  Motivated by recent applications of superdiffusive transport models to
shock-accelerated particle distributions in the heliosphere, we solve
analytically a one-dimensional fractional diffusion-advection equation for the
particle density. We derive an exact Fourier transform solution, simplify it in
a weak diffusion approximation, and compare the new solution with previously
available analytical results and with a semi-numerical solution based on a
Fourier series expansion. We apply the results to the problem of describing the
transport of energetic particles, accelerated at a traveling heliospheric
shock. Our analysis shows that significant errors may result from assuming an
infinite initial distance between the shock and the observer. We argue that the
shock travel time should be a parameter of a realistic superdiffusive transport
model.
