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 MittagLe er 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.

Generalization of Thermal and Mass Fluxes for the Flow of Differential Type Fluid with Caputo–Fabrizio Approach of Fractional Derivative

“…Equation ( 18) is solved with conditions (19), and its solution is expressed as Equation ( 20) is complicated, and it is not possible to invert the temperature field in t-domain by ordinary formula of Laplace inverse.…”

“…Nazar et al [18] discussed the double convectional flow via two approaches of noninteger operators and compared the obtained results of thermal, mass, and momentum profiles. Gomez et al [19] solved fractional diffusion-advection equation and obtained the analytical solution for supper diffusion. Tran et al [20] discussed the stabilities of fractional differential equation.…”