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Homederivation of heat equation

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# derivation of heat equation

Let us consider the heat conduction in a homogeneous matter with density $\varrho$ 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 simple closed surface in the matter and $v$ the spatial region restricted 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 constant (because the heat flows 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).

## Mathematics Subject Classification

35K05*no label found*35Q99

*no label found*

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