement that the value of at a point measures the difference between the value of and the value of at points nearby to, in the sense that the latter is encoded by the values of for small positive values of.
Following this observation, one may interpret the heat equation as imposing an infinitesimal averaging of a function. Given a solution of the heat equation, the value of for a small positive value of may be approximated as times the average value of the function over a sphere of very small radius centered at.

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 in the medium, it will spread out in all directions in the form of a diffusion wave. Unlike the elastic and electromagnetic waves, the speed of a diffusion wave drops with time: as it spreads over a larger region, the temperature gradient decreases, and therefore the heat flow decreases too.

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 x is a single coordinate and the heat flow towards is a scalar field. The equation becomes
Let be the internal energy per unit volume of the bar at each point and time. The rate of change in heat 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 centred at, one concludes that the rate at which heat changes at a given point is equal to the derivative of the heat flow at that point. 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.