846107.fig.001a
(a)
846107.fig.001b
(b)
846107.fig.001c
(c)
Figure 1: Plots of the 𝑡 derivatives of s i n ( 2 𝜋 𝑡 ) . The points are the 𝛼 = 0 , 0 . 2 5 , 0 . 5 0 , 0 . 7 5 , and 1st-order fractional derivatives at 𝑡 = 𝛼 . (a) Definition (2.13) with 𝑞 ( 𝑡 , 𝜎 ) = 𝑞 ( 𝑡 ) (thin line), 𝑞 ( 𝑡 , 𝜎 ) = 𝑞 ( 𝜎 ) (medium line), and 𝑞 ( 𝑡 , 𝜎 ) = 𝑞 ( 𝑡 𝜎 ) (thick line). Note that none of the definitions match the 1st-order derivative at 𝑡 = 1 . (b) Caputo-type VO operators (2.14) with 𝑞 ( 𝑡 , 𝜎 ) = 𝑞 ( 𝑡 ) (thin line), 𝑞 ( 𝑡 , 𝜎 ) = 𝑞 ( 𝜎 ) (medium line), and 𝑞 ( 𝑡 , 𝜎 ) = 𝑞 ( 𝑡 𝜎 ) (thick line). Note that the variant of (2.14) with argument 𝑡 matches the corresponding 𝛼 fractional derivatives at 𝑡 = 𝛼 . (c) Definition (2.11) (thick line) and Coimbra's operator (2.15) (thin line). The 𝑡 -derivative defined by (2.15) is equivalent to the corresponding fractional derivatives at all the points.