Surprising non-scaling behavior in fractional diffusion

It is well known that Brownian diffusion is characterized by a mean-squared displacement which varies linearly in time, r2 t and that anomalous diffusion, by a mean-square displacement which is nonlinear in time, r2 tα. It is generally accepted that fractional diffusion, which is used to describe anomalous diffusion, is also characterized by the same scaling behavior in the second moment, r2 tα. It is shown in this paper that this is not always so and that inward radial fractional diffusion in one, two and three dimensions does not exhibit this scaling, where as outward radial fractional diffusion in these cases does exhibit the scaling. The reason for this unexpected behavior is the existence of a characteristic length, and the separation of variables, which occurs because of the boundary conditions.