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

1 Department of Mathematics, Science Faculty, Firat University, 23119 Elazig, Turkey
2 Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara, Turkey
3 Institute of Space Sciences, Magurele-Bucharest, Romania

Special Issues: Recent Advances in Fractional Calculus with Real World Applications

In this manuscript the fractional form of wind-influenced projectile motion equations which have a significant place in physics is extensively investigated by preserving dimensionality of the physical quantities for fractional operators and features of wind-influenced projectile motion are computed analytically in view of Atangana-Baleanu (ABC) fractional derivative in Caputo sense. Moreover, ABC fractional derivative with (n + α)th-order and its Laplace transform (LT) are obtained, α ∈ [0, 1] and n ∈ N. A comparative analysis based on the classical case is carried out in order to shed more light on the potent of the ABC fractional operator. Hence we present the results for some values of α, k friction constant, different wind effects and different masses in 3D illustrations by comparing Caputo fractional operator. Thus, we can observe trajectory, time of flight, maximum height and range clearly. Moreover, the obtained results are shown to correspond to the classical case while the order α → 1.
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Keywords projectile motion; fractional model; Atangana-Baleanu fractional derivative; wind; drag

Citation: Ramazan Ozarslan, Erdal Bas, Dumitru Baleanu, Bahar Acay. Fractional physical problems including wind-influenced projectile motion with Mittag-Leffler kernel. AIMS Mathematics, 2020, 5(1): 467-481. doi: 10.3934/math.2020031


