Fractional diffusion: exact representations of spectral functions

For all the relevant transformed spaces, i.e. Fourier, Laplace and Fourier - Laplace, we present exact solutions of a fractional diffusion equation, describing random transport on fractals. The potential importance of such spectral representations lies in their applications to interpreting experimental measurements of anomalous diffusion processes. In contrast to the well known asymptotic results, the exact representations provide a much broader basis for comparison with data.