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March  2020, 13(3): 995-1006. doi: 10.3934/dcdss.2020058

## Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel

 1 Faculty of Science, Department of Mathematics-Computer Sciences, Necmettin Erbakan University, Konya, 42090, Turkey 2 Faculty of Sciences and Arts, Department of Mathematics, Balıkesir University, Balıkesir, 10145, Turkey

* Corresponding author: mehmetyavuz@erbakan.edu.tr

Received  August 2018 Revised  September 2018 Published  March 2019

In this manuscript, we have proposed a comparison based on newly defined fractional derivative operators which are called as Caputo-Fabrizio (CF) and Atangana-Baleanu (AB). In 2015, Caputo and Fabrizio established a new fractional operator by using exponential kernel. After one year, Atangana and Baleanu recommended a different-type fractional operator that uses the generalized Mittag-Leffler function (MLF). Many real-life problems can be modelled and can be solved by numerical-analytical solution methods which are derived with these operators. In this paper, we suggest an approximate solution method for PDEs of fractional order by using the mentioned operators. We consider the Laplace homotopy transformation method (LHTM) which is the combination of standard homotopy technique (SHT) and Laplace transformation method (LTM). In this study, we aim to demonstrate the effectiveness of the aforementioned method by comparing the solutions we have achieved with the exact solutions. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.

Citation: Mehmet Yavuz, Necati Özdemir. Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 995-1006. doi: 10.3934/dcdss.2020058
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##### References:
The solution function of (29) in the CFO sense for $x = 0.5$ (left) and $x = 1$ (right)
The solution function of (29) in the ABO sense for $x = 0.5$ (left) and $x = 1$ (right)
The solution of Eq. (37) in the CFO sense for various values of $\alpha .$
The solution function of (45) in the ABO sense for various values of $\alpha = 0.7$ (left) and $\alpha = 0.9$ (right)
Inaccuracy rates (%) of the mentioned method

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