The selection and application of variable order differential operators
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The selection and application of variable order differential operators


This work demonstrates the practicality of using variable order (VO) derivative operators for modeling the dynamics of complex systems. First we review the various candidate VO integral and derivative operator definitions proposed in the literature. We select a definition that is appropriate for physical modeling based on the following criteria: the VO operator must be able to return all intermediate values between 0 and 1 that correspond to the argument of the order of differentiation in addition to the integer order derivatives, and the derivative of a true constant function should be 0. Then we apply the chosen operator to 3 different problems: a stationary analysis of viscoelastic oscillators, the formulation of a Lagrangian equation of motion for a sedimenting particle in a viscous fluid, and the development of a constitutive equation for viscoelastic materials. In the first problem we obtain an analytical solution for the order of the operator and connect the meaning of functional order to the dynamic properties of a viscoelastic oscillator. We replace the multi-term differential equation for the viscoelastic oscillator with a single-term VO equation. We determine that the order of differentiation for a single operator describing all dynamic elements in the stationary equation of motion (mass, damping and spring) is equal to the 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 (π), so that the normalization constant is simply 2/π. For the second problem we focus on the transient equation of motion for a spherical particle sedimenting in a quiescent viscous liquid. In particular, we examine the various force terms in the equation of motion and propose a new form for the history drag acting on the particle at finite Reynolds numbers. This new form equates the history drag to the VO derivative of the velocity of the particle. Using numerical results from a finite element simulation of the particle we solve for order of the derivative q and evaluate how the order changes over time. Based on these results we propose a simple form for q and obtain a correlation for the history drag acting on the particle that is in good agreement with the numerical data for terminal Reynolds numbers ranging from 2.5 to 20. In the final problem we present a simple constitutive equation for linear viscoelastic materials strained at constant strain rates. We propose a relationship in which the stress is related to the q(t) derivative of strain, where q(t) in this case is a function of normalized time. This order function is postulated to be proportional to the rate of change of disorder within the material. From a statistical mechanics based theory, we find that q(t) is proportional to tInt. Using experimental data for an epoxy resin and carbon/epoxy composite undergoing compression, we determine the final form for the constitutive equation that models the linear viscoelastic deformation in time. The resulting dimensionless constitutive equation agrees well all the normalized data.

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