Abstract
An alternative construction for the space-time fractional diffusion-advection equation for the sedimentation phenomena is presented. The order of the derivative is considered as0<β,γ≤1for the space and time domain, respectively. The fractional derivative of Caputo type is considered. In the spatial case we obtain the fractional solution for the underdamped, undamped, and overdamped case. In the temporal case we show that the concentration has amplitude which exhibits an algebraic decay at asymptotically large times and also shows numerical simulations where both derivatives are taken in simultaneous form. In order that the equation preserves the physical units of the system two auxiliary parametersσxandσtare introduced characterizing the existence of fractional space and time components, respectively. A physical relation between these parameters is reported and the solutions in space-time are given in terms of the Mittag-Leffler function depending on the parametersβandγ. The generalization of the fractional diffusion-advection equation in space-time exhibits anomalous behavior.
Highlights
The Diffusion-Advection Equation (DAE) describes the evolution of a concentration profile due to diffusion and advection simultaneously; this equation describes physical phenomena where concentrations as mass, energy, or other physical quantities are transferred inside a physical system due to two contributions: diffusion and convection, in this equation the concentration-dependent diffusion coefficient
In the works of the authors mentioned above the pass from an ordinary derivative to a fractional one is direct, to be consistent with the dimensionality of the fractional differential equations (FDE) in the work [24]; the authors have proposed a systematic way to construct FDE for the physical systems analyzing the dimensionality of the ordinary derivative operator and trying to bring it to a fractional derivative operator consistently
We present the analysis of the DAE from the point of view of fractional calculus
Summary
The Diffusion-Advection Equation (DAE) describes the evolution of a concentration profile due to diffusion and advection simultaneously; this equation describes physical phenomena where concentrations as mass, energy, or other physical quantities are transferred inside a physical system due to two contributions: diffusion and convection, in this equation the concentration-dependent diffusion coefficient. Jespersen et al in [19] considered Levy flights subject to external force fields; the authors presented a Riesz/Weyl form of the DAE; the corresponding Fokker-Planck equation contains a fractional derivative in space. The fractional kinetic equations of the diffusion, DAE, and Fokker-Planck type are presented in [21]; the equations are derived from basic random walk models. The main purpose in this paper was to show an alternative solution of the DAE using the fractional derivative of Caputo type; this representation preserves the physical units of the system for any value taken by the exponent of the fractional derivative This alternative solution in the range 0 < β, γ ≤ 1 describes Levy flights (nonMarkovian version) and the phenomena of subdiffusion for space-time domain, respectively. The paper is organized as follows: second section introduction to fractional calculus, third section the fractional DAE, and the fourth section the conclusions
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
