This paper shows that we can improve the performance of the auto-adjustable electric machines if a fractional dynamic is considered in the algorithm of the controlling order. This structure is particularly interested in the separate control of active and reactive power of the double-fed induction generator (DFIG) of wind power conversion chain. Fractional regulators are used in the regulation o...

We define the complex powers of a densely defined operator A whose resolvent exists in a suitable region of the complex plane. Generally, this region is strictly contained in an angle and there exists α ∈ [0,∞) such that the resolvent of A is bounded by O((1 + |λ|)α) there. We prove that for some particular choices of a fractional number b, the negative of the fractional power (−A)b is the c.i....

Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. For processes lacking such scaling the corresponding description may be given by distributed-order equations. In the present paper we consider different forms of distributed-order fractional kinetic equations and investigate the effects descri...

In this paper, we give the sufficient conditions for the boundedness of the (fractional) Hausdorff operators on the Lebesgue spaces with power weights. In some special cases, these conditions are the same and also necessary. As an application, we obtain a better lower bound of fractional Hardy operators on the Lebesgue spaces compared with a result of the paper [25]. Mathematics subject classif...

The absorption of compressional and shear waves in many viscoelastic solids has been experimentally shown to follow a frequency power law. It is now well established that this type of loss behavior can be modeled using fractional derivatives. However, previous fractional constitutive equations for viscoelastic media are based on temporal fractional derivatives. These operators are non-local in ...

The one-dimensional chain of coupled oscillators with long-range power-law interaction is considered. The equation of motion in the infrared limit are mapped onto the continuum equation with the Riesz fractional derivative of order α, when 0 < α < 2. The evolution of soliton-like and breather-like structures are obtained numerically and compared for both types of simulations: using the chain of...

In this paper, we develop the unbounded extension of the Hille–Phillips functional calculus for generators of bounded groups. Mathematical applications include the generalised Lévy–Khintchine formula for subordinate semigroups, the analyticity of semigroups generated by fractional powers of group generators, where the power is not an odd integer, and a shifted abstract Grünwald formula. We also...

It has been shown that anomalous relaxation in dielectrics can be described in terms of equations with fractional derivatives. The solutions of the resulting equation with fractional derivatives are expressed by the Mittag –Leffler function and the Fox function. The conditions of a change from the Debye relaxation to “slow” (anomalous) relaxation with a power time dependence have been examined ...