On solutions of time-fractional advection-diffusion equation

In this paper, we present an attractive reliable numerical approach to find an approximate solution of the time-fractional advection-diffusion equation (FADE) under the Atangana-Baleanu derivative in Caputo sense (ABC) with Mittag-Leffler kernel. The analytic and approximate solutions of FADE have been determined by using reproducing kernel Hilbert space method (RKHSM). The most valuable advantage of the RKHSM is its ease of use and its quick calculation to obtain the numerical solution of the FADE. Our main tools are reproducing kernel theory, some important Hilbert spaces, and a normal basis. The convergence analysis of the RKHSM is studied. The computational results are compared with other results of an appropriate iterative scheme and also by using specific examples, these results clearly show: On the one hand, the effect of the ABC-fractional derivative with the Mittag-Leffler kernel in the obtained outcomes, and on the other hand, the superior performance of the RKHSM. From a numerical viewpoint, the RKHSM provides the solution's representation in a convergent series. Furthermore, the obtained results elucidate that the proposed approach gives highly accurate outcomes. It is worthy to observe that the numerical results of the specific examples show the efficiency and convenience of the RKHSM for dealing with various fractional problems emerging in the physical environment.