On the Selection and Meaning of Variable Order Operators for Dynamic Modeling

Figure 1

Plots of the 𝑡 derivatives of sin(2𝜋𝑡). The points are the 𝛼=0,0.25,0.50,0.75, 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.

(a)

(b)

(c)