  We study regularity and numerical methods for two-sided fractional diffusion
equations with a lower-order term.
  We show that the regularity of the solution in weighted Sobolev spaces can be
greatly improved compared to that in standard Sobolev spaces.
  With this regularity, we improve higher-order convergence of a spectral
Galerkin method. We present a spectral Petrov-Galerkin method and provide an
optimal error estimate for the Petrov-Galerkin method.
  Numerical results are presented to verify our theoretical convergence orders.