  • 1. K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dynam., 71 (2013), 613-619.    
  • 2. S. Ullah, M. A. Khan and M. Farooq, A fractional model for the dynamics of TB virus, Chaos Soliton. Fract., 116 (2018), 63-71.    
  • 3. I. Area, H. Batarfi, J. Losada, et al. On a fractional order Ebola epidemic model, Advances in Difference Equations, 2015 (2015), 278.
  • 4. H. Richard, Fractional calculus: an introduction for physicists, World Scientific, 2014.
  • 5. G. U. Varieschi, Applications of Fractional Calculus to Newtonian Mechanics, arXiv preprint arXiv:1712.03473, 2017.
  • 6. M. H. Heydari, Chebyshev cardinal functions for a new class of nonlinear optimal control problems generated by Atangana-Baleanu-Caputo variable-order fractional derivative, Chaos Soliton. Fract., 130 (2020), 109401.
  • 7. M. Hosseininia, M. H. Heydari, Meshfree moving least squares method for nonlinear variableorder time fractional 2D telegraph equation involving Mittag-Leffler non-singular kernel, Chaos Soliton. Fract., 127 (2019), 389-399.    
  • 8. M. Hosseininia, M. H. Heydari, Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag-Leffler nonsingular kernel, Chaos Soliton. Fract., 127 (2019), 400-407.    
  • 9. E. Bonyah, A. Atangana, A. A. Elsadany, A fractional model for predator-prey with omnivore, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013136.
  • 10. A. Al-khedhairi, A. A. Elsadany, A. Elsonbaty, Modelling immune systems based on Atangana-Baleanu fractional derivative, Chaos Soliton. Fract., 129 (2019), 25-39.    
  • 11. M. A. Khan, A. Khan, A. Elsonbaty, et al. Modeling and simulation results of a fractional dengue model, The European Physical Journal Plus, 134 (2019), 379.
  • 12. M. Yavuz, Characterizations of two different fractional operators without singular kernel, Math. Model. Nat. Pheno., 14 (2019), 302.
  • 13. M. Goyal, H. M. Baskonus, A. Prakash, An efficient technique for a time fractional model of lassa hemorrhagic fever spreading in pregnant women, The European Physical Journal Plus, 134 (2019), 482.
  • 14. T. Abdeljawad, Fractional operators with boundary points dependent kernels and integration by parts, Discrete Contin. Dyn. Syst. Ser. S, (2019), 351.
  • 15. E. Bas, R. Ozarslan, D. Baleanu, et al. Comparative simulations for solutions of fractional SturmLiouville problems with non-singular operators, Advances in Difference Equations, 2018 (2018), 350.
  • 16. A. Yusuf, M. Inc, A. I. Aliyu, et al. Efficiency of the new fractional derivative with nonsingular Mittag-Leffler kernel to some nonlinear partial differential equations, Chaos Soliton. Fract., 116 (2018), 220-226.    
  • 17. E. Bas, The Inverse Nodal problem for the fractional diffusion equation, Acta Scientiarum. Technology, 37 (2015), 251-257.    
  • 18. T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), 130.
  • 19. M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91 (1971), 134-147.    
  • 20. R. L. Magin, O. Abdullah, D. Baleanu, et al. Anomalous diffusion expressed through fractional order differential operators in the Bloch Torrey equation, J. Magn. Reson., 190 (2008), 255-270.    
  • 21. V. V. Uchaikin, Fractional derivatives for physicists and engineers, Berlin: Springer, 2013.
  • 22. R. L. Magin, Fractional calculus in bioengineering, Redding: Begell House, 2006.
  • 23. A. K. Golmankhaneh, A. M. Yengejeh and D. Baleanu, On the fractional Hamilton and Lagrange mechanics, Int. J. Theor. Phys., 51 (2012), 2909-2916.    
  • 24. D. G. Prakasha, P. Veeresha, H. M. Baskonus, Analysis of the dynamics of hepatitis E virus using the Atangana-Baleanu fractional derivative, The European Physical Journal Plus, 134 (2019), 241.
  • 25. W. Gao, B. Ghanbari, H. M. Baskonus, New numerical simulations for some real world problems with Atangana-Baleanu fractional derivative, Chaos Soliton. Fract., 128 (2019), 34-43.    
  • 26. M. Yavuz, N. Ozdemir, H. M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, 133 (2018), 215.
  • 27. J. F. Gomez-Aguilar, J. J. Rosales-García, J. J. Bernal-Alvarado, et al. Fractional mechanical oscillators, Revista mexicana de física, 58 (2012), 348-352.
  • 28. M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 1 (2015), 1-13.
  • 29. A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 757-763.    
  • 30. A. Ebaid, Analysis of projectile motion in view of fractional calculus, Appl. Math. Model., 35 (2011), 1231-1239.    
  • 31. A. O. Contreras, J. J. R. Garcia, L. M. Jimenez, et al. Analysis of projectile motion in view of conformable derivative, Open Phys., 16 (2018), 581-587.    
  • 32. B. Ahmad, H. Batarfi, J. J. Nieto, et al. Projectile motion via Riemann-Liouville calculus, Advances in Difference Equations, 2015 (2015), 63.
  • 33. J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 1 (2015), 87-92.
  • 34. J. Rosales, M. Guía, F. F. Gómez, et al. Two dimensional fractional projectile motion in a resisting medium, Open Phys., 12 (2014), 517-520.
  • 35. F. M. Alharbi, D. Baleanu, A. Ebaid, Physical properties of the projectile motion using the conformable derivative, Chinese J. Phys., 58 (2019), 18-28.    
  • 36. J. F. Gomez-Aguilar, R. F. Escobar-Jiménez, M. G. Lopez-Lopez, et al. Analysis of projectile motion: A comparative study using fractional operators with power law, exponential decay and Mittag-Leffler kernel, The European Physical Journal Plus, 133 (2018), 103.
  • 37. R. C. Bernardo, J. P. Esguerra, J. D. Vallejos, et al. Wind-influenced projectile motion, Eur. J. Phys., 36 (2015), 025016.
  • 38. E. Bas, R. Ozarslan, Real world applications of fractional models by Atangana-Baleanu fractional derivative, Chaos Soliton. Fract., 116 (2018), 121-125.    
  • 39. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
  • 40. S. Liang, R. Wu, L. Chen, Laplace transform of fractional order differential equations, Electron. J. Differ. Equ., 2015 (2015), 1-15.    


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