Fractional physical problems including wind-influenced projectile motion with Mittag-Leffler kernel

1.   Introduction

The reason of increasing popularity of fractional calculus is the natural appearance of its applications in diverse areas of applied sciences and engineering. Fractional differential equations (FDEs) involving real or complex order derivatives have proven to be a useful tool in modelling anomalous dynamics of various physical and biological processes. One of the most important tasks for fractional operators is to apply them to real world phenomena and to see the differences between them. Using real data, Diethelm in ^[1] has presented a Caputo fractional model for better understanding of the dynamics of a dengue fever outbreak. The authors in ^[2] have fractionalized a model in Caputo sense to get better dynamics of TB virus using real data whereas the dynamics of Ebola epidemic was better described in ^[3] using the Caputo fractional derivative. The fractional equivalent of various standard physics such as Schrödinger, frictional force, wave equation, harmonic oscillator, Dirac equation and projectile motion equation which all theoretical physics can be studied in via fractional calculus ^[4]. Chaotic systems, random walk problems, polymer material science and biophysics which are all applied physics and also be investigated by fractional calculus ^{[5,6,7,8,9,10,11,12,13]}. Furthermore, Bas et al., Yusuf et al. and Abdeljawad studied fractional derivatives in a different way in ^{[14,15,16,17,18]}.

In nature, numerous physical features possess an intrinsic fractional order types ^[19,20], for this reason fractional calculus has become so important instrument due to its efficiency in explaining real world phenomena more accurately. Fractional calculus stipulates a potent instrument for controlling memory and hereditary characteristics of several materials and processes ^[21]. This is one of the great significance of fractional calculus when compared with the ordinary calculus, whereas such control has no many effects. A lot of physical features have been analyzed via the concept of fractional calculus, with a very much amelioration over their integer order counterparts as well as more effective result when conferring with experimental data, i.e., in chemical, agricultural, biomedical, and also from physical perspectives such as Hamiltonian formulation and Lagrangian ^[22,23]. Owing to this impressive usefulness of fractional calculus, a number of fractional operators have been proposed to precisely model the memory effects dealing with variety of dynamical systems ^{[24,25,26,27]}. However, more works need to be done to properly expound such dynamical systems. Riemann-Liouville and Caputo are some of the well-known fractional derivatives with the limitations of kernels with the singular structure. It can be easily seen that these derivatives can not properly depict the whole of memory effect of a particular system. To expunge this deficit of kernel with singular nature, the authors in ^[28,29] introduced a fractional operator founded on the exponential and Mittag-Leffler functions, and thus, their definitions do not contain kernel of singular nature.

In light of the significance of the ABC fractional operator in engineering, science and in depicting the entire memory effect of the system, we feel motivated to investigate and analyze the projectile motion equations by means of ABC fractional derivative. Authors in ^{[30,31,32,33,34,35,36]} also studied fractional version of projectile motion. One of the most important problems in the field of physics is the projectile motion in a resistant medium. Furthermore, interesting structures of the projectile motion under wind effect are analyzed in ^[37].

It is known that the projectile motion has a movement with 2-dimension. Herein, we discuss such associated wind-influenced projectile motion. Projectile motion can be regarded as the movement of a launched object under gravitational force and symbols used are given by $g$ with $m/s^2$, mass $m$ with $kg$ and unheeding any of the corresponding external or resisting force $k$ with $s^{-1}$. Surmising that the particle begins at origin that is to say at $(x_0 = y_0 = 0\ m)$, containing initial velocity with an angle $\phi$, $\theta$ is the angle the wind makes with respect to the horizontal axis $x$, $U\ m/s$ is the wind speed, and modulus $\nu_0\ m/s$, then, in $xy$-plane one can present the classical equations of wind-influenced projectile motion ^[37] as

with the initial velocities of the projectile

The corresponding solutions of (1.1)-(1.2) and (1.1)-(1.3) can be presented as,

The manuscript has been prepared as follows: In section 2, essential definitions, properties and theorems associated with ABC fractional operator are presented. In section 3, $(n+\alpha)$th-order ABC operator is identified and its Laplace transform (LT) is proved for solving higher order linear initial value problems with Mittag-Leffler kernel when $\alpha\in [0, 1]$ and $n\in\mathbb{N}$. In section 4, we present some numerical results and compare obtained results with ABC and Caputo fractional derivatives for different values of $\alpha$ and $k$ friction coefficients, different wind effects and different masses in 3D illustration. Finally, some conclusion comments are given in section 5.

