⇦Chinese Physics Letters, 2017, Vol. 34, No. 2, Article code 020201 A New Fractional Model for the Falling Body Problem A. Ebaid1**, B. Masaedeh1, E. El-Zahar2,3 Affiliations 1Department of Mathematics, Faculty of Science, Tabuk University, P. O. Box 741, Tabuk 71491, Kingdom of Saudi Arabia 2Department of Mathematics, Faculty of Sciences and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj, 11942, Kingdom of Saudi Arabia 3Department of Basic Engineering Science, Faculty of Engineering, Shebin El-Kom, 32511, Menofia University, Egypt Received 5 October 2016 **Corresponding author. Email: aebaid@ut.edu.sa; halimgamil@yahoo.com Citation Text: Ebaid A, Masaedeh B and El-Zahar E 2017 Chin. Phys. Lett. 34 020201 Abstract Recently, a conformable fractional derivative has been proposed to calculate the derivative of non-integer order of time functions. It has been shown that this new fractional derivative definition obeys many advantages over the preceding definitions. For mathematical models in applied sciences and to preserve the dimensionality of the physical quantities, an auxiliary parameter ($\sigma$) which has the dimension of seconds should be implemented in the fractional derivative definition. We obtain analytic solutions for the resulting conformable fractional differential equations describing the vertical velocity and the height of the falling body. It is shown that the dimensions of velocity and height are always correct without any restrictions on the auxiliary parameter $\sigma$ which contradicts with the results in the literature when applying the Caputo definition to the same problem. This may open the door for many future works either to describe the role of such an auxiliary parameter or to derive a more suitable definition for the fractional derivative. DOI:10.1088/0256-307X/34/2/020201 PACS:02.30.Hq, 46.15.-x, 47.10.A- © 2017 Chinese Physics Society Article Text As a well known fact, the classical calculus was first introduced by Leibniz and Newton where derivatives of integer orders were clearly defined. However, the concept of the derivative of a non-integer order was introduced by L'Hôpital's letter to Leibniz in 1695 asking him about the derivative of order 1/2. The first application of fractional calculus was presented by Abel[1,2] in 1826 to solve an integral equation of the tautochrone problem. Since then, several definitions for the fractional derivative have been published.[3-7] Various applications of the fractional calculus in the applied sciences have been discussed by many researchers.[8-12] Recently, a new definition for the fractional derivative has been proposed by Khalil et al.[13] This new definition was applied to easily solve the time fractional heat equation.[14] A generalization of the results obtained by Khalil et al.[13] have been recently analyzed by Katugampola.[15] To fractionalize a mathematical model varying with time in Newtonian mechanics, an auxiliary parameter $\sigma$ should be imposed into the fractional derivative definition as follows:[16,17] $$ \frac{d}{dt}\rightarrow\frac{1}{\sigma^{1-\alpha}}\frac{d^{\alpha}}{dt^{\alpha}}, ~\alpha\in(0,1],~~ \tag {1} $$ where $\sigma$ has the dimension of seconds. Hence, the right-hand side of Eq. (1) is of dimension s$^{-1}$. The non-local time $\sigma$ is often called the cosmic time.