Projectile motion via Riemann-Liouville calculus
- Bashir Ahmad^{1}Email author,
- Hanan Batarfi^{1},
- Juan J Nieto^{1, 2},
- Óscar Otero-Zarraquiños^{2} and
- Wafa Shammakh^{1}
https://doi.org/10.1186/s13662-015-0400-3
© Ahmad et al.; licensee Springer. 2015
Received: 24 November 2014
Accepted: 2 February 2015
Published: 26 February 2015
Abstract
We present an analysis of projectile motion in view of fractional calculus. We obtain the solution for the problem using the Riemann-Liouville derivative, and then we compute some features of projectile motion in the framework of Riemann-Liouville fractional calculus. We compare the solutions using Caputo derivatives and Riemann-Liouville derivatives.
Keywords
MSC
1 Analysis of projectile motion in view of Riemann-Liouville fractional calculus
1.1 Introduction
In this paper we consider a projectile motion in view of Riemann-Liouville fractional calculus. The projectile motion is one of the simplest problems whose analogs are ubiquitous in physics. The purpose of this paper is to extend the Caputo approach of [1] to the Riemann-Liouville case. We obtain some new formulas, in particular, the trajectory using the Riemann-Liouville fractional derivative is different. We compare both approaches and indicate new directions of research.
The fractional calculus is an extension of the ordinary calculus and has a history of over 300 years old. It represents a generalization of the ordinary differentiation and integration to arbitrary order and fractional calculus has applications in various fields, e.g. physics, engineering or biology [2–4]. Differential equations of fractional order have assumed a relevant role in the most diverse areas of science and engineering. Some physical considerations in favor of the use of fractional models are given in [5] and fractional mechanics is presented in [6, 7].
Many times the authors replace the usual integer derivative by another derivative of fractional order. However, from the physical point of view that is not totally correct [8] and some dimensional correction in the new equation is necessary; for example, substituting a first order derivative \(D^{1} :=\frac{d}{dt}\) by \(\frac{1}{{\sigma }^{1-\alpha}}D^{\alpha}\) where σ has an appropriate dimension [9].
For some new directions in fractional calculus and fractional differential equations we refer the reader for example to [10–13].
1.2 Definitions and preliminaries
We recall some definitions of fractional calculus.
Let \(\alpha>0\), \(n-1<\alpha<n\), \(n\in\mathbb{N}\), \(n\ge1\).
1.2.1 Caputo fractional derivative
Consider the space \(AC^{n}[0,T]\) of functions with absolutely continuous derivatives up to order \(n-1\) and with absolutely continuous n-derivative.
1.2.2 Riemann-Liouville fractional derivative
Also \(D^{\alpha}t^{\alpha-j}=0\) for \(j=1,2,\ldots,[\alpha]+1\).
1.2.3 Mittag-Leffler function
Thus for \(0<\alpha<1\), the general solution of \({}^{c}D^{\alpha}f=0\) is a constant.
This is a crucial difference since \(t^{\alpha-1}\) has a singularity at \(t=0^{+}\).
1.3 Classical problem formulation of projectile motion
The range is the horizontal distance traveled by the projectile from the time it is fired until it lands. The maximum altitude is the height of the highest point in the trajectory. The time of flight is the amount of time the projectile spends in the air between when it is fired and when it lands.
1.4 Caputo fractional problem formulation
For \(\alpha=2\) we recover, of course, the classical case (1.9).
Physically, we can interpret the fractional derivatives of x and y, respectively, as the accelerations of the projectile in the horizontal and vertical directions, which reduce to the acceleration of the classical mechanics at \(\alpha\to2^{-}\).
1.5 Riemann-Liouville fractional problem formulation
Therefore \(D^{\alpha-1}x(t)=c_{1} \Gamma(\alpha)\).
We point out that the solution \(x(t)\), \(y(t)\) given by (1.16), (1.17) is qualitatively different from the solution (1.12). With this Riemann-Liouville approach we get new fractional trajectories.
1.6 Features of projectile motion in the fractional calculus
As we have recalled before, three quantities are particularly relevant for identifying, distinguishing, and analyzing trajectories in our setting: the range, the maximum altitude, and the time of flight.
We recall each of these quantities using the Caputo derivative [1] and compute them for the Riemann-Liouville derivative.
1.6.1 Trajectory
1.6.2 Range
The fractional projectile range is defined as the value of x at the impact point.
As \(\alpha\to2^{-}\), from (1.25) we obtain the range of the classical projectile (1.24).
1.6.3 Flight time
The fractional time of flight \(t_{\mathrm{F}\text{-flight}}\) is defined as the value of t at which the projectile hits the ground.
