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A Bit of History

  • Edmundo Capelas de OliveiraEmail author
Chapter
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Part of the Studies in Systems, Decision and Control book series (SSDC, volume 240)

Abstract

The purpose of this chapter is to sketch in chronological order the development of the fractional calculus, from its origins until recent times.

References

  1. 1.
    Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models. Imperial College, London (2010)Google Scholar
  2. 2.
    Kolwankar, K.M., Gangal, A.D.: Fractional differentiability of nowhere differentiable functions and dimensions. Chaos. 6, 505–513 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Frac. Differ. Appl. 2, 73–85 (2015)Google Scholar
  4. 4.
    Katugampola, U.N.: Correction to “what is a fractional derivative?” by Ortigueira and Machado [J. Comp. Phys. 293, 4–13 (2015), Special issue on Fractional PDEs]. J. Comp. Phys. 321, 1255–1257 (2016)Google Scholar
  5. 5.
    Ortigueira, M.D., Tenreiro Machado, J.A.: What is a fractional derivative? J. Comp. Phys. 293, 4–13 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Tarasov, V.E.: On chain rule for fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 30, 1–4 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Iyiola, O.S., Nwaeze, E.R.: Some new results on the new conformable fractional calculus with application using D’Alembert approach. Progr. Frac. Diff. Appl. 2, 115–122 (2016)CrossRefGoogle Scholar
  8. 8.
    Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model (2016). arXiv:1602.03408v1 [math.GM]
  9. 9.
    Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simulat. 44, 460–481 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mathai, A.M., Haubold, H.J.: Matrix Methods and Fractional Calculus. World Scientific, New Jersey (2017)Google Scholar
  11. 11.
    Tenreiro Machado, J.A., Kiryakova, V.: Historical survey: the chronicles of fractional calculus 20, 307–336 (2017)Google Scholar
  12. 12.
    Zhao, D., Luo, M.: General conformable fractional derivative and its physical interpretation. Calcolo 54, 903–917 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Oliveira, D.S., Capelas de Oliveira, E.: Hilfer-Katugampola fractional derivative (online). Comp. Appl. Math. 1–19 (2017)Google Scholar
  14. 14.
    Oliveira, D.S., Capelas de Oliveira, E.: On the generalized \((k,\rho )\)-fractional derivative. Progr. Fract. Differ. Appl. 2, 133–145 (2018)CrossRefGoogle Scholar
  15. 15.
    Sheikh, N.A., Ali, F., Saqib, M., Khan, I., Jan, S.A.A., Alshomrani, A.S., Alghamdi, S.S.: Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction. Results Phys. 7, 789–800 (2017)CrossRefGoogle Scholar
  16. 16.
    Yan, Y., Sun, Z.-Z., Zhang, J.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme. Commun. Comput. Phys. 22, 1028–1048 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kaplan, M., Bekir, A.: Construction of exact solutions to the space-time fractional differential equations via new approach. Optik 132, 1–8 (2017)CrossRefGoogle Scholar
  18. 18.
    Yang, X.-J., Tenreiro Machado, J.A.: A new fractional operator of variable order: Application in the description of anomalous diffusion. Phys. A Stat. Mech. Appl. 481, 276–283 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sousa, J.V.C., Capelas de Oliveira, E.: On the \(\psi \)-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simulat. 60, 72–91 (2018)Google Scholar
  20. 20.
    Kaplan, M., Akbulut, A.: Application of two different algorithms to the approximate long water wave equation with conformable fractional derivative. Arab. J. Basic Appl. Sci. 25, 77–84 (2018)CrossRefGoogle Scholar
  21. 21.
    Rogosin, S., Dubatovskaya, M.: Letnikov vs. Marchaud: A survey on two prominent constructions of fractional derivatives. Mathematics 6, 3 (2018).  https://doi.org/10.3390/math6010003MathSciNetCrossRefGoogle Scholar
  22. 22.
    Liang, X., Gao, F., Zhou, C-B., Wang, Z., Yang, X-J.: An anomalous diffusion model based on a new general fractional operator with the Mittag-Leffler function of Wiman type. Adv. Diff. Equat. Open Acess (2018)Google Scholar
  23. 23.
    Ferrari, F.: Weyl and Marchaud derivatives: a forgotten history. Mathematics 6, 6 (2018).  https://doi.org/10.3390/math6010006CrossRefGoogle Scholar
  24. 24.
    Evangelista, L.R., Lenzi, E.K.: Fractional diffusion equations and anomalous diffusion. Cambridge University Press, Cambridge (2018)Google Scholar
  25. 25.
    Ortigueira, M.D., Tenreiro Machado, J.A.: On fractional vectorial calculus. Bull. Pol. Ac. Tec. 66, 399–402 (2018)Google Scholar
  26. 26.
    Yang, X.-J., Gao, F., Srivastava, H.M.: A new computational approach for solving nonlinear local fractional PDEs. J. Comput. Appl. Math. 339, 285–296 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Agarwal, P., El-Sayed, A.A.: Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation. Phys. A Stat. Mech. Appl. 500, 40–49 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Cuahutenango-Barro, B., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: The fractional-time wave equation via Atangana-Baleanu fractional derivative. Chaos, Solitons and Fractals 115, 283–299 (2018)Google Scholar
  29. 29.
    Andrade, A.M.F., Lima,E.G., Dartora, C.A.: An Introduction to fractional calculus and its applications in electric circuit. Rev. Bras. Ens. Fis. 40, e3314 (2018)Google Scholar
  30. 30.
    Sun, H.G., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.Q.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simulat. 64, 213–231 (2018)CrossRefGoogle Scholar
  31. 31.
    Akman, T., Yildiz, B., Baleanu, D.: New discretization of Caputo-Fabrizio derivative. Comput. Appl. Math. 37, 3307–3333 (2018)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Ortigueira, M.D., Valério, D., Tenreiro Machado, J.A.: Variable order fractional systems. Commun. Nonlinear Sci. Num. Simulat. 71, 231–243 (2019)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhao, D., Luo, M.: Representations of acting processes and memory effects; general fractional derivative and its applications to theory of heat condition with finite wave speeds. Appl. Math. Comp. 346, 531–544 (2019)CrossRefGoogle Scholar
  34. 34.
    Capelas de Oliveira, E., Vaz Jr, J., Jarosz, S.: Fractional calculus via Laplace transform and its application in relaxation processes. Commun. Nonlinear Sci. Numer. Simulat. 69, 58–72 (2019)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sales Teodoro, G., Tenreiro Machado, J.A., Capelas de Oliveira, E.: A review of definitions of fractional derivatives and other operators. J. Comput. Phys. 388, 195–208 (2019)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.IMECCUniversidade Estadual de CampinasCampinasBrazil

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