## International Journal of Differential Equations

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

#### Abstract

We review the application of differential operators of noninteger order to the modeling of dynamic systems. We compare all the definitions of Variable Order (VO) operators recently proposed in literature and select the VO operator that has the desirable property of continuous transition between integer and non-integer order derivatives. We use the selected VO operator to connect the meaning of functional order to the dynamic properties of a viscoelastic oscillator. We conclude that the order of differentiation of a single VO operator that represents the dynamics of a viscoelastic oscillator in stationary motion is a normalized phase shift. The normalization constant is found by taking the difference between the order of the inertial term (2) and the order of the spring term (0) and dividing this difference by the angular phase shift between acceleration and position in radians ($\pi$), so that the normalization constant is simply $2/\pi$.

#### Article information

Source
Int. J. Differ. Equ., Volume 2010, Special Issue (2010), Article ID 846107, 16 pages.

Dates
Accepted: 8 October 2009
First available in Project Euclid: 26 January 2017

https://projecteuclid.org/euclid.ijde/1485399877

Digital Object Identifier
doi:10.1155/2010/846107

Mathematical Reviews number (MathSciNet)
MR2575291

Zentralblatt MATH identifier
1207.34011

#### Citation

Ramirez, Lynnette E. S.; Coimbra, Carlos F. M. On the Selection and Meaning of Variable Order Operators for Dynamic Modeling. Int. J. Differ. Equ. 2010, Special Issue (2010), Article ID 846107, 16 pages. doi:10.1155/2010/846107. https://projecteuclid.org/euclid.ijde/1485399877

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