Analytical expression of convection-diffusion equation with diffusion related to space and momentum in supernova remnants

Abstract Since the convection-diffusion equation was first used to discuss the transport properties of highenergy particles, the theory of shock particle acceleration has experienced rapid development. Scientists have proposed various acceleration mechanisms to explain the existence of these high-energy particles in supernova remnants, but the commonly accepted primary process is diffusion shock acceleration. The interaction between shock waves and particles can be approximated by the convection-diffusion equation. Previous research on the convection-diffusion equation focused on the correlation between diffusion coefficient and momentum, treating the diffusion coefficient as a spatial constant. However, in reality, the diffusion coefficient is related to both position and particle energy. So, in this article, we consider the position and energy dependence of the diffusion coefficient. By analyzing the actual physical environment in supernova remnants, we provide specific expressions of position and energy dependence, and use the (inverse) Laplace transform analysis method to obtain the specific analytical expression of the particle distribution function. Finally, our result is to present the particle distribution function at the positions before and after the shock wave in the form of the Whittaker M function, thus solving the more general particle propagation equation for diffusion shock wave acceleration – the convection-diffusion equation, which lays the foundation for further exploration of the radiation origin of supernova remnants.