# 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

$$-duc\varrho dv=-{u}_{t}^{\prime}dtc\varrho dv$$ |

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

$-dt{\displaystyle {\int}_{v}}c\varrho {u}_{t}^{\prime}\mathit{d}v.$ | (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

$$-k\nabla u\cdot d\overrightarrow{a}dt,$$ |

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

$$-dt{\oint}_{a}k\nabla u\cdot d\overrightarrow{a},$$ |

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

$-dt{\displaystyle {\int}_{v}}k\nabla \cdot \nabla udv=-dt{\displaystyle {\int}_{v}}k{\nabla}^{2}udv.$ | (2) |

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

$${\int}_{v}k{\nabla}^{2}udv={\int}_{v}c\varrho {u}_{t}^{\prime}\mathit{d}v.$$ |

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

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

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

${\alpha}^{2}{\nabla}^{2}u={\displaystyle \frac{\partial u}{\partial t}}.$ | (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 |
---|---|

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 |