^{1, 2}

^{1}

^{2}

The periodic solution of fractional oscillation equation with periodic input is
considered in this work. The fractional derivative operator is taken as

Fractional calculus has been used in the mathematical description of real problems arising in different fields of science. It covers the fields of viscoelasticity, anomalous diffusion, analysis of feedback amplifiers, capacitor theory, fractances, generalized voltage dividers, electrode-electrolyte interface models, fractional multipoles, fitting of experimental data, and so on [

The fractional differential and integral operators have been extensively applied to the field of viscoelasticity [

The theorem of existence and uniqueness of solutions for fractional differential equations has been presented in [

Fractional oscillators and fractional dynamical systems were investigated in [

Let

Let

It is well-known that the fractional oscillation equation

In this work, we consider the fractional oscillation equation with periodic input using the fractional derivative operator

For a classic undamped oscillation with the periodic input

If

If

In the next section, as a comparison we solve the fractional oscillation equation using the fractional derivative operator

We consider the periodic problem of linear fractional differential system. For nonlinear fractional differential system, the problem is more challenging and some contributions have been made. For example, Li and Ma [

In this section, we solve the fractional oscillation equation with the periodic input and initial conditions

In Figure

Solid line:

In Figure

Solid line:

The Mittag-Leffler functions in (

None of the three Mittag-Leffler functions in (

If

In this case, calculating the convolution

But for the fractional case,

We note that it is possible to obtain exact periodic solutions in impulsive fractional-order dynamical systems by choosing the correct impulses at the right moments of time [

We consider the fractional oscillation equation using the fractional derivative operator

We use the following Fourier transform and its inverse:

We rewrite the right hand side of (

Obviously, (

The curves of

Curves of

Curves of

The effects of the order

Curves of

Curves of

Similar to a damped oscillation with a periodic input in an integer-ordered case, we observe that the curves of

Curve of

The maximum amplitude is calculated to be

Surface of amplitude

The fractional oscillation equations with a harmonic periodic input are considered for the fractional derivative operators

This work was supported by the National Natural Science Foundation of China (no. 11201308) and the Innovation Program of Shanghai Municipal Education Commission (no. 14ZZ161).