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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 and Continuous Dynamical Systems - S, 2020, 13 (3) : 995-1006. doi: 10.3934/dcdss.2020058
##### References:
 [1] T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098–1107, arXiv: 1607.00262. doi: 10.22436/jnsa.010.03.20. [2] B. S. T. Alkahtani and A. Atangana, Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order, Chaos, Solitons & Fractals, 89 (2016), 539-546.  doi: 10.1016/j.chaos.2016.03.012. [3] B. S. T. Alkahtani and A. Atangana, Analysis of non-homogeneous heat model with new trend of derivative with fractional order, Chaos, Solitons & Fractals, 89 (2016), 566-571.  doi: 10.1016/j.chaos.2016.03.027. [4] B. S. T. Alkahtani, Chua's circuit model with Atangana–Baleanu derivative with fractional order, Chaos, Solitons & Fractals, 89 (2016), 547-551. [5] O. J. J. Alkahtani, Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model, Chaos, Solitons & Fractals, 89 (2016), 552-559.  doi: 10.1016/j.chaos.2016.03.026. [6] R. T. Alqahtani, Atangana-Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer, Journal of Nonlinear Sciences and Applications, 9 (2016), 3647-3654.  doi: 10.22436/jnsa.009.06.17. [7] F. A. M. N. Al-Salti and E. Karimov, Initial and boundary value problems for fractional differential equations involving Atangana-Baleanu derivative, preprint, arXiv: 1706.00740. [8] N. A. Asif, Z. Hammouch, M. B. Riaz and H. Bulut, Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, The European Physical Journal Plus, 133 (2018), 272. doi: 10.1140/epjp/i2018-12098-6. [9] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 22 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [10] A. Atangana and I. Koca, On the new fractional derivative and application to nonlinear Baggs and Freedman model, Journal of Nonlinear Sciences and Applications, 9 (2016), 2467-2480.  doi: 10.22436/jnsa.009.05.46. [11] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order, Chaos, Solitons & Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012. [12] A. Atangana and B. S. T. Alkahtani, New model of groundwater flowing within a confine aquifer: Application of Caputo-Fabrizio derivative, Arabian Journal of Geosciences, 9 (2016), 3647-3654.  doi: 10.22436/jnsa.009.06.17. [13] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 1-13. [14] M. Caputo, Linear models of dissipation whose Q is almost frequency independent I, Geophysical Journal International, 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x. [15] J. F. Gómez-Aguilar, R. F. Escobar-Jiménez, M. G. López-López and V. M. Alvarado-Martínez, Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media, Journal of Electromagnetic Waves and Applications, 30 (2016), 1937-1952.  doi: 10.1080/09205071.2016.1225521. [16] J. F. Gómez-Aguilar, V. F. Morales-Delgado, M. A. Taneco-Hernández, D. Baleanu, R. F. Escobar-Jiménez and M. M. Al Qurashi, Analytical solutions of the electrical RLC circuit via Liouville aputo operators with local and non-local kernels, Entropy, 18 (2016), 402. doi: 10.3390/e18080402. [17] J. Hristov, Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey's kernel to the Caputo-Fabrizio time-fractional derivative, Thermal Science, 20 (2016), 757-762.  doi: 10.2298/TSCI160112019H. [18] J. Hristov, Steady-state heat conduction in a medium with spatial non-singular fading memory: derivation of Caputo-Fabrizio space-fractional derivative with Jeffrey's kernel and analytical solutions, Thermal Science, 21 (2017), 827-839.  doi: 10.2298/TSCI160229115H. [19] I. Koca and A. Atangana, Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, Thermal Science, 21 (2017), 2299-2305.  doi: 10.2298/TSCI160209103K. [20] V. F. Morales-Delgado, J. F. Gómez-Aguilar, H. Yépez-Martínez, D. Baleanu, R. F. Escobar-Jimenez and V. H. Olivares-Peregrino, Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Advances in Difference Equations, 2016 (2016), Paper No. 164, 17 pp. doi: 10.1186/s13662-016-0891-6. [21] Z. Odibat and S. Momani, The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers & Mathematics with Applications, 58 (2009), 2199-2208.  doi: 10.1016/j.camwa.2009.03.009. [22] N. A. Sheikh, F. Ali, M. Saqib, I. Khan, S. A. A. Jan, A. S. Alshomrani and M. S. Alghamdi, Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results in Physics, 7 (2017), 789-800.  doi: 10.1016/j.rinp.2017.01.025. [23] N. A. Sheikh, F. Ali, M. Saqib, I. Khan and S. A. A. Jan, A comparative study of Atangana-Baleanu and Caputo-Fabrizio fractional derivatives to the convective flow of a generalized Casson fluid, The European Physical Journal Plus, 132 (2017), 54. doi: 10.1140/epjp/i2017-11326-y. [24] J. Singh, D. Kumar, Z. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048. [25] M. Yavuz and N. Ozdemir, European vanilla option pricing model of fractional order without singular kernel, Fractal and Fractional, 2 (2018), 3. doi: 10.3390/fractalfract2010003. [26] M. Yavuz, N. Ozdemir and H. M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, 133 (2018), 215. doi: 10.1140/epjp/i2018-12051-9.

