Cosmic rays are relativistic particles that hit continuously the Earth's atmosphere from outer space. Understanding the origin of these cosmic particles, and how they acquire their energy is one of the most important open problems in high energy astrophysics. 

It is widely believed that the majority of cosmic rays are accelerated at supernova remnants, via an acceleration mechanism that operates at shock waves. The spectrum of the accelerated particles can be obtained by solving the transport equation. This is a partial differential equation that describes the evolution in time of the energy spectrum of the accelerated cosmic rays. This equation can be solved analytically in the steady state (time independent) limit, while in order to find the time dependent solution one has to employ numerical methods, which are often time consuming. 

The goal of the stage is to search for accurate, approximate and analytic solutions of the time dependent problem.