# American Institute of Mathematical Sciences

• Previous Article
MHD natural convection boundary-layer flow over a semi-infinite heated plate with arbitrary inclination
• DCDS-S Home
• This Issue
• Next Article
A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative
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
##### References:

show all references

##### 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
 [1] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Zakia Hammouch, Dumitru Baleanu. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 975-993. doi: 10.3934/dcdss.2020057 [2] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Ebraheem O. Alzahrani. A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 937-956. doi: 10.3934/dcdss.2020055 [3] Pierre Aime Feulefack, Jean Daniel Djida, Atangana Abdon. A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3227-3247. doi: 10.3934/dcdsb.2018317 [4] Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031 [5] Ndolane Sene. Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020173 [6] Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 377-387. doi: 10.3934/dcdss.2020021 [7] Fırat Evirgen, Sümeyra Uçar, Necati Özdemir, Zakia Hammouch. System response of an alcoholism model under the effect of immigration via non-singular kernel derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020145 [8] Krunal B. Kachhia, Abdon Atangana. Electromagnetic waves described by a fractional derivative of variable and constant order with non singular kernel. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020172 [9] Yuxia Guo, Jianjun Nie. Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6873-6898. doi: 10.3934/dcds.2016099 [10] Badr Saad T. Alkahtani, Ilknur Koca. A new numerical scheme applied on re-visited nonlinear model of predator-prey based on derivative with non-local and non-singular kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 429-442. doi: 10.3934/dcdss.2020024 [11] Ravi Shanker Dubey, Pranay Goswami. Mathematical model of diabetes and its complication involving fractional operator without singular kernal. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020144 [12] G. M. Bahaa. Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 485-501. doi: 10.3934/dcdss.2020027 [13] Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032 [14] Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043 [15] Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250 [16] Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715 [17] Daniele Bartoli, Leo Storme. On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric. Advances in Mathematics of Communications, 2014, 8 (3) : 271-280. doi: 10.3934/amc.2014.8.271 [18] Sümeyra Uçar. Existence and uniqueness results for a smoking model with determination and education in the frame of non-singular derivatives. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020178 [19] Hayat Zouiten, Ali Boutoulout, Delfim F. M. Torres. Regional enlarged observability of Caputo fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 1017-1029. doi: 10.3934/dcdss.2020060 [20] Laurent Denis, Anis Matoussi, Jing Zhang. The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5185-5202. doi: 10.3934/dcds.2015.35.5185

2018 Impact Factor: 0.545

## Tools

Article outline

Figures and Tables