Special Solutions to the Space Fractional Diffusion Problem

Abstract

We derive a fundamental solution E to a space-fractional diffusion problem on the half-line. The equation involves the Caputo derivative. We establish properties of E as well as formulas for solutions to the Dirichlet and Neumann problems in terms of convolution of E with data. We also study integrability of derivative of solutions given in this way. We present conditions sufficient for uniqueness. Finally, we show the infinite speed of signal propagation.