The analytical analysis of fractional order Fokker-Planck equations

Hassan Khan; Umar Farooq; Fairouz Tchier; Qasim Khan; Gurpreet Singh; Poom Kumam; Kanokwan Sitthithakerngkiet; Hassan Khan; Umar Farooq; Fairouz Tchier; Qasim Khan; Gurpreet Singh; Poom Kumam; Kanokwan Sitthithakerngkiet

doi:10.3934/math.2022665

AIMS Mathematics

2022, Volume 7, Issue 7: 11919-11941. doi: 10.3934/math.2022665

Previous Article Next Article

Research article Special Issues

The analytical analysis of fractional order Fokker-Planck equations

1.
Department of Mathematics, Abdul Wali khan University Mardan, Pakistan
2.
Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
3.
Mathematics Department, King Saudi University, Riyadh, Saudi Arabia
4.
School of Mathematical Sciences, Dublin City University, Ireland
5.
Theoretical and Computational Science (TaCS) Center, Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok, Thailand
6.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
7.
Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok (KMUTNB), 1518, Wongsawang, Bangsue, Bangkok 10800, Thailand

Received: 06 January 2022 Revised: 19 March 2022 Accepted: 30 March 2022 Published: 20 April 2022
MSC : 26A33, 34A08

In the current note, we broaden the utilization of a new and efficient analytical computational scheme, approximate analytical method for obtaining the solutions of fractional-order Fokker-Planck equations. The approximate solution is obtained by decomposition technique along with the property of Riemann-Liouuille fractional partial integral operator. The Caputo-Riemann operator property for fractional-order partial differential equations is calculated through the utilization of the provided initial source. This analytical scheme generates the series form solution which is fast convergent to the exact solutions. The obtained results have shown that the new technique for analytical solutions is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology.
Citation: Hassan Khan, Umar Farooq, Fairouz Tchier, Qasim Khan, Gurpreet Singh, Poom Kumam, Kanokwan Sitthithakerngkiet. The analytical analysis of fractional order Fokker-Planck equations[J]. AIMS Mathematics, 2022, 7(7): 11919-11941. doi: 10.3934/math.2022665

Related Papers:

Abstract

In the current note, we broaden the utilization of a new and efficient analytical computational scheme, approximate analytical method for obtaining the solutions of fractional-order Fokker-Planck equations. The approximate solution is obtained by decomposition technique along with the property of Riemann-Liouuille fractional partial integral operator. The Caputo-Riemann operator property for fractional-order partial differential equations is calculated through the utilization of the provided initial source. This analytical scheme generates the series form solution which is fast convergent to the exact solutions. The obtained results have shown that the new technique for analytical solutions is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology.

