Semi-fractional diffusion equations

Abstract

It is well known that certain fractional diffusion equations can be solved by the densities of stable L\'evy motions. In this paper we use the classical semigroup approach for L\'evy processes to define semi-fractional derivatives, which allows us to generalize this statement to semistable L\'evy processes. A Fourier series approach for the periodic part of the corresponding L\'evy exponents enables us to represent semi-fractional derivatives by a Gr\"unwald-Letnikov type formula. We use this formula to calculate semi-fractional derivatives and solutions to semi-fractional diffusion equations numerically. In particular, by means of the Gr\"unwald-Letnikov type formula we provide a numerical algorithm to compute semistable densities.