Heat Equation

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heat equation

[′hēt i‚kwā·zhən]
A parabolic second-order differential equation for the temperature of a substance in a region where no heat source exists: ∂ t /∂τ = (kc)(∂2 t /∂ x 2+ ∂2 t /∂ y 2+ ∂ t 2/∂ z 2), where x, y, and z are space coordinates, τ is the time, t (x,y,z, τ) is the temperature, k is the thermal conductivity of the body, ρ is its density, and c is its specific heat; this equation is fundamental to the study of heat flow in bodies. Also known as Fourier heat equation; heat flow equation.

Heat Equation


a parabolic partial differential equation that describes the process of propagation of heat in a continuous medium, such as a gas, liquid, or solid. It is the basic equation in the mathematical theory of thermal conductivity.

The heat equation expresses the heat balance for a small element of volume of the medium; heat gains from sources and heat losses through the surface of the element are taken into account for heat transport by conduction. The equation has the following form for an isotropic nonhomogeneous medium:

Here, ρ is the density of the medium; cv is the specific heat of the medium at constant volume; t is time; x, y, and z are space coordinates; T = T(x, y, z, t) is the temperature, which is calculated by means of the heat equation; λ is the coefficient of thermal conductivity; and F = F(x, y, z, t) is the specified density of the heat sources. The magnitudes of ρ cv, and λ depend on the coordinates and, generally speaking, on the temperature. For an anisotropic medium, the heat equation contains in place of λ the thermal conductivity tensor λik, where i, k = 1,2,3.

In the case of an istropic homogeneous medium, the heat equation assumes the form

where Δ is the Laplace operator, a2 = λ/(ρcv) is the coefficient of thermal diffusivity, and f = F/(ρcv). In a stationary state, where the temperature does not vary with time, the heat equation becomes the Poisson equation or, when there are no heat sources, Laplace’s equation ΔT = 0. The basic problems for the heat equation are the Cauchy problem and the mixed boundary value problem (seeBOUNDARY VALUE PROBLEMS).

The heat equation was first studied by J. Fourier in 1822 and S. Poisson in 1835. Important results in the study of the heat equation were obtained by I. G. Petrovskii, A. N. Tikhonov, and S. L. Sobolev.


Carslaw, W. S. Teoriia teploprovodnosti. Moscow-Leningrad, 1947.
Vladimirov, V. S. Uravneniia matematicheskoi fiziki. Moscow, 1967.
Tikhonov, A. N., and A. A. Samarskii. Uravneniia matematicheskoi fiziki, 3rd ed. Moscow, 1966.


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