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March  2020, 13(3): 741-754. doi: 10.3934/dcdss.2020041

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 and Continuous Dynamical Systems - S, 2020, 13 (3) : 741-754. doi: 10.3934/dcdss.2020041
References:
[1]

T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016.

[2]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.

[3]

D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 444-462.  doi: 10.1016/j.cnsns.2017.12.003.

[4]

S. Bhalekar and V. Daftardar-Gejji, Convergence of the new iterative method, International Journal of Differential Equations, 2011 (2011), ArticleID 989065, 10 pages. doi: 10.1155/2011/989065.

[5]

M. Caputo, Elastic Dissipazione, ZaniChelli, Bologana, 1969.

[6]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85. 

[7]

V. Daftardar- Gejji and H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl., 316 (2006), 753-763.  doi: 10.1016/j.jmaa.2005.05.009.

[8]

R. S. Dubey, B. T. Alkhatani and A. Atangana, Analytic solution of Space-time fractional Fokker-Planck equation by homotopy perturbation sumudu transform method, Mathematical Problems in Engineering, 2015 (2015), Article ID 780929, 7 pages. doi: 10.1155/2015/780929.

[9]

I. S. Jesus and J. A. T Machado, Fractional control of heat diffusion systems, Nonlinear Dynamics, 54 (2008), 263-282.  doi: 10.1007/s11071-007-9322-2.

[10]

R. KhalilM. Al HoraniA. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.

[11]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[12]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Interoduction to Mathematical Models, World Scientific Publishing, 2010. doi: 10.1142/9781848163300.

[13]

G. M. Mittag-Leffler, Sur la nouvelle der fonction Eα(x), C.R. Acad. Sci. Paris (Ser.Ⅱ), 137 (1903), 554-558. 

[14]

Z. Odibat and S. Momani, Numerical solution of Fokker Planck equation with space- and time-fractional derivatives, Physics Letters A, 369 (2007), 349-358.  doi: 10.1016/j.physleta.2007.05.002.

[15] I. J. Podulbuny, Fractional Differential Equations, Academic Press, New York, 1999. 
[16]

A. Prakash and 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.

[17]

L. Yan, Numerical solution of fractional Fokker-Planck equations using iterative Laplace transform method, Abstract and Applied Analysis, 2013 (2013), Article ID 465160, 7 pages. doi: 10.1155/2013/465160.

show all references

References:
[1]

T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016.

[2]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.

[3]

D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 444-462.  doi: 10.1016/j.cnsns.2017.12.003.

[4]

S. Bhalekar and V. Daftardar-Gejji, Convergence of the new iterative method, International Journal of Differential Equations, 2011 (2011), ArticleID 989065, 10 pages. doi: 10.1155/2011/989065.

[5]

M. Caputo, Elastic Dissipazione, ZaniChelli, Bologana, 1969.

[6]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85. 

[7]

V. Daftardar- Gejji and H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl., 316 (2006), 753-763.  doi: 10.1016/j.jmaa.2005.05.009.

[8]

R. S. Dubey, B. T. Alkhatani and A. Atangana, Analytic solution of Space-time fractional Fokker-Planck equation by homotopy perturbation sumudu transform method, Mathematical Problems in Engineering, 2015 (2015), Article ID 780929, 7 pages. doi: 10.1155/2015/780929.

[9]

I. S. Jesus and J. A. T Machado, Fractional control of heat diffusion systems, Nonlinear Dynamics, 54 (2008), 263-282.  doi: 10.1007/s11071-007-9322-2.

[10]

R. KhalilM. Al HoraniA. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.

[11]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[12]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Interoduction to Mathematical Models, World Scientific Publishing, 2010. doi: 10.1142/9781848163300.

[13]

G. M. Mittag-Leffler, Sur la nouvelle der fonction Eα(x), C.R. Acad. Sci. Paris (Ser.Ⅱ), 137 (1903), 554-558. 

[14]

Z. Odibat and S. Momani, Numerical solution of Fokker Planck equation with space- and time-fractional derivatives, Physics Letters A, 369 (2007), 349-358.  doi: 10.1016/j.physleta.2007.05.002.

[15] I. J. Podulbuny, Fractional Differential Equations, Academic Press, New York, 1999. 
[16]

A. Prakash and 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.

[17]

L. Yan, Numerical solution of fractional Fokker-Planck equations using iterative Laplace transform method, Abstract and Applied Analysis, 2013 (2013), Article ID 465160, 7 pages. doi: 10.1155/2013/465160.

Figure 1.  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
Figure 2.  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
Figure 3.  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
Figure 4.  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
Figure 5.  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
Figure 6.  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
Table 1.  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
Table 2.  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
Table 3.  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
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