Arabian Journal of Mathematics

, Volume 7, Issue 2, pp 147–158 | Cite as

Non-differentiability and fractional differentiability on timescales

  • Mehdi Nategh
  • Abdolali Neamaty
  • Bahram Agheli
Open Access


This work deals with concepts of non-differentiability and a non-integer order differential on timescales. Through an investigation of a local non-integer order derivative on timescales, a mean value theorem (a fractional analog of the mean value theorem on timescales) is presented. Then, by illustrating a vanishing property of this derivative, its objectivity is discussed. As a first-hand result, the potentials and capability of this fractional derivative connected to nonsmooth analysis, including non-differentiable paths and a class of self-similar fractals, are stated. It is stated that the non-integer order derivative never vanishes almost everywhere. It has been shown that with the help of changing the order of differentiability on a q-timescale, the non-differentiability disappears.

Mathematics Subject Classification

26E70 26A33 28A80 



  1. 1.
    Albrecht-Buehler, G.: Fractal genome sequences. Gene 498(1), 20–27 (2012)CrossRefGoogle Scholar
  2. 2.
    Bastos, N.R.: Fractional calculus on time scales. Universidade de Aveiro, PhD thesis (2012)Google Scholar
  3. 3.
    Benkhettou, N.; da Cruz, A.M.B.; Torres, D.F.: A fractional calculus on arbitrary time scales: fractional differentiation and fractional integration. Signal Process. 107, 230–237 (2015)CrossRefGoogle Scholar
  4. 4.
    Bohner, M.; Peterson, A.C. (eds.).: Advances in Dynamic Equations on Time Scales. Springer Science & Business Media, New York (2002)Google Scholar
  5. 5.
    Bohner, M.; Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Springer Science & Business Media, New York (2001)Google Scholar
  6. 6.
    Carvalho, A.: Fractal geometry of Weierstrass-type functions. Fractals 17(01), 23–37 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, W.: Time-space fabric underlying anomalous diffusion. Chaos Solitons Fract. 28(4), 923–929 (2006)Google Scholar
  8. 8.
    Chen, D.X.: Nonlinear oscillation of a class of second-order dynamic equations on time scales. Appl. Math. Sci. 6(60), 2957–2962 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ciurdariu, L.: Integral inequalities on time scales via isotonic linear functionals. Appl. Math. Sci. 9(134), 6655–6668 (2015)Google Scholar
  10. 10.
    Ernst, T.: A Comprehensive Treatment of q-calculus. Springer Science & Business Media (2012)Google Scholar
  11. 11.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, England (1990)Google Scholar
  12. 12.
    Golmankhaneh, A.K.; Baleanu, D.: Fractal calculus involving gauge function. Commun. Nonlinear Sci. Numer. Simul. 37, 125–130 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Guidolin, D.; Marinaccio, C.; Tortorella, C.; Ruggieri, S.; Rizzi, A.; Maiorano, E.; Specchia, G.; Ribatti, D.: A fractal analysis of the spatial distribution of tumoral mast cells in lymph nodes and bone marrow. Exp. Cell Res. 339(1), 96–102 (2015)CrossRefGoogle Scholar
  14. 14.
    Havlin, S.; Buldyrev, S.V.; Goldberger, A.L.; Mantegna, R.N.; Ossadnik, S.M.; Peng, C.K.; Simons, M.; Stanley, H.E.: Fractals in biology and medicine. Chaos Solitons Fract. 6, 171–201 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hilger, S.: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 18(1–2), 18–56 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hilger, S.: Differential and difference calculus-unified!. Nonlinear Anal. Theory Methods Appl. 30(5), 2683–2694 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Humphrey, J.A.C.; Schuler, C.A.; Rubinsky, B.: On the use of the Weierstrass-Mandelbrot function to describe the fractal component of turbulent velocity. Fluid Dyn. Res. 9(1–3), 81–95 (1992)CrossRefGoogle Scholar
  18. 18.
    Jiang, S.; Zheng, Y.: An analytical model of thermal contact resistance based on the Weierstrass-Mandelbrot fractal function. In: Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 224(4), 959–967 (2010)Google Scholar
  19. 19.
    Kac, V.; Cheung, P.: Quantum Calculus. Springer, Berlin (2002)Google Scholar
  20. 20.
    Liang, Y.; Allen, Q.Y.; Chen, W.; Gatto, R.G.; Colon-Perez, L.; Mareci, T.H.; Magin, R.L.: A fractal derivative model for the characterization of anomalous diffusion in magnetic resonance imaging. Commun. Nonlinear Sci. Numer. Simul. 39, 529–537 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Neamaty, A.; Nategh, M.; Agheli, B.: Time-space fractional Burger’s equation on time scales. J. Comput. Nonlinear Dyn. 12(3), 031022 (2017)CrossRefGoogle Scholar
  22. 22.
    Samko, S.G.; Kilbas, A.A.; Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993)zbMATHGoogle Scholar
  23. 23.
    Stiassnie, M.: A look at fractal functions through their fractional derivatives. Fractals 5(04), 561–564 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Taylor, M.E.: Measure Theory and Integration. American Mathematical Society, Rhode Island (2006)Google Scholar
  25. 25.
    Thim, J.: Continuous nowhere differentiable functions. Lule\(\dot{a}\) University. Master Thesis (2003)Google Scholar
  26. 26.
    Williams, P.A.: Unifying fractional calculus with time scales. University of Melbourne, PhD thesis (2012)Google Scholar
  27. 27.
    Yang, X.J.; Baleanu, D.; Srivastava, H.M.: Local Fractional Integral Transforms and Their Applications. Elsevier, London (2015)Google Scholar
  28. 28.
    Zhang, L.; Yu, C.; Sun, J.Q.: Generalized weierstrass-mandelbrot function model for actual stocks markets indexes with nonlinear characteristics. Fractals 23(02), 1550006 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zygmund, A.: Trigonometric Series. Volume I & II combined, 3rd edn. Cambridge University Press, Cambridge (2002)Google Scholar

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Mehdi Nategh
    • 1
    • 2
  • Abdolali Neamaty
    • 2
  • Bahram Agheli
    • 3
  1. 1.Department of Mathematics and StatisticsMissouri S & TRollaUSA
  2. 2.Department of Mathematics and StatisticsUniversity of MazandaranBabolsarIran
  3. 3.Department of Mathematics, Qaemshahr BranchIslamic Azad UniversityQaemshahrIran

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