An Especial Fractional Oscillator

We propose a peculiar fractional oscillator. By assuming that the motion takes place in a complex media where the level of fractionality is low, we find that the time rate of change of the energy of this system has an oscillatory behavior. 1. Introduction In complex media such as glasses, liquid crystals, polymers, and biopolymers, the dynamical variable of interest often obeys fractional differential equations [1–6]. For instance, the mean squared displacement of a Brownian particle is given by ; this linear dependence on time is referred to as normal diffusion. In complex media this kind of behavior is often violated, leading to anomalous diffusion. For a subdiffusive process , with . For this process, fractional dynamic equations emerge naturally in the physical concept of continuous time random walks [6, 7]. Fractional differential equations have many applications in applied science and engineering [8–11]. Fractional differential equations have been investigated in pure sciences, such as pure mathematics [12]. As a fractional generalization of the oscillation phenomena, one can consider [13] The case with corresponds to attenuated oscillation phenomenon [14–19]. With the initial conditions and , the solution is , the so-called Mittag-Leffler function. The case with corresponds to amplified oscillation phenomenon. With the initial conditions , , and , the solution is . In [14] it has been discarded on the ground that it signifies the instability of the system. In a recent study [11] it has been proven that the Mittag-Leffler function of this order is a Nussbaum function. Hence it may have applications in the control theory of electrical engineering. In another study [12] general equations of the type have been considered; however in this work the emphasis is on the existence and uniqueness of solutions. There are various representations of the Mittag-Leffler function:(i)series representation [13],(ii)integral representation [14],(iii)approximate representation for a medium with low level of fractionality [17, 19, 20]. In this work we use the approximate representation to probe fractional differential equations of order . The plan of this paper is as follows. In Section 2 we describe this approximate representation. In Section 3 we obtain the solution for (1) and we compare it with the exact results of [11, 14]. In Section 4 we propose a new fractional oscillator of the form where . We obtain the solution for this damped oscillator, and we discuss the time rate of change of this oscillator. Finally in Section 5 we present our conclusions. 2. The