On the fractional model of Fokker-Planck equations with two different operator

Zeliha Korpinar; Mustafa Inc; Dumitru Baleanu; Zeliha Korpinar; Mustafa Inc; Dumitru Baleanu

doi:10.3934/math.2020015

AIMS Mathematics

2020, Volume 5, Issue 1: 236-248. doi: 10.3934/math.2020015

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On the fractional model of Fokker-Planck equations with two different operator

1.
MusAlparslan University, Faculty of Economic and Administrative Sciences, Department of Administration, 49250, Muş/Turkiye
2.
Fırat University, Science Faculty, Department of Mathematics, 23119 Elazığ/Turkiye
3.
Department of Mathematics, CankayaUniversity, 06530 Balgat, Ankara, Turkey
4.
Institute of Space Sciences, Magurele-Bucharest, Romania

Received: 06 August 2019 Accepted: 09 October 2019 Published: 05 November 2019
MSC : 35C08, 76M60

In this paper, the fractional model of Fokker-Planck equations are solved by using Laplace homotopy analysis method (LHAM). LHAM is expressed with a combining of Laplace transform and homotopy methods to obtain a new analytical series solutions of the fractional partial differential equations (FPDEs) in the Caputo-Fabrizio and Liouville-Caputo sense. Here obtained solutions are compared with exact solutions of these equations. The suitability of the method is removed from the plotted graphs. The obtained consequens explain that technique is a power and efficient process in investigation of solutions for fractional model of Fokker-Planck equations.
- Laplace homotopy analysis method,
- fractional model of Fokker-Planck equations,
- Caputo-Fabrizio derivative,
- series solution
Citation: Zeliha Korpinar, Mustafa Inc, Dumitru Baleanu. On the fractional model of Fokker-Planck equations with two different operator[J]. AIMS Mathematics, 2020, 5(1): 236-248. doi: 10.3934/math.2020015