2.   Preliminaries

In here we provide some important definitions, properties and theorems that will serve as a tool to the main results of the manuscript.

Definition 2.1. ^[38] The Atangana-Baleanu left and right fractional derivatives in Caputo form involving Mittag-Leffler function are given by

where $f\in H^1(a, b), a < b, \alpha\in [0, 1]$ and $B(\alpha)$ is a normalization function that satisfy $B(\alpha) > 0$, $B(0) = B(1) = 1.$

Theorem 2.1. ^[38] ABC derivative has the following LT, $0 < \alpha\leq1$,

Definition 2.2. ^[39] Mittag-Leffler function is expressed as

Definition 2.3. ^[39] For $\alpha\in[n-1, n)$, the Caputo fractional derivative is defined as

Theorem 2.2. ^[40] The Laplace transform of Caputo fractional derivative is given by

Definition 2.4. ^[40] Let $f, g:[0, \infty)\rightarrow\mathbb{R}$ and their convolution can be expressed as

Property 2.1. ^[40] The LT has the following property,

Property 2.2. ^[40] The inverse LT of some specific functions as below:

ⅰ) $\mathcal{L}^{-1}\Bigg\{\frac{s^\alpha}{s(s^\alpha+a)}\Bigg\} = E_\alpha(-at^\alpha)$.

ⅱ) $\mathcal{L}^{-1}\Bigg\{\frac{a}{s(s^\alpha+a)}\Bigg\} = 1-E_\alpha(-at^\alpha)$.

ⅲ) $\mathcal{L}^{-1}\Bigg\{\frac{1}{(s^\alpha+a)}\Bigg\} = t^{\alpha-1}E_{\alpha, \alpha}(-at^\alpha)$.

3.   Main results

In this portion, the main findings and results of the paper such as definition of ABC fractional derivative with $(n+\alpha)th$-order and its corresponding Laplace transform, projectile motion in a resistant medium with ABC fractional derivative will be presented.

Definition 3.1. Let $0 < \alpha \le 1$, the definition of ABC fractional derivative with $(n+\alpha)$th-order is defined as following formula

Theorem 3.1. If f(t) satisfies equation (3.1), then the LT of (3.1) is as following equality,

Proof. Let $0 < \alpha \le 1$, taking the LT of both sides of (3.1) and performing necessary operations,

thus, last equation is obtained and this completes the proof.

3.1.   Wind-influenced fractional projectile motion with Mittag-Leffler kernel

There exist some resistances from the practical point of view whose effects can be modelled with fractional operators and a drag force. To this aim, we utilize ABC fractional operator to model projectile motion equations under wind effect. We consequently obtain some novel exact expressions for this equation.

At first, let us give the fractional form of ordinary derivative

where $K$ is a dimension ($s^{-1}$). Starting from this, we can give the fractional version of (1.1) in the ABC sense

with the initial conditions

Taking the LT of both sides of (3.5), we have

thus one can attain

and this yields

where $B(\alpha)$ is a normalization constant such that $B(0) = B(1) = 1$. If we use formula (3.2), we have

Performing similar operations to the equation (3.6), we have

Note that the results obtained above (1.4) can be found for all classical cases while the order $\alpha \rightarrow 1$.

Moreover one can obtain flight time $T$ from $y(T) = 0$,

We can calculate it for approximate value of Mittag-Leffler function

from here similarly range $R$ can be calculated from $x(T) = R$ for the approximate value of Mittag-Leffler function.

3.2.   Wind-influenced fractional projectile motion with singular kernel

Now, let's apply Caputo fractional operator to model projectile motion equations under wind effect.

Taking the LT of both sides of (3.15) and (3.16) with the initial conditions (3.7)-(3.8) by Theorem (2.2), and subsequently applying inverse LT we have vertical and horizontal displacements and velocities

Following the same procedure for wind-influenced projectile motion with ABC fractional derivative, we can find range and time of flight for the equations (3.15) and (3.16).

Note that, from the above obtained results, the results for classical cases represented in Eqs. (1.4) and (1.5) can be obtained for the projectile motion in resistive medium by taking limit $\alpha \rightarrow 1$.

