  In this report we investigate the regularity of the solution to the
fractional diffusion, advection, reaction equation on a bounded domain in
$\mathbb{R}^{1}$. The analysis is performed in the weighted Sobolev spaces,
$H_{(a , b)}^{s}(\mathrm{I})$. Three different characterizations of $H_{(a ,
b)}^{s}(\mathrm{I})$ are presented, together with needed embedding theorems for
these spaces. The analysis shows that the regularity of the solution is bounded
by the endpoint behavior of the solution, which is determined by the parameters
$\alpha$ and $r$ defining the fractional diffusion operator. Additionally, the
analysis shows that for a sufficiently smooth right hand side function, the
regularity of the solution to fractional diffusion reaction equation is lower
than that of the fractional diffusion equation. Also, the regularity of the
solution to fractional diffusion advection reaction equation is two orders
lower than that of the fractional diffusion reaction equation.