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Abstract

In this paper, the fractional model of Fokker-Planck equations are solved by using Laplace homotopy analysis method (LHAM). LHAM is expressed with a combining of Laplace transform and homotopy methods to obtain a new analytical series solutions of the fractional partial differential equations (FPDEs) in the Caputo-Fabrizio and Liouville-Caputo sense. Here obtained solutions are compared with exact solutions of these equations. The suitability of the method is removed from the plotted graphs. The obtained consequens explain that technique is a power and efficient process in investigation of solutions for fractional model of Fokker-Planck equations.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[2]	I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999.
[3]	J. Sabatier, O. P. Agrawal, J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007.
[4]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Switzerland, 1993.
[5]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.
[6]	J. Losada, J. J. Nieto, Properties of the new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87-92.
[7]	A. Atangana, B. S. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), 4439-4453.
[8]	A. Atangana, S. Qureshi, Modeling attractors of chaotic dynamical systems with fractal-fractional operators, Chaos, Solitons and Fractals, 123 (2019), 320-337. doi: 10.1016/j.chaos.2019.04.020
[9]	F. Tchier, M. Inc, Z. S. Korpinar, et al. Solution of the time fractional reaction-diffusion equations with residual power series method, Adv. Mech. Eng., 8 (2016), 1-10.
[10]	A. I. Aliyu, M. Inc, A. Yusuf, et al. A fractional model of vertical transmission and cure of vector-borne diseases pertaining to the Atangana-Baleanu fractional derivatives, Chaos, Solitons and Fractals, 116 (2018), 268-277.
[11]	R. T. Alqahtani, Fixed-point theorem for Caputo-Fabrizio fractional Nagumo equation with nonlinear diffusion and convection, J. Nonlinear Sci. Appl., 9 (2016), 1991-1999. doi: 10.22436/jnsa.009.05.05
[12]	M. Inc, Z. S. Korpinar, M. M. Al Qurashi, et al. Anew method for approximate solution of some nonlinear equations: Residual power series method, Adv. Mech. Eng., 8 (2016), 1-7.
[13]	Z. Korpinar, On numerical solutions for the Caputo-Fabrizio fractional heat-like equation, Therm. Sci., 22 (2018), S87-S95.
[14]	M. S. Mohamed, K. A. Gepreel, F. A. Al-Malki, et al. Approximate solutions of the generalized Abel's integral equations using the extension Khan's homotopy analysis transformation method, J. Appl. Math., 2015 (2015), 357861.
[15]	V. G. Gupta, P. Kumar, Approximate solutions of fractional linear and nonlinear differential equations using Laplace homotopy analysis method, Int. J. Nonlinear Sci., 19 (2015), 113-120.
[16]	M. Dehghan, J. Manafian, A. Saadatmandi, The solution of the linear fractional partial differential equations using the homotopy analysis method, Zeitschrift für Naturforschung-A, 65 (2010), 935-949.
[17]	H. Xu, J. Cang, Analysis of a time fractional wave-like equation with the homotopy analysis method, Phys. Lett. A, 372 (2008), 1250-1255. doi: 10.1016/j.physleta.2007.09.039
[18]	H. Jafari, S. Das, H. Tajadodi, Solving a multi-order fractional differential equation using homotopy analysis method, Journal of King Saud University-Science, 23 (2011), 151-155. doi: 10.1016/j.jksus.2010.06.023
[19]	K. Diethelm, N. J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154 (2004), 621-640.
[20]	E. F. Goufo, M. K. P.Doungmo, J. N. Mwambakana, Duplication in a model of rock fracture with fractional derivative without singular kernel, Open Math., 13 (2015), 839-846.
[21]	V. F. Morales-Delgado, J. F. Gómez-Aguilar, H. Yépez-Martnez, et al. Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Adv. Differ. Equ-NY, 2016 (2016), 164.
[22]	A. Prakash, H. Kaur, Numerical solution for fractional model of Fokker-Planck equation by using q-HATM, Chaos, Solitons and Fractals, 105 (2017), 99-110. doi: 10.1016/j.chaos.2017.10.003
[23]	M. Magdziarz, A. Weron, K.Weron, Fractional Fokker-Planck dynamics: stochastic representation and computer simulation, Phys. Rev. E, 75 (2007), 016708.
[24]	M. Magdziarz, A.Weron, Competition between subdiffusion and Levy flights: a Monte Carlo approach, Phys. Rev. E, 75 (2007), 056702.
[25]	M. Magdziarz, Stochastic representation of subdiffusion processes with time-dependent drift, Stoch. Proc. Appl., 119 (2009), 3238-3252. doi: 10.1016/j.spa.2009.05.006
[26]	L. Yan, Numerical solutions of fractional Fokker-Planck equations using iterative Laplace transform method, Abstr. Appl. Anal., 2013 (2013), 1-7.
[27]	A. Yildirim, Analytical approach to Fokker-Planck equation with space-and time-fractional derivatives by homotopy perturbation method, Journal of King Saud University-Science, 22 (2010), 257-264. doi: 10.1016/j.jksus.2010.05.008
[28]	S. Kumar, Numerical computation of time-fractional Fokker- Planck equation arising in solid state physics and circuit theory, Zeitschrift fur Naturforschung, 68 (2013), 777-784. doi: 10.5560/zna.2013-0057
[29]	Z. Odibat, S. Momani, Numerical solution of Fokker-Planck equation with space and time-fractional derivatives, Phys. Lett. A, 369 (2007), 349-358. doi: 10.1016/j.physleta.2007.05.002
[30]	G. Harrison, Numerical solution of the Fokker-Planck equation using moving finite elements, Numer. Meth. Part. D. E., 4 (1988), 219-232. doi: 10.1002/num.1690040305
[31]	J. Yao, A. Kumar, S. Kumar, A fractional model to describe the Brownian motion of particles and its analytical solution, Adv. Mech. Eng., 7 (2015), 1-11.
[32]	T. Körpinar, R. C. Demirkol, Z. Körpinar, Soliton propagation of electromagnetic field vectors of polarized light ray traveling in a coiled optical fiber in Minkowski space with Bishop equations, Eur. Phys. J. D, 73 (2019), 203.
[33]	T. Körpinar, R. C. Demirkol, Z. Körpinar, Soliton propagation of electromagnetic field vectors of polarized light ray traveling in a coiled optical fiber in the ordinary space, Int. J. Geom. Methods M., 16 (2019), 1950117.
[34]	T. Körpınar, R. C. Demirkol, Electromagnetic curves of the linearly polarized light wave along an optical fiber in a 3D semi-Riemannian manifold, J. Mod. Optic., 66 (2019), 857-867. doi: 10.1080/09500340.2019.1579930

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