[16] The fractional derivative $\frac{d^{\alpha}}{dt^{\alpha}}$ can then be replaced by any one of the well-known definitions of the fractional derivatives. In Refs. [16,17], the fractional derivative $\frac{d^{\alpha}}{dt^{\alpha}}$ was replaced with the Caputo definition and then applied to the falling body problem. The obtained results by Ref. [17] have led to a restriction on $\sigma$ to keep the dimensionality correct for the solution of the vertical velocity and distance. The objective of this work is then to show that such a restriction on $\sigma$ can be avoided by using the conformable fractional derivative.[13] Khalil et al.[13] obtained the following formula for the $\alpha$-derivative of a function $f(t)$, $$ \frac{d^{\alpha}}{dt^{\alpha}}f(t)=t^{n+1-\alpha}\frac{d^{n+1}}{dt^{n+1}}f(t), ~\alpha\in(n,n+1],~~ \tag {2} $$ where $f(t)$ is $(n + 1)$-differentiable at $t>0$. The expression in Eq. (2) has been also derived via a generalized form of the definition suggested in Ref. [13]. Therefore, for $\alpha\in(0,1]$ we have $$ \frac{d^{\alpha}}{dt^{\alpha}}f(t)=t^{1-\alpha}\frac{d}{dt}f(t).~~ \tag {3} $$ Substituting Eq. (3) into Eq. (1) yields $$ \frac{d}{dt}\rightarrow\frac{t^{1-\alpha}}{\sigma^{1-\alpha}} \frac{d}{dt},~\alpha\in(0,1].~~ \tag {4} $$ This result will be used in the current work to analyze the motion of a falling body in a resisting medium. It will be shown that the analytic solutions for the vertical velocity and distance reduce to the classical ones as $\alpha\rightarrow 1$. Moreover, the correctness of dimensionality of the physical quantities is actually guaranteed by Eqs. (4) and (8). Consider the motion of an object of mass $m$ falling through the air from a height $h$ in the Earth's gravitational field with initial velocity $v_0$. The classical equation of motion for the particle is given by[17] $$ m\frac{dv}{dt}=-mg-mkv,~~ \tag {5} $$ where $k$ is a positive constant and its dimensionality is the inverse of seconds, i.e., $[k]=$s$^{-1}$. The initial conditions are given as $$ v(0)=v_0,~ z(0)=h,~~ \tag {6} $$ where $z(t)$ is the vertical distance of the particle at arbitrary time $t$ and $\frac{dz(t)}{dt}=v(t)$. On applying Eq. (4) to the velocity $v$ of the particle, it then follows $$ \frac{dv}{dt}\rightarrow\frac{t^{1-\alpha}}{\sigma^{1-\alpha}}\frac{dv}{dt}.~~ \tag {7} $$ Therefore, Eq. (5) becomes $$ m\frac{t^{1-\alpha}}{\sigma^{1-\alpha}}\frac{dv}{dt}=-mg-mkv,~~ \tag {8} $$ which can be rewritten in the form of $$ \frac{dv}{dt}+(k\sigma^{1-\alpha}t^{-1+\alpha})v=-(g\sigma^{1-\alpha})t^{-1+\alpha}.~~ \tag {9} $$ The analytic solution of this first-order linear ordinary differential equation can be obtained as follows: $$ v(t)=-\frac{g}{k}+c e^{-{\it \Omega} t^{\alpha}},~~ \tag {10} $$ where $c$ is an integration constant and ${\it \Omega}$ is defined as $$ {\it \Omega}=\frac{k\sigma^{1-\alpha}}{\alpha}.~~ \tag {11} $$ Applying the first initial conditions in Eq. (6) leads to $$ c=v_0+\frac{g}{k}.~~ \tag {12} $$ Hence, the vertical velocity is given as $$ v(t)=-\frac{g}{k}+\Big(v_0+\frac{g}{k}\Big)e^{-{\it \Omega} t^{\alpha}}.~~ \tag {13} $$ On implementing Eq. (4) once more for $z(t)$, Eq. (13) becomes $$ \frac{t^{1-\alpha}}{\sigma^{1-\alpha}}\frac{dz}{dt}= -\frac{g}{k}+\Big(v_0+\frac{g}{k}\Big)e^{-{\it \Omega} t^{\alpha}},~~ \tag {14} $$ which can be simplified to $$ \frac{dz}{dt}=-At^{-1+\alpha}+Bt^{-1+\alpha}e^{-{\it \Omega} t^{\alpha}},~~ \tag {15} $$ where $A$ and $B$ are defined by $$ A=\frac{g\sigma^{1-\alpha}}{k},~ B=\sigma^{1-\alpha}\Big(v_0+\frac{g}{k}\Big).~~ \tag {16} $$ On solving the differential Eq. (15) according to the second initial condition in Eq. (6), we finally obtain $$ z(t)=h-\frac{A}{\alpha}t^{\alpha}+\frac{B}{\alpha{\it \Omega}} (1-e^{-{\it \Omega} t^{\alpha}}).