derivation of heat equation

Let us consider the heat conduction in a ϱ and specific heat capacity c.  Denote by  u(x,y,z,t)  the temperature in the point  (x,y,z)  at the time t.  Let a be a surface in the matter and v the spatial region by it.

When the growth of the temperature of a volume element dv in the time dt is du, the element releases the amount


of heat, which is the heat flux through the surface of dv.  Thus if there are no sources and sinks of heat in v, the heat flux through the surface a in dt is

-dtvcϱut𝑑v. (1)

On the other hand, the flux through da in the time dt must be proportional to a, to dt and to the derivative of the temperature in the direction of the normal line of the surface element da, i.e. the flux is


where k is a positive (because the heat always from higher temperature to lower one).  Consequently, the heat flux through the whole surface a is


which is, by the Gauss’s theorem, same as

-dtvkudv=-dtvk2udv. (2)

Equating the expressions (1) and (2) and dividing by dt, one obtains


Since this equation is valid for any region v in the matter, we infer that


Denoting  kcϱ=α2,  we can write this equation as

α22u=ut. (3)

This is the differential equationMathworldPlanetmath of heat conduction, first derived by Fourier.


  • 1 K. Väisälä: Matematiikka IV.  Handout Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
Title derivation of heat equation
Canonical name DerivationOfHeatEquation
Date of creation 2013-03-22 18:45:04
Last modified on 2013-03-22 18:45:04
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Derivation
Classification msc 35K05
Classification msc 35Q99
Related topic DerivationOfWaveEquation