Abstract
In this paper, wave propagation is considered in a medium described by a fractional-order model, which is formulated with the use of the two-sided fractional derivative of Ortigueira and Machado. Although the relation of the derivative to causality is clearly specified in its definition, there is no obvious relation between causality of the derivative and causality of the transfer function induced by this derivative. Hence, causality of the system is investigated; its output is an electromagnetic signal propagating in media described by the time-domain two-sided fractional derivative. It is demonstrated that, for the derivative order in the range [1,+∞), the transfer function describing attenuated signal propagation is not causal for any value of the asymmetry parameter of the derivative. On the other hand, it is shown that, for derivative orders in the range (0,1), the transfer function is causal if and only if the asymmetry parameter is equal to certain specific values corresponding to the left-sided Grünwald–Letnikov derivative. The results are illustrated by numerical simulations and analyses. Some comments on the Kramers–Krönig relations for logarithm of the transfer function are presented as well.
Highlights
Fractional-order (FO) Maxwell’s equations [1,2,3,4] represent a generalization of classical electromagnetism with the use of FO derivatives, which provides new interesting solutions constituting intermediate cases between the ones already existing in physics
Fractal Fract. 2021, 5, 10 the FO derivative should satisfy the semigroup property and the trigonometric functions invariance [9], we are able to demonstrate that causal solutions to this problem are obtained only for the derivative parameters corresponding to the left-sided Grünwald–Letnikov fractional derivative
Signal propagation is analysed in terms of causality for the media described by FO model (FOM), based on the two-sided Ortigueira–Machado derivative
Summary
Fractional-order (FO) Maxwell’s equations [1,2,3,4] represent a generalization of classical electromagnetism with the use of FO derivatives, which provides new interesting solutions constituting intermediate cases between the ones already existing in physics. Several attempts have been made [6,7,8,9], it is not clear which definition of the FO derivative should be used in electrical sciences. In this paper, we employ a very general definition of the FO derivative, i.e., the two-sided Ortigueira–. Machado derivative [6,11], which unites the ideas of forward and backward differentiations, and employs two parameters, i.e., the derivative order and the asymmetry parameter. This definition of the FO derivative covers the cases of the left- and right-sided
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