Partial differential equation and heat conduction

The aim of this seminar is to describe the problem affecting heat transfer equation in a relativistic frame and the efforts to solve it. The first part is dedicated to introduce the tools needed to practically understand the mathematical language of partial differential equation, by using some simple and solved models. The second part explains, in terms of previous section, how heat conduction law violates the principles of relativity and takes account on what are the classical and modern ways to correct that equation. The last section put in evidence the actual absence of formal derivation of Fourier’s law in classical mechanics, leading to a bigger question: how can we generalized a law from a Newtonian frame to a relativistic one even if we do not know how to prove it?