Abstract.
The concept of fractal medium is present in several real-world problems, for instance, in the geological formation that constitutes the well-known subsurface water called aquifers. However, attention has not been quite devoted to modeling for instance, the flow of a fluid within these media. We deem it important to remind the reader that the concept of fractal derivative is not to represent the fractal sharps but to describe the movement of the fluid within these media. Since this class of ordinary differential equations is highly complex to solve analytically, we present a novel numerical scheme that allows to solve fractal ordinary differential equations. Error analysis of the method is also presented. Application of the method and numerical approximation are presented for fractal order differential equation. The stability and the convergence of the numerical schemes are investigated in detail. Also some exact solutions of fractal order differential equations are presented and finally some numerical simulations are presented.
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F. Ali, N.A. Sheikh, I. Khan, M. Saqib, J. Magn. & Magn. Mater. 423, 327 (2017)
F. Ali, M. Saqib, I. Khan, N.A. Sheikh, Eur. Phys. J. Plus 131, 377 (2016)
F. Ali, S.A.A. Jan, I. Khan, M. Gohar, N.A. Sheikh, Eur. Phys. J. Plus 131, 310 (2016)
M. Al-Refai, Y. Luchko, Appl. Math. Comput. 257, 40 (2015)
A. Atangana, B.S.T. Alkahtani, Entropy 17, 4439 (2015)
A. Atangana, B. Dumitru, Therm. Sci. 20, 763 (2016)
A. Atangana, I. Koca, Chaos, Soliton Fractals 89, 447 (2016)
A. Atangana, Appl. Math. Comput. 273, 948 (2016)
M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)
E.F. Doungmo Goufo, M.K. Pene, N. Jeanine, Open Math. 13, 839 (2015)
R. Gnitchogna, A. Atangana, Numer. Methods Part. Differ. Equ. (2017) https://doi.org/10.1002/num.22216
D.W. Brzezinski, Appl. Math. Nonlinear Sci. 1, 23 (2016)
R.B. Gnitchogna, A. Atangana, Int. J. Math. Mod. Methods Appl. Sci. 9, 105 (2015)
M.T. Gencoglu, H.M. Baskonus, H. Bulut, AIP Conf. Proc. 1798, 020103 (2017)
S. Kumar, X.B. Yin, D. Kumar, Adv. Mech. Eng. 7, 1 (2015)
K.M. Owolabi, Chaos, Solitons Fractals 93, 89 (2016)
K.M. Owolabi, Commun. Nonlinear Sci. Numer. Simul. 44, 304 (2017)
W. Chen, Chaos, Soliton Fractals 28, 9239 (2016)
R. Kanno, Physica A 248, 165 (1998)
W. Chen, H.G. Sun, X. Zhang, D. Korosak, Comput. Math. Appl. 59, 1754 (2010)
J.H. Cushman, D.O.’ Malley, M. Park, Phys. Rev. E 79, 032101 (2009)
F. Mainardi, A. Mura, G. Pagnini, Int. J. Differ. Equ. 29, 104505 (2010)
W. Chen, X.D. Zhang, D. Korosak, Int. J. Nonlinear Sci. Numer. 11, 3 (2010)
A. Atangana, Chaos, Soliton Fractals 102, 396 (2017)
A. Latif, Fixed Point Theor. Appl. 2009, 170140 (2009)
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Atangana, A., Jain, S. A new numerical approximation of the fractal ordinary differential equation. Eur. Phys. J. Plus 133, 37 (2018). https://doi.org/10.1140/epjp/i2018-11895-1
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DOI: https://doi.org/10.1140/epjp/i2018-11895-1