# Heat Equation

Also found in: Wikipedia.

## heat equation

[′hēt i‚kwā·zhən]
(thermodynamics)
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.

### REFERENCES

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.

D. N. ZUBAREV

References in periodicals archive ?
He covers fundamental materials of Riemannian geometry; the space of Riemannian metrics, and continuity of the eigenvalues; Cheeger and Yau estimates on the minimum positive eigenvalue; the estimation of the kth eigenvalue and Lichnerowicz-Obata's theorem; the Payne, P lya, and Weinberger type inequalities for the Dirichlet eigenvalues; the heat equation and the set of lengths of closed geodesics; and negative curvature manifolds and the spectral rigidity theorem.
It is based on the heat equation in a solid region where the phase change interface is regarded as a moving boundary (Stefan, 1891).
The heat transfer inside the solid during quenching process could be described by heat equation and corresponding boundary conditions, assuming no heat generation during quenching process:
More precisely the PI plans to exploit the relation between the sub-Riemannian distance and the properties of the kernel of the corresponding hypoelliptic heat equation and to study controllability properties of the Schroedinger equation.
We show for the heat equation that when we consider finite time intervals, the Dirichlet-Neumann and Neumann-Neumann methods converge superlinearly for an optimal choice of the relaxation parameter, similar to the case of Schwarz waveform relaxation algorithms.
This paper deals with the output regulation problem for set-point control of a one-dimensional heat equation via pointwise in-domain (or interior) actuation.
Adomian Decomposition method was successfully applied to nonlinear differential delay equations ,a non-linear dynamic systems, the heat equation [8,9], the wave equation , coupled non-linear Partial differential equations [11,12], linear and non-linear integro-differential equations .
Key words: V-groove joints, polyimide, robotic leg, heat equation and virtual work.
Abstract: The present paper explores theoretical aspects of solving the heat equation in the monodimensional case.
Consider the initial value problem for a semilinear heat equation
When they are linearized, type I is the same as the classical heat equation (based on Fourier's law) whereas the linearized versions of type-II and type-III theories permit propagation of thermal waves at finite speed.
The work in the aspect originated in Li and Yau's paper , in which they proved a differential Harnack inequality for positive solutions to the heat equation on Riemannian manifolds with a fixed metric.

Site: Follow: Share:
Open / Close