On the Recursive Fractional Variable-Order Derivative: Equivalent Switching Strategy, Duality, and Analog Modeling

Dominik Sierociuk,Michal Macias,Wiktor Malesza

doi:10.1007/s00034-014-9895-1

Dominik Sierociuk, Michal Macias + Show 1 more

Open Access

https://doi.org/10.1007/s00034-014-9895-1

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Abstract

In this paper, a switching strategy for recursive fractional variable-order derivative is proposed. This strategy can be interpreted as an explanation of order switching mechanism for this particular type of derivative. Additionally, important properties of variable fractional order derivatives, required for prove the main result, are introduced both in a difference equation and a matrix form. Duality between the recursive and standard variable-order derivative is detailed derived. Based on the switching scheme, an analog realization of the recursive variable-order derivative definition is presented. Experimental results obtained for the analog realization are compared to the numerical results.

Highlights

Introduction to Fractional CalculusFractional calculus is a generalization of traditional integer order integration and differentiation actions onto non-integer order
In order to prove the above stated problem, we present below the required background from fractional calculus and introduce necessary properties of recursive fractional variable-order derivative
The experimental results compared to numerical implementation of the 4th type of variable-order derivative definition are presented in Figs. 18 and 19

Summary

Introduction

Fractional calculus is a generalization of traditional integer order integration and differentiation actions onto non-integer order. The idea of such a generalization has been mentioned in 1695 by Leibniz and L’Hospital. In the end of 19th century, Liouville and Riemann introduced first definition of fractional derivative. Only just in late 60’ of the 20th century, this idea drew attention of engineers.

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