Analytical solution to nonlinear thermal diffusion: Kirchhoff versus Cole-Hopf transformations

The Kirchhoff transformation is the classical method of solution to the nonlinear thermal diffusion problem with temperature dependent properties. It essentially converts the nonlinear problem into a linear one if the thermal diffusivity is approximately constant. Unfortunately, with the only exception of an exponential dependence of the thermal conductivity on temperature, all other thermal conductivity functions produce an inconvenient form for the inverse transform. This paper shows that the Kirchhoff transformation is a particular consequence of the more general Cole-Hopf transformation. However, the classical presentation of the Kirchhoff transformation in terms of a definite integral is more restrictive than the result obtained from the Cole-Hopf transformation and it is this restrictiveness that causes the practical inconvenience in the form of the inverse transform. It is shown that a more compact and practically convenient form of the inverse transform can be obtained by using directly the result from the Cole-Hopf transformation, hence, making its application more attractive.