derivation of heat equation

Let us consider the heat conduction in a $\varrho$ 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

 $-du\;c\,\varrho\,dv\;=\;-u^{\prime}_{t}\,dt\,c\,\varrho\,dv$

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

 $\displaystyle-dt\int_{v}c\varrho u^{\prime}_{t}\,dv.$ (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

 $-k\,\nabla{u}\cdot d\vec{a}\;dt,$

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

 $-dt\oint_{a}k\nabla{u}\cdot d\vec{a},$

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

 $\displaystyle-dt\int_{v}k\,\nabla\cdot\nabla{u}\,dv\;=\;-dt\int_{v}k\,\nabla^{% 2}u\,dv.$ (2)

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

 $\int_{v}k\,\nabla^{2}u\,dv\;=\;\int_{v}c\,\varrho u^{\prime}_{t}\,dv.$

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

 $k\,\nabla^{2}u\;=\;c\,\varrho u^{\prime}_{t}.$

Denoting  $\displaystyle\frac{k}{c\varrho}=\alpha^{2}$,  we can write this equation as

 $\displaystyle\alpha^{2}\nabla^{2}u\;=\;\frac{\partial u}{\partial t}.$ (3)

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

References

• 1 K. Väisälä: Matematiikka IV.  Handout Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
Title derivation of heat equation DerivationOfHeatEquation 2013-03-22 18:45:04 2013-03-22 18:45:04 pahio (2872) pahio (2872) 9 pahio (2872) Derivation msc 35K05 msc 35Q99 DerivationOfWaveEquation