Cauchy and source problems for an advection-diffusion equation with Atangana-Baleanu derivative on the real line

In this paper, a linear advection-diffusion equation involving Atangana-Baleanu derivative described with the Mittag-Leffler kernel is considered on the real line. Different kinds of diffusive transports in the nature obey the exponential/generalized exponential and Mittag-Leffler functions rather than the power law. By this reality, the current study is devoted to investigate the fundamental solutions of the Cauchy and source problems. For this purpose, Laplace and exponential Fourier transforms are applied. The results are achieved in terms of one and two-parameter Mittag-Leffler functions. The results show that the Atangana-Baleanu derivative is an effective alternative to Caputo derivative to model the diffusion with advection processes because the continuous structure of Mittag-Leffler kernel removes the computational complexities. Thus, it is rather practical to achieve analytical solutions. (C) 2018 Elsevier Ltd. All rights reserved.