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    Investigation of the fractional coupled viscous Burgers' equation involving Mittag-Leffler kernel
    (Elsevier, 2019) Sulaiman, Tukur Abdulkadir; Yavuz, Mehmet; Bulut, Hasan; Baskonus, Haci Mehmet
    This study investigates the fractional coupled viscous Burgers' equation under the Liouville-Caputo, Atangana-Baleanu and Yang-Srivastava-Machado fractional derivatives. With the help of fixed-point theorem, and using the Atangana-Baleanu fractional derivative with Mittag-Leffler kernel type kernel, we proved the existence and uniqueness of the studied model. The Laplace Homotopy perturbation method (LPM) defined with the Liouville-Caputo, Atangana-Baleanu and Yang-Srivastava-Machado operators is used in obtaining the exact solutions of the nonlinear model. The numerical simulations of the obtained solutions are performed. We have seen the effect of the various parameters and variables on the displacement in Figs. 1-6. (C) 2019 Elsevier B.V. All rights reserved.
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    Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel
    (Springer Heidelberg, 2018) Yavuz, Mehmet; Ozdemir, Necati; Baskonus, Haci Mehmet
    In this paper, time-fractional partial differential equations (FPDEs) involving singular and nonsingular kernel are considered. We have obtained the approximate analytical solution for linear and nonlinear FPDEs using the Laplace perturbation method (LPM) defined with the Liouville-Caputo (LC) and Atangana-Baleanu (AB) fractional operators. The AB fractional derivative is defined with the Mittag-Leffler function and has all the properties of a classical fractional derivative. In addition, the AB operator is crucial when utilizing the Laplace transform (LT) to get solutions of some illustrative problems with initial condition. We show that the mentioned method is a rather effective and powerful technique for solving FPDEs. Besides, we show the solution graphs for different values of fractional order a, distance term x and time value t. The classical integer-order features are fully recovered if a is equal to 1.