# American Institute of Mathematical Sciences

• Previous Article
Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative
• DCDS-S Home
• This Issue
• Next Article
Extension of triple Laplace transform for solving fractional differential equations

## Comparative study of fractional Fokker-Planck equations with various fractional derivative operators

 Department of Mathematical Sciences, P. D. Patel Institute of Applied Sciences, Charotar University of Science and Technology (CHARUSAT), Changa, Anand-388421, Gujarat, India

Received  June 2018 Revised  September 2018 Published  March 2019

This paper presents a comparative study of fractional Fokker-Planck equations with various fractional derivative operators such as Caputo fractional derivative, Atangana-Baleanu fractional derivative and conformable fractional derivative. The new iterative method has been successively applied for finding approximate analytical solutions of the fractional Fokker-Planck equations with various fractional derivative operators. This method gives an analytical solution in the form of a convergent series with easily computable components. The behavior of solutions and the effects of different values of fractional order are shown graphically for various fractional derivative operators. Some examples are given to show ability of the method for solving the fractional Fokker-Planck equations.

Citation: Krunal B. Kachhia. Comparative study of fractional Fokker-Planck equations with various fractional derivative operators. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020041
##### References:

show all references

##### References:
Behavior of $u(x,t)$ corresponding to the values $\alpha = 0.3$, $\alpha = 0.6$ and $\alpha = 0.9$ for $B(\alpha) = 1$ and $t = 5$ from left to right
Behavior of $u(x,t)$ corresponding to the values $\alpha = 0.5$ for Caputo fractional derivative, Atangana-Baleanu fractional derivative and conformable fractional derivative from left to right
Behavior of $u(x,t)$ corresponding to the values $\alpha = 0.3$, $\alpha = 0.6$ and $\alpha = 0.9$ for $B(\alpha) = 1$ and $t = 5$ from left to right
Behavior of $u(x,t)$ corresponding to the values $\alpha = 0.3$, $\alpha = 0.6$ and $\alpha = 0.9$ for $B(\alpha) = 1$ and $t = 5$ from left to right
Behavior of $u(x, t)$ corresponding to the values $(\alpha = 0.3, \beta = 0.8)$, $(\alpha = 0.7, \beta = 0.4)$ and $(\alpha = 0.9, \beta = 0.9)$ for $B(\alpha) = 1$ and $t = 5$ from left to right
Behavior of $u(x, t)$ corresponding to the values $\alpha = 0.5, \beta = 0.5$ for Caputo fractional derivative, Atangana-Baleanu fractional derivative and conformable fractional derivative from left to right
Comparison of $u(x,t)$ with different fractional differential operators at different values of $\alpha$ when $x = 2,t = 3$
 $\alpha$ Caputo derivative Atangana-Baleanu derivative conformable derivative 0.25 8.8128 6.6843 40.2414 0.5 11.9088 8.9088 20.9282 0.75 14.7781 12.6030 17.3163 1 17 17 17
 $\alpha$ Caputo derivative Atangana-Baleanu derivative conformable derivative 0.25 8.8128 6.6843 40.2414 0.5 11.9088 8.9088 20.9282 0.75 14.7781 12.6030 17.3163 1 17 17 17
Comparison of $u(x,t)$ with different fractional differential operators at different values of $\alpha$ when $x = 2,t = 3$
 $\alpha$ Caputo derivative Atangana-Baleanu derivative conformable derivative 0.25 17.6255 12.2796 80.4828 0.5 23.8176 15.8632 41.8564 0.75 29.5563 23.3458 34.6326 1 34 34 34
 $\alpha$ Caputo derivative Atangana-Baleanu derivative conformable derivative 0.25 17.6255 12.2796 80.4828 0.5 23.8176 15.8632 41.8564 0.75 29.5563 23.3458 34.6326 1 34 34 34
Comparison of $u(x,t)$ with different fractional differential operators at different values of $\alpha$ when $x = 2,t = 3$
 $\alpha,\beta$ Caputo derivative Atangana-Baleanu derivative conformable derivative $\alpha=0.9,\beta=0.2$ 5.1777 5.0846 5.2498 $\alpha=0.7,\beta=0.4$ 6.7396 6.0633 7.5260 $\alpha=0.5,\beta=0.6$ 8.1927 6.5332 11.8531 $\alpha=1,\beta=1$ 14.5 14.5 14.5
 $\alpha,\beta$ Caputo derivative Atangana-Baleanu derivative conformable derivative $\alpha=0.9,\beta=0.2$ 5.1777 5.0846 5.2498 $\alpha=0.7,\beta=0.4$ 6.7396 6.0633 7.5260 $\alpha=0.5,\beta=0.6$ 8.1927 6.5332 11.8531 $\alpha=1,\beta=1$ 14.5 14.5 14.5
 [1] 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, 2018, 0 (0) : 937-956. doi: 10.3934/dcdss.2020055 [2] 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 [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] 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, 2018, 0 (0) : 975-993. doi: 10.3934/dcdss.2020057 [5] Ruiyang Cai, Fudong Ge, Yangquan Chen, Chunhai Kou. Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019033 [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, 2018, 0 (0) : 377-387. doi: 10.3934/dcdss.2020021 [7] Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215 [8] Mehmet Yavuz, Necati Özdemir. Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 995-1006. doi: 10.3934/dcdss.2020058 [9] 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 [10] Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025 [11] Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 723-739. doi: 10.3934/dcdss.2020040 [12] Gary Lieberman. Oblique derivative problems for elliptic and parabolic equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2409-2444. doi: 10.3934/cpaa.2013.12.2409 [13] Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615 [14] Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037 [15] Gaohang Yu. A derivative-free method for solving large-scale nonlinear systems of equations. Journal of Industrial & Management Optimization, 2010, 6 (1) : 149-160. doi: 10.3934/jimo.2010.6.149 [16] Dong-Hui Li, Xiao-Lin Wang. A modified Fletcher-Reeves-Type derivative-free method for symmetric nonlinear equations. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 71-82. doi: 10.3934/naco.2011.1.71 [17] Fangfang Dong, Yunmei Chen. A fractional-order derivative based variational framework for image denoising. Inverse Problems & Imaging, 2016, 10 (1) : 27-50. doi: 10.3934/ipi.2016.10.27 [18] Ekta Mittal, Sunil Joshi. Note on a $k$-generalised fractional derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 797-804. doi: 10.3934/dcdss.2020045 [19] A. M. Bagirov, Moumita Ghosh, Dean Webb. A derivative-free method for linearly constrained nonsmooth optimization. Journal of Industrial & Management Optimization, 2006, 2 (3) : 319-338. doi: 10.3934/jimo.2006.2.319 [20] Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030

2018 Impact Factor: 0.545