4.   Numerical discussion

In the portion, we provide the physical features and performances of the underlying wind-influenced projectile equations involving ABC and Caputo fractional operators. We suppose $K = k$ and $B(\alpha) = 1$ in our results. We illustrate the governing wind-influenced projectile motion equation with fractional ABC derivative with the order $(\alpha+1)$ and under different $\alpha$ orders, different initial velocities, different air drags, different wind effects and different angles in 3-D figures. $\theta$ is the angle the wind makes with respect to the horizontal axis $x$, $U\ m/s$ is the wind speed, $U = 0$ shows no-wind position, and $t$ is the time (second).

5.   Conclusions

We have studied wind-influenced projectile motion via ABC fractional derivative and have presented projectile motion equations in wind influenced medium with the aid of Laplace transform of ABC fractional operator. Due to the advantages of ABC fractional derivative, we have used this fractional derivative for wind-influenced projectile motion with air drag. It is well-known that preserving the dimensionality in physical quantities has so significance and so it has been preserved in this study. Moreover, we have expressed $(n+\alpha)$th-order ABC fractional derivative and demonstrate its Laplace transform. The attained results via fractional operators are close with that of the classical versions. We illustrate wind-influenced projectile motion with ABC-fractional derivative under different wind effects, different angles, different orders, different velocities, different masses and air drags in 3-D figures.

From Figure 7 and Figure 5, it is evident that the fractional parameter $\alpha$ serves as the resisting parameter and resistivity of the medium varies inversely with the increasing values of the fractional parameter, so for small value of alpha height as well as the range of the projectile is least.

● Figure 1 shows the effect of tailwind on the projectile motion under different wind angles between $[0, \frac{\pi}{2}]$ and so the particle moves forward. We observe that if the wind effects with $\theta = \frac{\pi}{4}$, range will be maximum and if the wind effects with $\theta = \frac{\pi}{2}$, range will be minimum relative to the other wind effects and it will reach to maximum height.

● Figures 2 and 3 show the effect of headwind on the projectile motion under different wind angles between $(\frac{\pi}{2}, \frac{3\pi}{2})$. We observe that if the wind effects with $\theta = \frac{5\pi}{6}$, the particle moves backward, and max-height increases. If the wind effects with $\theta = \frac{7\pi}{6}$, the particle moves backward, and max-height decreases.

● Figure 4 shows the effect of headwind on the projectile motion under different wind angles between $(\frac{3\pi}{2}, 2\pi)$ and so the particle moves forward. We observe that if the wind effects with $\theta = \frac{33\pi}{18}$, range decreases and max-height decreases.

● Figure 5 shows the effect of headwind on the projectile motion under different fractional orders with $\theta = \frac{\pi}{3}$, so the particle moves forward and max-height increases as the order increases. If the wind angle was headwind, then it would move backward.

● Figure 6 shows the effect of headwind on the projectile motion under different launch angles. We observe that time of flight and max-height increase as the angle increases. The particle reaches max range in $\phi = \frac{\pi}{4}$ and reaches max-height in $\phi = \frac{\pi}{2}$.

● Figures 7, 8, 9 show the effect of headwind on the projectile motion under different orders, launch velocities and masses for vertical displacement.

● Figure 10 shows the effect of tailwind on the projectile motion under different wind speeds while $\theta = \frac{33\pi}{18}$, so the particle moves forward and max-height decreases as the velocity increases.

● Figure 11 shows the effect of headwind on the projectile motion under different wind speeds while $\theta = \frac{7\pi}{6}$, so the particle moves backward and max-height decreases as the velocity increases.

● Figure 12 shows the effect of headwind on the projectile motion under different wind speeds while $\theta = \frac{5\pi}{6}$, so the particle moves backward and max-height increases as the velocity increases.

● Figure 13 shows the effect of tailwind on the projectile motion under different wind speeds while $\theta = \frac{\pi}{6}$, so the particle moves forward and max-height increases as the velocity increases.

● Figure 14 shows the effect of mass on the projectile motion while $\theta = \frac{\pi}{6}, \phi = \frac{\pi}{4}$, and we observe that the particle moves forward, max-height increases as the mass increases.

● Figure 15 shows the effect of drag force on the projectile motion while $\theta = \frac{\pi}{6}, \phi = \frac{\pi}{4}$, and we observe that range and max-height increase as drag force decreases.

● Figures 16, 17, and 18 show the comparison of wind-influenced projectile motion with ABC, Caputo and classical cases while $\alpha \rightarrow 1$, and trajectories obtained by ABC and Caputo tend to converge to the trajectories by classical derivatives as $\alpha \rightarrow 1$.

Conflict of interest

The authors declare no conflict of interest in this paper.