Fractional model equation for anomalous diffusion

In recent years the phenomenon of anomalous diffusion has attracted more and more attention. One of the main impulses was initiated by de Gennes' idea of the “ant in the labyrinth”. Several authors presented asymptotic probability density functions for the location of a random walker on a fractal object. As this density function and the time dependence of its second moment are now well established, a modified diffusion equation providing the correct result is formulated. The parameters of this fractional partial differential equation are uniquely determined by the fractal Hausdorff dimension of the underlying object and the anomalous diffusion exponent. The presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters. A closed form solution is given in terms of Fox's H-function. In the asymptotic case a “halved” diffusion equation can be established. Furthermore, the differences to equations considered previously are discussed.