Eur. Phys. J. Plus (2018) 133: 215
https://doi.org/10.1140/epjp/i2018-12051-9

Regular Article

Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel

¹ Necmettin Erbakan University, Faculty of Science, Konya, Turkey
² Balikesir University, Faculty of Art and Science, Balikesir, Turkey
³ Munzur University, Faculty of Engineering, Tunceli, Turkey

^* e-mail: hmbaskonus@gmail.com

Received: 7 February 2018
Accepted: 29 April 2018
Published online: 6 June 2018

Abstract

In this paper, time-fractional partial differential equations (FPDEs) involving singular and non-singular 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 , distance term x and time value t . The classical integer-order features are fully recovered if is equal to 1.