Advertisement

A critical analysis of the conformable derivative

  • Ahmed A. AbdelhakimEmail author
  • José A. Tenreiro Machado
Original Paper
  • 78 Downloads

Abstract

We prove that conformable “fractional” differentiability of a function \(f:[0,\infty [\,\longrightarrow \mathbb {R}\) is nothing else than the classical differentiability. More precisely, the conformable \(\alpha \)-derivative of f at some point \(x>0\), where \(0<\alpha <1\), is the pointwise product \(x^{1-\alpha }f^{\prime }(x)\). This proves the lack of significance of recent studies of the conformable derivatives. The results imply that interpreting fractional derivatives in the conformable sense alters fractional differential problems into differential problems with the usual integer-order derivatives that may no longer properly describe the original fractional physical phenomena. A general fractional viscoelasticity model is analysed to illustrate this state of affairs. We also test the modelling efficiency of the conformable derivative on various fractional models. We find that, compared with the classical fractional derivative, the conformable framework results in a substantially larger error.

Keywords

Fractional derivative Fractional differential equations Fractional analysis Viscoelasticity 

Mathematics Subject Classification

26A33 34A08 74D05 

Notes

Acknowledgements

The authors are grateful for the valuable comments of the anonymous referees that improved the presentation of this manuscript.

Compliance with ethical standards

Conflict of interest

The authors declares that there is no conflict of interest.

Human participants or animals

The study does not involve any human participants or animals and does not require any form of a consent.

References

  1. 1.
    Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Almeida, R., Bastos, N.R.O., Teresa, M.: Modeling some real phenomena by fractional differential equations. Math. Methods Appl. Sci. 39(16), 4846–4855 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Almeida, R.: What is the best fractional derivative to fit data? Appl. Anal. Discrete Math. 11(2), 358–368 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Anderson, D.R., Ulness, D.J.: Properties of the Katugampola fractional derivative with potential application in quantum mechanics. J. Math. Phys. 56, 6 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bagley, R.L., Torvik, P.J.: On the fractional calculus model of viscoelastic behavior. J. Rheol. 30(1), 133–155 (1986)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bayour, B., Torres, D.F.M.: Existence of solution to a local fractional nonlinear differential equation. J. Comput. Appl. Math 312, 127–133 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91(8), 134–147 (1971)CrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, C., Jiang, Y.-L.: Simplest equation method for some time-fractional partial differential equations with conformable derivative. Comput. Math. Appl. 75(8), 2978–2988 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    de Oliveira, E.C., Machado, J.A.T.: A review of definitions for fractional derivatives and integral. Math. Problems Eng. 2014, 1–6 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Eslami, M., Rezazadeh, H.: The first integral method for Wu–Zhang system with conformable time-fractional derivative. Calcolo 53(3), 475–485 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gieseking, E.: Newton’s law of cooling. An experimental investigation (2014). http://jwilson.coe.uga.edu/EMAT6680Fa2014/Gieseking/Exploration 2012/Newton
  12. 12.
    Giusti, A.: A comment on some new definitions of fractional derivative. Nonlinear Dyn. 93(3), 1757–1763 (2018)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hosseini, K., Bekir, A., Ansari, R.: New exact solutions of the conformable time-fractional Cahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method. Optik 132, 203–209 (2017)CrossRefGoogle Scholar
  14. 14.
    Katugampola, U.N.: A new fractional derivative with classical properties. e-print arXiv:1410.6535
  15. 15.
    Katugampola, U.N.: Correction to “What is a fractional derivative ?” By Ortigueira and Machado. J. Comput. Phys. 321, 1255–1257 (2016). (J. Comput. Phys. 293, 413 (2015). Special issue on fractional PDEs)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Khalil, R., Horani, M.A., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  18. 18.
    Morales-Delgado, V.F., Gmez-Aguilar, J.F., Escobar-Jimnez, R.F., Taneco-Hernndez, M.A.: Fractional conformable derivatives of Liouville–Caputo type with low-fractionality. Physica A 503, 424–438 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ortigueira, M.D., Machado, J.A.T.: What is a fractional derivative? J. Comput. Phys. 293(15), 4–13 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ortigueira, M.D., Machado, J.A.T.: Which derivative? Fractal Fractional 1(1), 3 (2017)CrossRefGoogle Scholar
  21. 21.
    Ortigueira, M.D., Machado, J.A.T.: A critical analysis of the Caputo–Fabrizio operator. Commun. Nonlinear Sci. Numer. Simul. 59, 608–611 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Roderic, S.: Lakes, Viscoelastic Materials. Cambridge University Press, Cambridge (2009)Google Scholar
  23. 23.
    Tarasov, V.E.: No violation of the Leibniz rule. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 18(11), 2945–2948 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tarasov, V.E.: No Nonlocality. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 62, 157–163 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tariq, H., Akram, G.: New traveling wave exact and approximate solutions for the nonlinear Cahn–Allen equation: evolution of a nonconserved quantity. Nonlinear Dyn. 88(1), 581–594 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    United Nations, The World at Six Billion Off Site, Table 1, World Population From Year 0 to Stabilization, 5 (1999)Google Scholar
  27. 27.
    Ward, I.M., Sweeney, J.: Mechanical Properties of Solid Polymers. Wiley, New York (2012)CrossRefGoogle Scholar
  28. 28.
    Yang, S., Wang, L., Zhang, S.: Conformable derivative: application to non-Darcian flow in low-permeability porous media. Appl. Math. Lett. 79, 105–110 (2018)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zhou, H.W., Yang, S., Zhang, S.Q.: Conformable derivative approach to anomalous diffusion. Physica A 491, 1001–1013 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Ahmed A. Abdelhakim
    • 1
    Email author
  • José A. Tenreiro Machado
    • 2
  1. 1.Mathematics Department, Faculty of ScienceAssiut UniversityAssiutEgypt
  2. 2.Department of Electrical EngineeringInstitute of Engineering of Polytechnic of PortoPortoPortugal

Personalised recommendations