  In this article we investigate the solution of the steady-state fractional
diffusion equation on a bounded domain in $\real^{1}$. From an analysis of the
underlying model problem, we postulate that the fractional diffusion operator
in the modeling equations is neither the Riemann-Liouville nor the Caputo
fractional differential operators. We then find a closed form expression for
the kernel of the fractional diffusion operator which, in most cases,
determines the regularity of the solution. Next we establish that the Jacobi
polynomials are pseudo eigenfunctions for the fractional diffusion operator. A
spectral type approximation method for the solution of the steady-state
fractional diffusion equation is then proposed and studied.
