1.Introduction

The calculus of differentiation and integration is known as the fractional calculus. The fractional derivatives for the first time were proposed by Gottfried Wilhelm Leibniz in (1695) ¹. Now the fractional calculus has been considered as a new tool for modeling the complex systems ^2-16. Since then, the fractional derivative was examined for various functions. The fractional derivative of the exponential function and the power function, respectively, are obtained by Liouville in (1832) and Riemann in (1847) ¹. Many researchers consider an integral form for the fractional derivative and the two most popular types of fractional derivatives are Riemann-Louville and Caputo. The fractional derivative has many interesting and unexpected properties; for example, under special conditions, the derivative of a constant can be nonzero, such as the case of the Riemann-Liouville fractional derivative. On the other hand, the Caputo derivative of a constant, as the ordinary derivative, vanishes. For further information on fractional calculus, the interested reader is referred to Refs. ^17-21.

Different definitions of fractional derivatives can be proposed, each with remarkable properties ^22-26, all of them valid and mathematically acceptable.

In Ref. ³⁷, the authors proposed a new fractional differential equation to describe the mechanical oscillations of a simple system and they analyzed the systems mass-spring and spring-damper. In Ref. ³⁸, the authors proposed a fractional differential equation to describe the vertical motion of a body through the air. Two-dimensional projectile motion in a free and in a resistive medium were investigated using the so-called conformable derivative in Ref. ³⁹. The motion of

a projectile by using the Riemann-Liouville fractional derivative and the Caputo approach is studied in Refs. ^40,41.

Recently, using fractional calculus, the dynamics of a particle have been studied for resisted horizontal motion within a viscoelastic medium and in the presence of a uniform force ²². Moreover, in Ref. ²³, in the framework of conformable fractional quantum mechanics, the three-dimensional fractional harmonic oscillator is studied and by using an effective and efficient formalism, Schrödinger equation, probability density, probability flux and continuity equation have been investigated and in Ref. ²⁴. Fractional calculus has also been studied for the Dirac equation, the resulting wave function, and the energy eigenvalue equation.

There are different methods to solve fractional differential equations analytically. One of the most common, simple, and practical methods used is the Laplace transform ²⁵. In this paper, the Laplace transform of fractional operators is represented, and some related formula is introduced. Fractional calculus has been considered for modeling viscoelastic systems that cover various fields and subjects ²². Here, we show that the proposed fractional model has a better result as compared to that of the non-fractional models have shown for probing the different aspects of mechanical physics.

This work is organized as follows. We first review the fractional calculus in Sec. 2. Next, we investigate dynamics of a particle within a viscoelastic medium in Sec. 3. In Sec. 4, by considering a retarding force proportional to the fractional velocity, vertical motion of a body in a resisting medium is studied and in the last section, to provide a better understanding of the motion of a projectile in a resisting viscoelastic medium, we will discuss it under the condition that there exists a retarding force proportional to the fractional velocity.

2.Introduction to Fractional Calculus

The Riemann-Louville fractional integral is defined as

where 0<α<1 and f(x) is a continuous function. Also the Caputo fractional derivative is introduced as ²²

where n=[ν]+1 and [x] implies a Gauss symbol.

The Laplace transform of Caputo fractional derivative can be represented by the following form

and by inserting α = 1 and m = 2 in Eq (1), we have

The Mittag-Leffler functions and the generalized Mittag- Leffler functions for α´,β´ > 0 and z∈C are defined as ³⁴

For α',β'>0, a∈R and sα'>a inverse Laplace transform formula has the form

3.Resisted motion of a particle in a viscoelastic medium

Now let us investigate the dynamics of a particle in a viscoelastic medium. In reality, per cycle of motion the part of the energy is destroyed. In the other words, the measure of damping is determined by the amount of energy lost.

Experimentally, we can consider the horizontal motion in a viscoelastic medium as the simplest example of the resisted motion of a particle. By considering a general order of viscoelastic damping, the frictional force takes the following form

In order to be consistent with the time dimensionality, we consider, the fractional derivative operator as

where C₁ represents the fractional time in the system ³⁷. Then, in Eq. (8), we change C to (C/C11-α).

