  In this paper we investigate the numerical approximation of the fractional
diffusion, advection, reaction equation on a bounded interval. Recently the
explicit form of the solution to this equation was obtained. Using the explicit
form of the boundary behavior of the solution and Jacobi polynomials, a
Petrov-Galerkin approximation scheme is proposed and analyzed. Numerical
experiments are presented which support the theoretical results, and
demonstrate the accuracy and optimal convergence of the approximation method.