~~ \tag {17} $$ The analytic solutions obtained by Eqs. (13) and (17) should be reduced to the corresponding solutions in the classical Newtonian mechanics when $\alpha\rightarrow 1$. In addition, if the acceleration due to gravity is measured in $ms^{-2}$, then the fractional vertical velocity in Eq. (13) must have dimension $ms^{-1}$ and the fractional height in Eq. (17) must have dimension $m$. These issues are addressed in the following. Let us first investigate the solutions (13) and (17) as $\alpha\rightarrow 1$. In this case we obtain ${\it \Omega}\rightarrow k$ from Eq. (11). Therefore, Eq. (13) reduces to $$ v(t)=-\frac{g}{k}+\Big(v_0+\frac{g}{k}\Big)e^{-k t},~~ \tag {18} $$ which is the analytic expression for velocity in the case of the classical Newtonian mechanics.[17] Moreover, the constants $A$ and $B$ in Eq. (16) become $$ A\rightarrow\frac{g}{k},~ B\rightarrow v_0+\frac{g}{k}.~~ \tag {19} $$ Hence, the fractional vertical distance (17) reduces to $$ z(t)=h-\frac{gt}{k}+\frac{1}{k}\Big(v_0+\frac{g}{k}\Big)(1-e^{-k t}),~~ \tag {20} $$ which is also the analytic expression for the vertical distance in the classical Newtonian mechanics.[17] Regarding the dimensions of the fractional forms of $v(t)$ and $z(t)$ in Eqs. (13) and (17), respectively, it must be the first time to specify the dimensions of the constants ${\it \Omega}$, $A$, and $B$ as indicated by $$\begin{align} [{\it \Omega}]=\,&s^{-\alpha},\\ [A]=\,&ms^{-\alpha},\\ [B]=\,&ms^{-\alpha}.~~ \tag {21} \end{align} $$ By this, it should be noted that ${\it \Omega} t^{\alpha}$ is a dimensionless quantity, i.e., a scalar quantity for $$ [{\it \Omega} t^{\alpha}]=s^{-\alpha}\times s^{\alpha}=1.~~ \tag {22} $$ Then $v(t)$ in Eq. (13) always has dimension $ms^{-1}$ for all positive values of the auxiliary parameter $\sigma$ without any restrictions on this parameter. To clarify this point, let us recall the analytic solution in Eq. (17) in terms of the Mittag–Leffler function as $$ v(t)=-\frac{g}{k}+\Big(v_0+\frac{g}{k}\Big)E_{\alpha}(-k\sigma^{1-\alpha}t^{\alpha}).~~ \tag {23} $$ To keep the dimensionality of $v(t)$ in Eq. (23), the magnitude $k\sigma^{1-\alpha}t^{\alpha}$ should be a scalar quantity. This can be achieved by setting the following restriction[17] $$ k\sigma=\alpha,~~ \tag {24} $$ and this leads to $$ 0 < \sigma\le\frac{1}{k}.~~ \tag {25} $$ Our point of view can be further confirmed by discussing the dimensions of each term in the solution $z(t)$ given by Eq. (17) as follows: $$\begin{align} [h]=\,&m,\\ \Big[\frac{A}{\alpha}t^{\alpha}\Big]=\,&ms^{-\alpha}\times s^{\alpha}=m,\\ \Big[\frac{B}{\alpha{\it \Omega}}\Big]=\,&\frac{ms^{-\alpha}}{s^{-\alpha}}=m,~~ \tag {26} \end{align} $$ where $ (1-e^{-{\it \Omega} t^{\alpha}})$ is a scalar by Eq. (22). Accordingly, the current vertical distance solution $z(t)$ has dimension $m$ whatever the value of $\sigma$. However, the corresponding solution in Eq. (17) has dimension $m$ under the restriction Eq. (24). More details about this issue can be summarized as follows. The relation (24) seems also to be needed to keep parameter $\sigma$, involved in the definition of conformable fractional derivative, irrelevant to the obtained results. However, we believe that the relation (24) is not acceptable for the following reasons. First, this relation is not unique, where it can be generalized and replaced by setting $k\sigma=r\alpha$ providing that $r$ is a real number. Accordingly, a question arises here: what is the value of $r$ that should be selected in this case? Secondly, it was also observed from Eq. (24) that the auxiliary parameter $\sigma$ is related to the parameter $k$ in the case of a resistant medium. Here a further question arises: what is the corresponding relation in the case of no resistance? Such a relation cannot be found. In addition, it is clear from Eq. (24) that in the absence of the resistance, i.e., $k\rightarrow 0$, then $\alpha\rightarrow 0$ and therefore the definition Eq. (1) fails. The above conclusion can be confirmed by studying the falling body problem in a vacuum, where there are no external parameters. In the literature, only Refs. [17,18] were available on studying the falling body problem in a resistant medium by using the fractional calculus. However, in Ref. [18] the authors replaced the integer derivative by a fractional one on a purely mathematical or heuristic basis, where the auxiliary parameter $\sigma$ was ignored. This was not completely correct from the physical and engineering points of view because the physical parameters contained in the differential equation should not have the dimensionality measured in the laboratory.[17] Therefore, Ref. [17] is the only available reference in which the auxiliary parameter $\sigma$ was used to fractionalize the falling body problem. By this, we have compared our results with only those of Ref. [17]. In summary, the conformable fractional derivative has been applied to solve the falling body problem. An auxiliary parameter $\sigma$ was implemented to formulate the fractional derivatives of time functions. The analytic solutions for the fractional vertical velocity $v(t)$ and fractional height $z(t)$ have been obtained. In addition, the correctness of dimensionality of the fractional vertical velocity and the fractional height is actually guaranteed by Eqs. (4) and (8). Hence, we have to search for a new role for the auxiliary parameter $\sigma$ when applying the conformable fractional derivative on scientific problems of a time nature. References INVESTIGATION OF THE STABILITY OF SOLUTIONS OF PARTIAL DIFFERENTIAL-DIFFERENCE EQUATIONS WITH LEADING TERMComplex and higher order fractional curl operator in electromagneticsOn the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motionAnalysis of projectile motion in view of fractional calculusA new definition of fractional derivativeA falling body problem through the air in view of the fractional derivative approach
[1]Miller K S and Ross B 1993 An Introduction to the Fractional Calculus and Fractional Differential Equations (New York: John Wiley Sons)
[2]Machado J T, Kiryakova V and Mainardi F 2011 Commun. Nonlin. Sci. Num. Simul. 16 1140
[3]Riemann B 1953 Versuch Einer Allgemeinen Auffassung der Integration und Differentiation (New York: Dover)
[4] Letnikov A V 1967 Sbornik: Math. 3 1 (in Russian)
[5]Weyl H 1917 Naturforschende Gesellschaft Zürich 62 296
[6]Riesz M 1949 Acta Math. 81 1
[7]Riesz M 1939 Bull. La Société Mathématique France 67 153
[8] Naqvi Q A and Abbas M 2004 Opt. Commun. 241 349
[9] Jumarie G 2005 Appl. Math. Lett. 18 817
[10] Ebaid A 2011 Appl. Math. Modell. 35 1231
[11]Ebaid A, El-Sayed D M M and Aljoufi M D 2012 Appl. Math. Sci. 6 4075
[12]Hardy G H 2013 J. Lond. Math. Soc. I 20 48
[13] Khalil R, Al Horani M, Yousef A and Sababheh M 2014 J. Comput. Appl. Math. 264 65
[14]Çenesiz Y and Kurt 2015 Recent Advances in Applied Mathematics, Modelling and Simulation (Florence: WSEAS press) pp 195–199
[15]Katugampola U N 2014 arXiv:1410.6535v2
[16]Gómez-Aguilara G F, Rosales-García J J and Bernal-Alvarado J J 2012 Rev. Mex. Fís. 58 348
[17]Juan R G J, Guia C M, Martinez O J and Baleanu D 2013 Proceed. Rom. Acad. Ser. A 14 42
[18] Kwok S F 2005 Physica A 350 199