Semi-fractional diffusion equations

TL;DR

This paper introduces semi-fractional derivatives based on semistable Lévy processes, generalizing fractional diffusion equations, and provides numerical methods for their computation and solutions.

Contribution

It develops a new class of semi-fractional derivatives using semigroup theory and Fourier series, extending fractional diffusion models to semistable Lévy processes.

Findings

Derived Gr"unwald-Letnikov type formula for semi-fractional derivatives

Presented a numerical algorithm for semistable density computation

Demonstrated solutions to semi-fractional diffusion equations numerically

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.