Again, it is verified that the classical flight time \(t_{\mathrm{C}\text{-flight}}\) can be obtained from (1.28) at \(\alpha\to2^{-}\) to get (1.27).
1.6.4 Maximum height
2 Relationship between the ranges
In this section we present two main results: one of them comparing the range of a projectile using the fractional calculus by the Caputo derivative with the range in the classical case (Theorem 1 of [1]), and the other comparing again the classical range with the fractional range, but using this time the new formulas obtained for the Riemann-Liouville derivative.
Theorem 2.1
([1])
- 1.
\(R_{\mathrm{F}\text{-}\mathrm{C}}=R_{\mathrm{C}}\) if \(\mu=1\), i.e., \(v_{0}\sin\phi=[\frac {2^{\alpha-1}}{\Gamma(\alpha+1)}]^{\frac{1}{2-\alpha}}g\).
- 2.
\(R_{\mathrm{F}\text{-}\mathrm{C}}>R_{\mathrm{C}}\) if \(\mu>1\), i.e., \(v_{0}\sin\phi>[\frac {2^{\alpha-1}}{\Gamma(\alpha+1)}]^{\frac{1}{2-\alpha}}g\).
- 3.
\(R_{\mathrm{F}\text{-}\mathrm{C}}< R_{\mathrm{C}}\) if \(\mu<1\), i.e., \(v_{0}\sin\phi<[\frac {2^{\alpha-1}}{\Gamma(\alpha+1)}]^{\frac{1}{2-\alpha}}g\).
We now give the corresponding result for the Riemann-Liouville range without proof, since it is similar to that of Theorem 2.1.
Theorem 2.2
- 1.
\(R_{\mathrm{F}\text{-}\mathrm{RL}}=R_{\mathrm{C}}\) if \(\upsilon=1\), i.e., \(v_{0}\sin\phi= (\frac{2\Gamma(\alpha)}{\alpha^{\alpha-1}} )^{\frac{1}{\alpha -2}}\frac{1}{g}\).
- 2.
\(R_{\mathrm{F}\text{-}\mathrm{RL}}>R_{\mathrm{C}}\) if \(\upsilon>1\), i.e., \(v_{0}\sin\phi> (\frac{2\Gamma(\alpha)}{\alpha^{\alpha-1}} )^{\frac{1}{\alpha -2}}\frac{1}{g}\).
- 3.
\(R_{\mathrm{F}\text{-}\mathrm{RL}}< R_{\mathrm{C}}\) if \(\upsilon<1\), i.e., \(v_{0}\sin\phi< (\frac{2\Gamma(\alpha)}{\alpha^{\alpha-1}} )^{\frac{1}{\alpha -2}}\frac{1}{g}\).
3 Maximum projectile range
In applications, the maximum projectile range and the required optimal projection angle are of considerable interest (e.g. in situations for which the projectile serves as a delivery system [1, 14, 15]). In order to maximize \(R_{\mathrm{F}}\), it is necessary to optimize the projection angle ϕ. This is developed below.
Theorem 3.1
(see [1], Theorem 3)
We now present a new result on the optimal angle with the Riemann-Liouville approach.
Theorem 3.2
Proof
On the other hand, \(1<\alpha\le2^{-}\) implies that \(1\le\frac{1}{\sqrt {\alpha-1}}<\infty\), i.e., \(\frac{\pi}{4}\le\phi_{\mathrm{max}}<\frac {\pi}{2}\).
When in this equation we let \(\alpha\to2^{-}\), we obtain \(\phi _{\mathrm{max}}=\frac{\pi}{4}\), which is the optimal projection angle in the classical case.
We get from it the expression of the classical maximum range \(R_{\mathrm{C}\text{-}\mathrm{max}}=\frac{v_{0}^{2}}{g}\) as \(\alpha\to2^{-}\). □
4 Conclusions
We have studied the motion of a projectile using the Riemann-Liouville fractional derivative. We have compared the trajectory, range, flight time, maximum height, maximum projectile range, and optimal angle with the results obtained previously for the fractional Caputo derivative.
Some relevant qualitative differences between the Caputo and the Riemann-Liouville approach are indicated. For example, the maximum projectile range is increasing with the order of derivative for the Caputo approach and, by contrast, it is decreasing with the order of the derivative for our Riemann-Liouville approach.
In the future we suggest to study the motion of a projectile in a resistant medium via the fractional calculus approach.
Declarations
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (88-130-35-HiCi). The authors, therefore, acknowledge technical and financial support of KAU.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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