Analytical solutions for the fractional diffusion-advection equation describing super-diffusion

Abstract

This paper presents the alternative construction of the diffusion-advection equation in the range (1; 2). The fractional derivative of the Liouville-Caputo type is applied. Analytical solutions are obtained in terms of Mittag-Leffler functions. In the range (1; 2) the concentration exhibits the superdiffusion phenomena and when the order of the derivative is equal to 2 ballistic diffusion can be observed, these behaviors occur in many physical systems such as semiconductors, quantum optics, or turbulent diffusion. This mathematical representation can be applied in the description of anomalous complex processes.