  This paper explicitly computes the transition densities of a spectrally
negative stable process with index greater than one, reflected at its infimum.
First we derive the forward equation using the theory of sun-dual semigroups.
The resulting forward equation is a boundary value problem on the positive
half-line that involves a negative Riemann-Liouville fractional derivative in
space, and a fractional reflecting boundary condition at the origin. Then we
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.