References

[1]	N. H. Dinh, S. J. Lee, J. Y. Kim, K. K. Choi, Study on seismic performance of a mold transformer through shaking table tests, Appl. Sci., 10 (2020), 361. https://doi.org/10.3390/app10010361 doi: 10.3390/app10010361
[2]	P. Zhou, W. Zhu, Function projective synchronization for fractional-order chaotic systems, Nonlinear Anal.- Real, 12 (2011), 811–816. https://doi.org/10.1016/j.nonrwa.2010.08.008 doi: 10.1016/j.nonrwa.2010.08.008
[3]	J. Singh, D. Kumar, D. Baleanu, On the analysis of fractional diabetes model with exponential law, Adv. Differ. Equ., 2018 (2018), 1–15. https://doi.org/10.1186/s13662-018-1680-1 doi: 10.1186/s13662-018-1680-1
[4]	M. Higazy, Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic, Chaos Soliton. Fract., 138 (2020), 110007. https://doi.org/10.1016/j.chaos.2020.110007 doi: 10.1016/j.chaos.2020.110007
[5]	F. Guo, H. Peng, B. Zou, R. Zhao, X. Liu, Localisation and segmentation of optic disc with the fractional-order Darwinian particle swarm optimisation algorithm, IET Image Process., 12 (2018), 1303–1312. https://doi.org/10.1049/iet-ipr.2017.1149 doi: 10.1049/iet-ipr.2017.1149
[6]	P. Veeresha, D. G. Prakasha, H. M. Baskonus, New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, Chaos: Interdiscip. J. Nonlinear Sci., 29 (2019), 013119. https://doi.org/10.1063/1.5074099 doi: 10.1063/1.5074099
[7]	K. A. Abro, Role of fractal-fractional derivative on ferromagnetic fluid via fractal Laplace transform: A first problem via fractal-fractional differential operator, Eur. J. Mech. B-Fluid., 85 (2021), 76–81. https://doi.org/10.1016/j.euromechflu.2020.09.002 doi: 10.1016/j.euromechflu.2020.09.002
[8]	A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59 (2010), 1326–1336. https://doi.org/10.1016/j.camwa.2009.07.006 doi: 10.1016/j.camwa.2009.07.006
[9]	Y. Ferdi, Some applications of fractional order calculus to design digital filters for biomedical signal processing, J. Mech. Med. Biol., 12 (2012), 1240008. https://doi.org/10.1142/S0219519412400088 doi: 10.1142/S0219519412400088
[10]	T. P. Stefaski, J. Gulgowski, Electromagnetic-based derivation of fractional-order circuit theory, Commun. Nonlinear Sci., 79 (2019), 104897.
[11]	D. G. Prakasha, P. Veeresha, H. M. Baskonus, Analysis of the dynamics of hepatitis E virus using the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus, 134 (2019), 1–11. https://doi.org/10.1140/epjp/i2019-12590-5 doi: 10.1140/epjp/i2019-12590-5
[12]	M. A. Khan, S. Ullah, M. Farooq, A new fractional model for tuberculosis with relapse via Atangana-Baleanu derivative, Chaos Soliton. Fract., 116 (2018), 227–238.
[13]	M. A. Khan, S. Ullah, K. O. Okosun, K. Shah, A fractional order pine wilt disease model with Caputo-Fabrizio derivative, Adv. Differ. Equ., 2018 (2018), 1–18. https://doi.org/10.1186/s13662-018-1868-4 doi: 10.1186/s13662-018-1868-4
[14]	C. A. Valentim, J. A. Rabi, S. A. David, Fractional mathematical oncology: On the potential of non-integer order calculus applied to interdisciplinary models, Biosystems, 2021, 104377. https://doi.org/10.1016/j.biosystems.2021.104377 doi: 10.1016/j.biosystems.2021.104377
[15]	K. Kavitha, V. Vijayakumar, R. Udhayakumar, K. S. Nisar, Results on the existence of Hilfer fractional neutral evolution equations with infinite delay via measures of noncompactness, Math. Meth. Appl. Sci., 44 (2021), 1438–1455. https://doi.org/10.1002/mma.6843 doi: 10.1002/mma.6843
[16]	V. Mehandiratta, M. Mehra, G. Leugering, Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge: A study of fractional calculus on metric graph, Netw. Heterog. Media, 2021. https://doi.org/10.3934/nhm.2021003 doi: 10.3934/nhm.2021003
[17]	V. E. Tarasov, V. V. Tarasova, Economic dynamics with memory: Fractional calculus approach, Walter de Gruyter GmbH and Co KG, 2021.
[18]	E. Hernández-Balaguera, Coulostatics in bioelectrochemistry: A physical interpretation of the electrode-tissue processes from the theory of fractional calculus, Chaos Soliton. Fract., 145 (2021), 110787.
