The observed power laws in space and time profiles of energetic particles in the heliosphere can be the result of an underlying superdiffusive transport behavior. Such anomalous, non-Gaussian transport regimes can arise, for example, as a consequence of intermittent structures in the solar wind. Non-diffusive transport regimes may also play a critical role in other astrophysical environments such as supernova remnant shocks. To clarify the role of superdiffusion in the transport of particles near shocks, we study the solutions of a fractional diffusion-advection equation to investigate this issue. A fractional generalization of the Laplace operator, the Riesz derivative, provides a model of superdiffusive propagation. e vy motion. The expected power law profiles of particles upstream of the plasma shock, where particles are injected, can be reproduced with this approach. The method provides a full, time-dependent solution of the fractional diffusion-advection equation. The developed models enable a quantitative comparison to energetic particle properties based on a comprehensive, superdiffusive transport equation and allow for an application in a number of scenarios in astrophysics and space science.