Modelling fluid flow and heat transfer inside a mini- or micro-channel constitutes a challenge because it requires taking into account many effects that do not occur in traditional macrostructured systems. A semi-analytical heat transfer model that takes into account conduction and advection in the fluid as well as conduction in the solid walls (conjugate heat transfer) of a flat mini-channel is first derived and verified. It is based on Fourier transforms of the temperature and normal flux in the direction of the Poiseuille flow. It allows to relate its bulk temperature Tb(x) to external surface sources by two transfer functions without the use of any internal heat transfer coefficient distribution, whatever the location of these sources. The second part of the paper is devoted to the use of this model in an inverse way, that is to retrieve the Tb(x) distribution starting from the additional observation of the noised synthetic temperature profiles over the external faces of both walls of the channel. Estimations of the average velocity and of the external heat transfer coefficient are first implemented. The temperature and flux distributions over the internal faces of the walls are estimated by an inverse method then, before a reconstruction of the internal bulk temperature profile.