[19]	O. Darrigol, Worlds of flow: A history of hydrodynamics from the Bernoullis to Prandtl, Oxford University Press, 2005.
[20]	H. Risken, Fokker-planck equation, Springer, Berlin, Heidelberg, 1996, 63–95.
[21]	F. Herau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal., 244 (2007), 95–118.
[22]	F. Urena, L. Gavete, A. G. Gomez, J. J. Benito, A. M. Vargas, Non-linear Fokker-Planck equation solved with generalized finite differences in 2D and 3D, Appl. Math. Comput., 368 (2020), 124801.
[23]	J. Liu, J. Zhang, X. Zhang, Semi-discretized numerical solution for time fractional convection diffusion equation by RBF-FD, Appl. Math. Lett., 128 (2022), 107880. https://doi.org/10.1016/j.aml.2021.107880 doi: 10.1016/j.aml.2021.107880
[24]	W. Li, Y. Pang, Application of Adomian decomposition method to nonlinear systems, Adv. Differ. Equ., 2020 (2020), 1–17. https://doi.org/10.1186/s13662-020-2529-y doi: 10.1186/s13662-020-2529-y
[25]	M. Tatari, M. Dehghan, M. Razzaghi, Application of the Adomian decomposition method for the Fokker-Planck equation, Math. Comput. Model., 45 (2007), 639–650. https://doi.org/10.1016/j.mcm.2006.07.010 doi: 10.1016/j.mcm.2006.07.010
[26]	R. S. Dubey, B. S. T. Alkahtani, A. Atangana, Analytical solution of space-time fractional Fokker-Planck equation by homotopy perturbation Sumudu transform method, Math. Probl. Eng., 2015. https://doi.org/10.1155/2015/780929 doi: 10.1155/2015/780929
[27]	L. Yan, Numerical solutions of fractional Fokker-Planck equations using iterative Laplace transform method, Abstr. Appl. Anal., 2013 (2013), Hindawi. https://doi.org/10.1155/2013/465160 doi: 10.1155/2013/465160
[28]	M. I. SYAM, On approximate solutions of Fokker-Planck equation by the modified residual power series method, J. Fract. Calc. Appl., 10 (2019), 225–232.
[29]	W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47 (2009), 204–226. https://doi.org/10.1137/080714130 doi: 10.1137/080714130
[30]	Z. Odibat, S. Momani, Numerical solution of Fokker-Planck equation with space-and time-fractional derivatives, Phys. Lett. A, 369 (2007), 349–358. https://doi.org/10.1016/j.physleta.2007.05.002 doi: 10.1016/j.physleta.2007.05.002
[31]	J. Singh, H. K. Jassim, D. Kumar, An efficient computational technique for local fractional Fokker-Planck equation, Physica A, 555 (2020), 124525. https://doi.org/10.1016/j.physa.2020.124525 doi: 10.1016/j.physa.2020.124525
[32]	C. Li, D. Qian, Y. Chen, On Riemann-Liouville and caputo derivatives, Discrete Dyn. Nat. Soc., 2011. https://doi.org/10.1155/2011/562494 doi: 10.1155/2011/562494
[33]	R. S. Dubey, B. S. T. Alkahtani, A. Atangana, Analytical solution of space-time fractional Fokker-Planck equation by homotopy perturbation Sumudu transform method, Math. Probl. Eng., 2015. https://doi.org/10.1155/2015/780929 doi: 10.1155/2015/780929
[34]	Z. Odibat, S. Momani, Numerical solution of Fokker-Planck equation with space-and time-fractional derivatives, Phys. Lett. A, 369 (2007), 349–358. https://doi.org/10.1016/j.physleta.2007.05.002 doi: 10.1016/j.physleta.2007.05.002
[35]	A. Khalouta, A. Kadem, A new approximate analytical method and its convergence for nonlinear time-fractional partial differential equations, Sci. Iran., 28 (2021), 3315–3323.
[36]	A. Yusuf, B. Acay, U. T. Mustapha, M. Inc, D. Baleanu, Mathematical modeling of pine wilt disease with Caputo fractional operator, Chaos Soliton. Fract., 143 (2021), 110569. https://doi.org/10.1016/j.chaos.2020.110569 doi: 10.1016/j.chaos.2020.110569
[37]	U. Farooq, H. Khan, F. Tchier, E. Hincal, D. Baleanu, H. B. Jebreen, New approximate analytical technique for the solution of time fractional fluid flow models, Adv. Differ. Equ., 2021 (2021), 1–20. https://doi.org/10.1186/s13662-021-03240-z doi: 10.1186/s13662-021-03240-z

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)