High-energy astrophysics concerns itself with the origin of energetic particles nd photons in the universe. Some astrophysical objects which have been dentified as sources of energetic particles include the sun; flare stars; SNRs (supernova remnants); pulsars and AGN (active galactic nuclei). The search for an extraterrestrial source of radiation began in 1911 when the ustrian physicist Victor Hess initiated a series of balloon flights to determine he radiation intenstity variation as a function of height above the surface f the Earth. He showed th a t the intensity of naturally occurring radiation ncreases with sufficient height, which can be explained by a decreased shielding f an extraterrestrial source of radiation by the remaining column of air bove the balloon. Millikan christened the observed high-energy particles osmic Rays (CRs) at a later date. It is now an established fact that cosmic rays do not constitute a local phenomenon and that in fact the majority must originate from distant sources either in our own Galaxy or beyond. The purpose of this thesis is to investigate particle acceleration at the shock fronts associated with expanding SNRs. Most models of galactic CR acceleration have focused on SNRs based on the tota l energy input required. The phenomenon being investigated requires a knowledge of CRs themselves - their propagation and development through the atmosphere (scattering, collisions resulting in fragmentation into their constituent parts); shock dynamics - how a shock front is formed, jump conditions for particles across a shock front; SNRs - their formation and expansion characteristics; and also particle acceleration - acceleration times, diffusive lengths, scattering mechanisms etc. The Bell-Lucek hypothesis, which suggests th a t diffusive shock acceleration (DSA), the conventional process of particle acceleration in shocks, may also result in an amplification of the highly tangled magnetic field around the shock, is investigated using the “box” model. The equations in question were solved using a C programme th a t solves systems of equations using the Runge-Kutta method.