*6.5. Application to Representing Generalized Mittag-Leffler Function (1)*

The previous research (Mainardi [25], Achar et al. [33], Uchaikin ([38], Chapter 7)) presented the free response to fractional oscillators of Class I type by using a kind of special function, called the generalized Mittag-Leffler function, see (32). The novelty of our result presented in Theorem 10 is in that Equation (172) or (173) is consistent with the representation style in engineering by using elementary functions. Thus, we obtain novel representations of the generalized Mittag-Leffler functions as follows.

**Corollary 13.** *The generalized Mittag-Leffler function in the form*

$$\mathbf{x}\_{1}(t) = \mathbf{x}\_{10}E\_{\mathbf{a},1}\left[-\left(\omega\_{n}t\right)^{\mathbf{a}}\right] + \nu\_{10}tE\_{\mathbf{a},2}\left[-\left(\omega\_{n}t\right)^{\mathbf{a}}\right], 1<\mathbf{a} \le \mathbf{2}, t \ge 0,\tag{187}$$

*is the solution to fractional oscillators in Class I (Mainardi [25], Achar et al. [33], Uchaikin ([38], Chapter 7)). It can be expressed by the one in (172). That is, for 1 < α* ≤ *2, t* ≥ *0,*

*Symmetry* **2018**, *10*, 40

$$\begin{split} x\_{1}(t) &= \mathbf{x}\_{10}E\_{a,1}\left[ -\left(\omega\_{n}t\right)^{a} \right] + \nu\_{10}tE\_{a,2}\left[ -\left(\omega\_{n}t\right)^{a} \right] \\ &= e^{-\frac{\omega n \sin\frac{\pi\pi}{4}}{2\left|\cos\frac{\pi\pi}{4}\right|}t} \\ &= e^{-\frac{\omega n \sin\frac{\pi\pi}{4}}{2\left|\cos\frac{\pi\pi}{4}\right|}t} \left[ \\ &\quad + \left[ \frac{\nu\_{10} + \frac{\omega^{2}\sin\frac{\pi\pi}{4}}{2}\left|\cos\frac{\pi\pi}{4}\right|}{\sqrt{1 - \frac{\omega^{2}\sin\frac{\pi\pi}{4}}{4\left|\cos\frac{\pi\pi}{4}\right|}}} \right] \end{split} \tag{188}$$

The proof of Corollary 13 is straightforward from (172). When *v*10= 0 in (187), we obtain a corollary below.

**Corollary 14.** *The generalized Mittag-Leffler function given by*

$$\mathbf{x}\_1(t) = \mathbf{x}\_{10} E\_{\mathbf{u},1} \left[ - \left( \omega\_\mathbf{u} t \right)^a \right], 1 < a \le 2, t \ge 0,\tag{189}$$

*can be expressed by the elementary functions, for 1 < α* ≤ *2, t* ≥ *0, in the form*

$$\mathbf{x}\_{1}(t) = \mathbf{x}\_{10}E\_{a,1}\left[-\left(\omega\_{\mathrm{ft}}t\right)^{a}\right] = e^{-\frac{\omega\sin\frac{\alpha\pi}{2}t}{2\left[\cos\frac{\alpha\pi}{2}\right]}t} \left[\begin{array}{c} \mathbf{x}\_{10}\cos\left(\frac{\omega\_{\mathrm{w}}}{\sqrt{\omega^{a-2}\left|\cos\frac{\alpha\pi}{2}\right|}}\sqrt{1-\frac{\omega^{a}\sin^{2}\frac{\alpha\pi}{2}}{4\omega\_{\mathrm{w}}^{2}\left|\cos\frac{\alpha\pi}{2}\right|}}t\right) \\\\ + \left(\frac{\frac{\omega^{a}\sin\frac{\alpha\pi}{2}}{2\omega\_{\mathrm{w}}\left|\cos\frac{\alpha\pi}{2}\right|}x\_{10}}{\sqrt{1-\frac{\omega^{a}\sin^{2}\frac{\alpha\pi}{2}}{4\omega\_{\mathrm{w}}^{2}\left|\cos\frac{\alpha\pi}{2}\right|}}t}\right)\sin\left(\frac{\omega\_{\mathrm{w}}\sqrt{1-\frac{\omega^{a}\sin^{2}\frac{\alpha\pi}{2}}{4\omega\_{\mathrm{w}}^{2}\left|\cos\frac{\alpha\pi}{2}\right|}}t}{\sqrt{\omega^{a-2}\left|\cos\frac{\alpha\pi}{2}\right|}}\right) \end{array}\right].\tag{190}$$

**Proof.** If *v*10 = 0 in (187), (188) becomes (189). The proof completes. 

If *x*10= 0 in (187), we obtain another corollary as follows.

**Corollary 15.** *The generalized Mittag-Leffler function expressed by*

$$\mathbf{x}\_1(t) = \upsilon\_{10} t E\_{\mathbf{a},2} \left[ - (\omega\_n t)^a \right], 1 < a \le 2, t \ge 0,\tag{191}$$

*can be represented, for 1 < α* ≤ *2, t* ≥ *0, by the elementary functions in the form*

$$\psi\_1(t) = \upsilon\_{10} t E\_{a,2} \left[ - \left( \omega\_n t \right)^a \right] = e^{-\frac{\omega \sin \frac{\omega t}{2}}{2 \left[ \cos \frac{\omega t}{2} \right]} t} \left[ \left( \frac{\upsilon\_{10}}{\sqrt{1 - \frac{\omega^a \sin^2 \frac{4 \pi}{2}}{4 \omega\_n^2 \left| \cos \frac{4 \pi}{2} \right|}}} \right) \sin \left( \frac{\omega\_n \sqrt{1 - \frac{\omega^a \sin^2 \frac{4 \pi}{2}}{4 \omega\_n^2 \left| \cos \frac{4 \pi}{2} \right|}}}{\sqrt{\omega^{a-2} \left| \cos \frac{4 \pi}{2} \right|}} t \right) \right]. \tag{192}$$

**Proof.** When *x*10 = 0 in (187), (188) becomes (192). The proof finishes. 

### **7. Impulse Responses to Three Classes of Fractional Oscillators**

In this section, we shall present the impulse responses to three classes of fractional oscillators using elementary functions.

In Section 4, we have proved that

$$H\_{yj}(\omega) = H\_{xj}(\omega), \; j = 1, 2, 3, \; j$$

where *Hyj*(*ω*) is the frequency response function solved directly from a *j*th fractional oscillator while *Hxj*(*ω*) is the one derived from its equivalent oscillator. Doing the inverse Fourier transform on the both sides above, therefore, we have

$$h\_{yj}(t) = h\_{xj}(t), \; j = \ 1, \; 2, \; 3,$$

where *hyj*(*t*) is the impulse response obtained directly from the *j*th fractional oscillator but *hxj*(*t*) is the one solved from its equivalent one. In that way, therefore, we may establish the theoretic foundation for representing the impulse responses to three classes of fractional oscillators by using elementary functions.

The main highlight presented in this section is to propose the impulse responses to three classes of fractional oscillators in the closed analytic form expressed by elementary functions. As a by product, we shall represent a certain generalized Mittag-Leffler functions using elementary functions.
