Distributed-Order Dynamic Systems

Abstract

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Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed orders

“…In this paper, we use the following definition of distributed-order differential/integral operators [8]:…”

Chaos synchronization in fractional differential systems

“…But, in recent decades, this has changed. It was found that fractional calculus is useful, even powerful, for modelling viscoelasticity [4], electromagnetic waves [5], boundary layer effects in ducts [6], quantum evolution of complex systems [7], distributed-order dynamical systems [8] and others. That is, the fractional differential systems are more suitable to describe physical phenomena that have memory and genetic characteristics.…”

Fractional rheological models for thermomechanical systems. Dissipation and free energies

“…Materials whose constitutive equations can be described by a fractional derivative [5] are of increasing interest in recent years (see [30,29]). It is well known that materials with such properties can be considered in the class of materials with memory and may describe elastic, fluid and electromagnetic materials, but also other kinds of phenomena, such as heat flux models.…”

“…The creep function for such models also has power law form. Such experimental backing has motivated many studies of materials with fading memory given by a fractional derivative, including [6,22,4,14,17,30,23] and in the frequency domain [19,36]. Many experimental observations on a variety of materials subject to a constant load show plastic behavior, which can be described by the fractional derivative approach.…”