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

Krunal B. Kachhia

doi:10.3934/dcdss.2020041

Article Contents

2020, Volume 13, Issue 3: 741-754. Doi: 10.3934/dcdss.2020041

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

Krunal B. Kachhia

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: December 2019

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Abstract

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.

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Mathematics Subject Classification: Primary: 34A08, 35R11, 26A33; Secondary: 65J15.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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