Heat equation

Given an open subset U of Rn and a subinterval I of R, one says that a function u : U × I → R is a solution of the heat equation if where (x1, ..., xn, t) denotes a general point of the domain.

It is typical to refer to t as time and x1, ..., xn as spatial variables, even in abstract contexts where these phrases fail to have their intuitive meaning.

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 u(x, y, z, t) of three spatial variables (x, y, z) and time variable t. One then says that u is a solution of the heat equation if in which α 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 u(x, y, z, t) being the temperature at the point (x, y, z) and time t. If the medium is not homogeneous and isotropic, then α would not be a fixed coefficient, and would instead depend on (x, y, z); the equation would also have a slightly different form.

The diffusivity constant, α, is often not present in mathematical studies of the heat equation, while its value can be very important in engineering.

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.

This equation therefore describes the end result in all thermal problems in which a source is switched on (for example, an engine started in an automobile), 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 (as again, with an automobile in which the engine has been running for long enough).

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.

The coefficient α in the equation takes into account the thermal conductivity, specific heat, and density of the material.

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.

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

If the medium is a thin rod of uniform section and material, the position x is a single coordinate and the heat flow

This derivation assumes that the material has constant mass density and heat capacity through space as well as time.

becomes Note that the state equation, given by the first law of thermodynamics (i.e. conservation of energy), is written in the following form (assuming no mass transfer or radiation).

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.

Although they are not diffusive in nature, some quantum mechanics problems are also governed by a mathematical analog of the heat equation (see below).

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.

The following solution technique for the heat equation was proposed by Joseph Fourier in his treatise Théorie analytique de la chaleur, published in 1822.

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.

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.

This solution is obtained from the preceding formula as applied to the data f(x, t) suitably extended to R × [0,∞), so as to be an odd function of the variable x, that is, letting f(−x, t) := −f(x, t) 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, t) = 0.

Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry.

A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, inspiring the introduction of the Ricci flow by Richard Hamilton in 1982 and culminating in the proof of the Poincaré conjecture by Grigori Perelman in 2003.

Following Robert Richtmyer and John von Neumann's introduction of artificial viscosity methods, solutions of heat equations have been useful in the mathematical formulation of hydrodynamical shocks.

Physically, the evolution of the wave function satisfying Schrödinger's equation might have an origin other than diffusion[citation needed].

Diffusion problems dealing with Dirichlet, Neumann and Robin boundary conditions have closed form analytic solutions (Thambynayagam 2011).

The heat equation is also widely used in image analysis (Perona & Malik 1990) and in machine learning as the driving theory behind scale-space or graph Laplacian methods.

This method can be extended to many of the models with no closed form solution, see for instance (Wilmott, Howison & Dewynne 1995).