Regularity, and spectral type approximation, of the solution to the fractional order diffusion equation in 1-d.

Few results are known on the regularity of the solution to fractional order differential equations. In this presentation we will investigate the regularity of the solution to the fractional order diffusion equation in 1-d. The fractional order diffusion operator we investigate is motivated by considering the 1-d heat equation. For this operator we are able to give a precise characterization of the regularity of the solution. Next, for this fractional diffusion operator, we investigate generalized eigenfunctions and eigenvalues. Using these generalized eigenfunctions, a spectral type approximation scheme for the fractional order diffusion equation is proposed and analyzed. Numerical results are presented illustrating the regularity of the solution, and demonstrating the spectral type approximation method.