Chin. Phys. Lett.  2017, Vol. 34 Issue (2): 020201    DOI: 10.1088/0256-307X/34/2/020201
 GENERAL |
A New Fractional Model for the Falling Body Problem
A. Ebaid1**, B. Masaedeh1, E. El-Zahar2,3
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
Cite this article:
A. Ebaid, B. Masaedeh, E. El-Zahar 2017 Chin. Phys. Lett. 34 020201
 Download: PDF(278KB)   PDF(mobile)(277KB)   HTML Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks
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.
Received: 05 October 2016      Published: 25 January 2017
 PACS: 02.30.Hq (Ordinary differential equations) 46.15.-x (Computational methods in continuum mechanics) 47.10.A- (Mathematical formulations)
 TRENDMD: URL: http://cpl.iphy.ac.cn/10.1088/0256-307X/34/2/020201       OR      http://cpl.iphy.ac.cn/Y2017/V34/I2/020201
 Service E-mail this article E-mail Alert RSS Articles by authors A. Ebaid B. Masaedeh E. El-Zahar
 [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
Related articles from Frontiers Journals
 [1] SHEN Ming, ZHAO Liang. Oscillating Quintom Model with Time Periodic Varying Deceleration Parameter[J]. Chin. Phys. Lett., 2014, 31(1): 020201 [2] TIAN Rui-Lan, WU Qi-Liang, LIU Zhong-Jia, and YANG Xin-Wei. Dynamic Analysis of the Smooth-and-Discontinuous Oscillator under Constant Excitation[J]. Chin. Phys. Lett., 2012, 29(8): 020201 [3] K. Fakhar, A. H. Kara. The Reduction of Chazy Classes and Other Third-Order Differential Equations Related to Boundary Layer Flow Models[J]. Chin. Phys. Lett., 2012, 29(6): 020201 [4] LI Xian-Feng**, Andrew Y. -T. Leung, CHU Yan-Dong. Symmetry and Period-Adding Windows in a Modified Optical Injection Semiconductor Laser Model[J]. Chin. Phys. Lett., 2012, 29(1): 020201 [5] XIA Li-Li . Poisson Theory and Inverse Problem in a Controllable Mechanical System[J]. Chin. Phys. Lett., 2011, 28(12): 020201 [6] GUO Bo-Ling, LING Li-Ming, ** . Rogue Wave, Breathers and Bright-Dark-Rogue Solutions for the Coupled Schrödinger Equations[J]. Chin. Phys. Lett., 2011, 28(11): 020201 [7] ZHANG Yi** . The Method of Variation of Parameters for Solving a Dynamical System of Relative Motion[J]. Chin. Phys. Lett., 2011, 28(10): 020201 [8] CAO Qing-Jie, **, HAN Ning, TIAN Rui-Lan . A Rotating Pendulum Linked by an Oblique Spring[J]. Chin. Phys. Lett., 2011, 28(6): 020201 [9] YAN Lu, SONG Jun-Feng, QU Chang-Zheng** . Nonlocal Symmetries and Geometric Integrability of Multi-Component Camassa–Holm and Hunter–Saxton Systems[J]. Chin. Phys. Lett., 2011, 28(5): 020201 [10] R. C. Aziz, I. Hashim** . Liquid Film on Unsteady Stretching Sheet with General Surface Temperature and Viscous Dissipation[J]. Chin. Phys. Lett., 2010, 27(11): 020201 [11] TIAN Rui-Lan, CAO Qing-Jie, LI Zhi-Xin. Hopf Bifurcations for the Recently Proposed Smooth-and-Discontinuous Oscillator[J]. Chin. Phys. Lett., 2010, 27(7): 020201 [12] LIU Fu-Hao, ZHANG Qi-Chang, TAN Ying. Analysis of High Codimensional Bifurcation and Chaos for the Quad Bundle Conductor's Galloping[J]. Chin. Phys. Lett., 2010, 27(4): 020201 [13] LIU Fu-Hao, ZHANG Qi-Chang, WANG Wei. Analysis of Hysteretic Strongly Nonlinearity for Quad Iced Bundle Conductors[J]. Chin. Phys. Lett., 2010, 27(3): 020201 [14] Osama Yusuf Ababneh, Rokiah@Rozita Ahmad. Construction of Third-Order Diagonal Implicit Runge-Kutta Methods for Stiff Problems[J]. Chin. Phys. Lett., 2009, 26(8): 020201 [15] MEI Feng-Xiang, SHANG Mei. large Last Multiplier of Generalized Hamilton System[J]. Chin. Phys. Lett., 2008, 25(11): 020201
Viewed
Full text

Abstract