On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel

In this paper, we give analytical solutions of a fractional-time wave equation with memory effect and frictional memory kernel of Mittag-Leffler type via the Atangana-Baleanu fractional order derivative. The method of separation of variables and the Laplace transform has been used to obtain the exact solutions for the fractional order wave equations. Additionally, we present analytical solutions considering the Caputo-Fabrizio fractional derivative with exponential kernel. We showed that the solutions obtained via Caputo-Fabrizio fractional order derivative were a particular case of the solutions obtained with the new fractional derivative based in the Mittag-Leffler law. (C) 2018 Elsevier Ltd. All rights reserved.