Stable Lévy motions are now used in many areas of science and engineering to model anomalous super-diffusion, where a plume of particles spreads faster than the traditional Brownian motion model predicts. Their probability densities solve a partial differential equation that involves fractional derivatives. When the process is reflected to stay in the positive the real line, it becomes a Markov process. In this talk, we explicitly compute the transition densities of this Markov process, and their governing equation. This Kolmogorov forward equation involves a fractional boundary condition to enforce the no-flux boundary. This seems to be the first known derivation of a fractional reflecting boundary condition. We then apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time.