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

• 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.

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

 Citation: • • 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

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

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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Tables(3)

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