New Idea of Atangana‐Baleanu Time‐fractional Derivative to Advection‐diffusion Equation

SummaryThe 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 α = 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.