show all references

##### References:
 [1] T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098–1107, arXiv: 1607.00262. doi: 10.22436/jnsa.010.03.20. [2] B. S. T. Alkahtani and A. Atangana, Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order, Chaos, Solitons & Fractals, 89 (2016), 539-546.  doi: 10.1016/j.chaos.2016.03.012. [3] B. S. T. Alkahtani and A. Atangana, Analysis of non-homogeneous heat model with new trend of derivative with fractional order, Chaos, Solitons & Fractals, 89 (2016), 566-571.  doi: 10.1016/j.chaos.2016.03.027. [4] B. S. T. Alkahtani, Chua's circuit model with Atangana–Baleanu derivative with fractional order, Chaos, Solitons & Fractals, 89 (2016), 547-551. [5] O. J. J. Alkahtani, Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model, Chaos, Solitons & Fractals, 89 (2016), 552-559.  doi: 10.1016/j.chaos.2016.03.026. [6] R. T. Alqahtani, Atangana-Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer, Journal of Nonlinear Sciences and Applications, 9 (2016), 3647-3654.  doi: 10.22436/jnsa.009.06.17. [7] F. A. M. N. Al-Salti and E. Karimov, Initial and boundary value problems for fractional differential equations involving Atangana-Baleanu derivative, preprint, arXiv: 1706.00740. [8] N. A. Asif, Z. Hammouch, M. B. Riaz and H. Bulut, Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, The European Physical Journal Plus, 133 (2018), 272. doi: 10.1140/epjp/i2018-12098-6. [9] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 22 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [10] A. Atangana and I. Koca, On the new fractional derivative and application to nonlinear Baggs and Freedman model, Journal of Nonlinear Sciences and Applications, 9 (2016), 2467-2480.  doi: 10.22436/jnsa.009.05.46. [11] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order, Chaos, Solitons & Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012. [12] A. Atangana and B. S. T. Alkahtani, New model of groundwater flowing within a confine aquifer: Application of Caputo-Fabrizio derivative, Arabian Journal of Geosciences, 9 (2016), 3647-3654.  doi: 10.22436/jnsa.009.06.17. [13] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 1-13. [14] M. Caputo, Linear models of dissipation whose Q is almost frequency independent I, Geophysical Journal International, 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x. [15] J. F. Gómez-Aguilar, R. F. Escobar-Jiménez, M. G. López-López and V. M. Alvarado-Martínez, Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media, Journal of Electromagnetic Waves and Applications, 30 (2016), 1937-1952.  doi: 10.1080/09205071.2016.1225521. [16] J. F. Gómez-Aguilar, V. F. Morales-Delgado, M. A. Taneco-Hernández, D. Baleanu, R. F. Escobar-Jiménez and M. M. Al Qurashi, Analytical solutions of the electrical RLC circuit via Liouville aputo operators with local and non-local kernels, Entropy, 18 (2016), 402. doi: 10.3390/e18080402. [17] J. Hristov, Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey's kernel to the Caputo-Fabrizio time-fractional derivative, Thermal Science, 20 (2016), 757-762.  doi: 10.2298/TSCI160112019H. [18] J. Hristov, Steady-state heat conduction in a medium with spatial non-singular fading memory: derivation of Caputo-Fabrizio space-fractional derivative with Jeffrey's kernel and analytical solutions, Thermal Science, 21 (2017), 827-839.  doi: 10.2298/TSCI160229115H. [19] I. Koca and A. Atangana, Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, Thermal Science, 21 (2017), 2299-2305.  doi: 10.2298/TSCI160209103K. [20] V. F. Morales-Delgado, J. F. Gómez-Aguilar, H. Yépez-Martínez, D. Baleanu, R. F. Escobar-Jimenez and V. H. Olivares-Peregrino, Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Advances in Difference Equations, 2016 (2016), Paper No. 164, 17 pp. doi: 10.1186/s13662-016-0891-6. [21] Z. Odibat and S. Momani, The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers & Mathematics with Applications, 58 (2009), 2199-2208.  doi: 10.1016/j.camwa.2009.03.009. [22] N. A. Sheikh, F. Ali, M. Saqib, I. Khan, S. A. A. Jan, A. S. Alshomrani and M. S. Alghamdi, Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results in Physics, 7 (2017), 789-800.  doi: 10.1016/j.rinp.2017.01.025. [23] N. A. Sheikh, F. Ali, M. Saqib, I. Khan and S. A. A. Jan, A comparative study of Atangana-Baleanu and Caputo-Fabrizio fractional derivatives to the convective flow of a generalized Casson fluid, The European Physical Journal Plus, 132 (2017), 54. doi: 10.1140/epjp/i2017-11326-y. [24] J. Singh, D. Kumar, Z. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048. [25] M. Yavuz and N. Ozdemir, European vanilla option pricing model of fractional order without singular kernel, Fractal and Fractional, 2 (2018), 3. doi: 10.3390/fractalfract2010003. [26] M. Yavuz, N. Ozdemir and H. M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, 133 (2018), 215. doi: 10.1140/epjp/i2018-12051-9.
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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