New idea of Atangana-Baleanu time-fractional derivative to advection-diffusion equation

The analytical study of one-dimensional generalized fractional advection-diffusion equation with a time-dependent concentration source on the boundary is carried out. The generalization consists into considering the advection-diffusion equation with memory based on the time-fractional Atangana-Baleanu derivative with Mittag-Leffler kernel. Analytical solution of the fractional differential advection-diffusion equation along with initial and boundary value conditions has been determined by employing Laplace transform and finite sine-Fourier transform. On the basis of the properties of Atangana-Baleanu fractional derivatives and the properties of Mittag-Leffler functions, the general solution is particularized for the fractional parameter alpha = 1 in order to find solution of the classical advection-diffusion process. The influence of memory parameter on the solute concentration has been investigated using the analytical solution and the software Mathcad. From this analysis, it is found that for a constant concentration's source on the boundary, the solute concentration is increasing with fractional parameter, and therefore, an advection-diffusion process described by Atangana-Baleanu time-fractional derivative leads to a smaller solute concentration than in the classical process.