In this case, the Newtonian equation satisfies the equation of motion as follows

with the following initial conditions

On the other hand we know that Fs=Lxt, therefore we have

then by substituting Eq. (1) and Eq. (2) into Eq. (12), we find the following relation:

which, upon substitution of Eq. (11), becomes

where

Therefore, we can write

which can be compared with Eq. (5) to obtain the following parameters:

after which the solution for x(t) reads

In Fig. 1, x (t) with three different values of α as a function of t with parameters C = 0.8, C₁ = 1.2, m = 1 and V₀ = 10 has been plotted. Using the equation above, the velocity can be written as

Figure 1 Horizontal component of the position x(t) as a function for time t, given by Eq. (18). 

In Fig. 2, we have plotted V(t) with three different values of α as a function of t with parameters C = 0.9, C₁ = 1.2, m = 1 and V₀ = 10. Then, we have obtained the acceleration as

Figure 2 Velocity V (t) as a function of time t, given by Eq. (19). 

Special cases:1) For α=1/2

Recalling Eqs. (4) and (17) and substituting into Eq. (18), the obtained solution becomes

So, velocity and acceleration can be calculated as follows:

in view of other special approaches as follows 2) For C = 0, Eq. (21) leads

4.Vertical motion of a body in a resisting medium

Now let us consider the vertical motion of a body in a resisting medium in which there exists a retarding force proportional to the fractional velocity. In this case, we consider that the body is projected downward with zero initial velocity v(0) = 0 in a uniform gravitational field. Then the equation of motion is given by

with the following initial condition

Taking the Laplace transform of both side of the Eq. (25), we get

Solving the Eq. (27) with respect to f (s), we have

which can be rewritten as

Using the inverse Laplace transform yt=L-1F(s), we have

For the special case when α = 1, we obtain

which can be expanded in series as

so that in the limit of C → 0,

On the other hand, for α = 1/2 we will have

For simplicity above equation can be written as

where, after using Γ(3)=2!, the result is read as

5.Motion of a projectile in a resisting medium

In this section we are interested in considering motion of a projectile in a resisting viscoelastic medium in which there exists a retarding force proportional to the fractional velocity. In this case we have the following equations

with the initial conditions

Taking the Laplace transform of both on both sides of Eq. (39), we can find

where F(s) and G(s) are Laplace transforms of x(t) and y(t) respectively. Using the inverse Laplace transform and properties of Mittag-Leffler function, we have

In Fig. 3, we have plotted x(t) with three different values of α as a function of t with parameters C = 0.8, C₁ = 1.2, m = 1, θ=π/6 and V₀ = 10. Also, in Fig. 4, we have plotted 𝑦(t) with three different values of α as a function of t with parameters C = 0.8, C₁ = 1.2, m = 1, g=10, θ=π/6 and V₀ = 10.

Figure 3 Horizontal component of the position x(t) as a function of time t, given by Eq. (41). 

Figure 4 Vertical component of the position y(t) as a function of time t, given by Eq. (42). 

Differentiating x(t) and y(t) with respect to the time, the velocity can be calculated as

and

If we denote the range and the time required for the entire trajectory by R´ and T´ respectively, the following representation is obtained

Now consider the case that α = 1- ε and ε is sufficiently small. In this case we have

By using Eqs. (45) and (46), up to a first order in ε, we have

which, when α goes to 1, can be simplified into

The range is obtained from the relation R´= x(T´) as

2 which can be reduced, when α goes to 1, to we can also have

Therefore, the change due to the fractional resistance is given by

Thus, the range becomes larger for the fractional resistance when compared with the linear resistance case.

6.Conclusion

In this article, we have considered fractional calculus as a new tool in studying interesting aspects of classical mechanics. First, we have briefly discussed the basic concepts of fractional calculus and we have presented an interpretation of fractional derivative and solution of fractional equations analytically. Then, by considering the modeling of viscoelastic systems within the fractional calculus framework, we have investigated applications of this approach in three different problems in classical mechanics including the study of resisted motion of a particle in a viscoelastic medium, the vertical motion of a body in a resisting medium and the motion of a projectile in a resisting medium. The obtained results satisfy the ordinary results of classical mechanics in. It has also been proved that the ordinary solutions are obtained provided the fractional effects are removed. Thus, the results demonstrate that the proposed fractional model presents an enhanced description as compared to that of the non-fractional models have shown when probing the different aspects of mechanical physics.