Fractional Bateman equations in the Atangana-Baleanu sense

Abstract Considering experimental evidence of fractional behavior for alpha-type radioactive decay, this paper is placed in the context of the fractional Bateman equations with several memory orders. Recent papers studied these equations for Caputo/Riemann-Liouville fractional calculus and suggested the investigation of alternative fractional operators. We focus on the Atangana-Baleanu (AB) formulae, specifically on the associated Volterra integral equations to preserve mass balance. On the one hand, we deal with the associated mathematics of the Bateman model: we solve it in terms of multivariate Mittag-Leffler functions, by means of the Mikusiński operational method, which is more general than the Laplace-transform technique. On the other hand, we address the physical implications of the AB calculus, discussing relevant issues and deficiencies that indicate alternative lines of research.