Advertisement

Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena

  • Abdon AtanganaEmail author
  • J. F. Gómez-Aguilar
Regular Article
Part of the following topical collections:
  1. Focus Point on Modelling Complex Real-World Problems with Fractal and New Trends of Fractional Differentiation

Abstract.

To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature. We demonstrate the failure of the semi-group principle in modeling real-world problems. We use natural phenomena to illustrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall. We present statistical properties of generator for each fractional derivative, including Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu ones. The Atangana-Baleanu and Caputo-Fabrizio fractional derivatives show crossover properties for the mean-square displacement, while the Riemann-Liouville is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the Caputo Fabrizio kernel has a steady state between the transition. Only the Atangana-Baleanu kernel is a crossover for the waiting time distribution from stretched exponential to power law. A new criterion was suggested, namely the Atangana-Gómez fractional bracket, that helps describe the energy needed by a fractional derivative to characterize a 2-pletic manifold. Based on these properties, we classified fractional derivatives in three categories: weak, mild and strong fractional differential and integral operators. We presented some applications of fractional differential operators to describe real-world problems and we proved, with numerical simulations, that the Riemann-Liouville power-law derivative provides a description of real-world problems with much additional information, that can be seen as noise or error due to specific memory properties of its power-law kernel. The Caputo-Fabrizio derivative is less noisy while the Atangana-Baleanu fractional derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state. The study suggests that the properties of associativity and commutativity or the semi-group principle are just irrelevant in fractional calculus. Properties of classical derivatives were established for the ordinary calculus with no memory effect and it is a failure of mathematical investigation to attempt to describe more complex natural phenomena using the same notions.

References

  1. 1.
    J.R. Kantor, The Scientific Evolution of Psychology (Principia Press, Chicago, 1963)Google Scholar
  2. 2.
    P. Green, Alexander of Macedon (University of California Press Ltd., Oxford, 1991)Google Scholar
  3. 3.
    R. Sorabji (Editor), Aristotle Transformed (Bloomsburg Academic, London, 1990)Google Scholar
  4. 4.
    J. Filonik, Athenian Impiety Trials: A Reappraisal (Dike, 2013)Google Scholar
  5. 5.
    A.G. Kurosh, Mat. Sbornik. 20, 237 (1947)MathSciNetGoogle Scholar
  6. 6.
    D. Eisenhardt, Learn Mem. 21, 534 (2014)CrossRefGoogle Scholar
  7. 7.
    I. Yaglom, Complex Numbers in Geometry (Academic Press, N.Y., 1968) pp. 195-219, translated by E. Primrose from 1963 Russian original, appendix: Non-Euclidean geometries in the plane and complex numbersGoogle Scholar
  8. 8.
    J. Briggs, Fractals: The Patterns of Chaos (Thames and Hudson, London, 1992)Google Scholar
  9. 9.
    T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore/New Jersey, 1992)Google Scholar
  10. 10.
    H. Takayasu, Fractals in the Physical Sciences (Manchester University Press, Manchester, 1990)Google Scholar
  11. 11.
    R.D. Knight, J. Geom. 83, 137 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    W. Blaschke, Differentialgeometrie der Kreise und Kugeln, in Vorlesungen über Differentialgeometrie, Grundlehren der Mathematischen Wissenschaften (Springer, Berlin, 1929)Google Scholar
  13. 13.
    M.D. Ortigueira, J.A.T. Machado, J. Comput. Phys. 293, 4 (2015)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    V.E. Tarasov, Nonlinear Sci. Numer. Simul. 18, 2945 (2013)CrossRefGoogle Scholar
  15. 15.
    G.B. Folland, Advanced Calculus (Prentice Hall, 2002)Google Scholar
  16. 16.
    M. Caputo, Geophys. J. Int. 13, 529 (1967)ADSCrossRefGoogle Scholar
  17. 17.
    A. Atangana, B. Dumitru, Therm. Sci. 20, 763 (2016)CrossRefGoogle Scholar
  18. 18.
    M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 2, 1 (2016)CrossRefGoogle Scholar
  19. 19.
    C.G. Galizia, S.L. McIlwrath, R. Menzel, Cell Tissue Res. 295, 383 (1999)CrossRefGoogle Scholar
  20. 20.
    R.D. Schafer, An Introduction to Nonassociative Algebras, Vol. 22 (Academic Press, 1966)Google Scholar
  21. 21.
    S. Okubo, Introduction to Octonion and Other Non-associative Algebras in Physics (Cambridge University Press, 1995)Google Scholar
  22. 22.
    E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies, 4th ed. (Dover Publications, New York, 1937)Google Scholar
  23. 23.
    R. Broucke, Astrophys. Space Sci. 72, 33 (1980)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    A. Biswas, V. Shapiro, Graph. Models 66, 133 (2004)CrossRefGoogle Scholar
  25. 25.
    F. Cantrijn, A. Ibort, M. De-Leon, J. Aust. Math. Soc. A 66, 303 (1999)CrossRefGoogle Scholar
  26. 26.
    J. Baez, A. Hoffnung, C. Rogers, Commun. Math. Phys. 293, 701 (2010)ADSCrossRefGoogle Scholar
  27. 27.
    K.C. Louden, Compiler Construction: Principles and Practice (1997)Google Scholar
  28. 28.
    M. Pitkänen, Prespacetime J. 7, 66 (2016)Google Scholar
  29. 29.
    H. Couclelis, N. Gale, Geogr. Ann. Ser. B Human Geogr. 68, 1 (1986)CrossRefGoogle Scholar
  30. 30.
    M. Chaichian, P. Presnajder, A. Tureanu, Phys. Rev. Lett. 94, 151602 (2005)ADSCrossRefGoogle Scholar
  31. 31.
    S. Doplicher, K. Fredenhagen, J.E. Roberts, Commun. Math. Phys. 172, 187 (1995)ADSCrossRefGoogle Scholar
  32. 32.
    M.R. Douglas, N.A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001)ADSCrossRefGoogle Scholar
  33. 33.
    V. Moretti, Rev. Math. Phys. 15, 1171 (2003)MathSciNetCrossRefGoogle Scholar
  34. 34.
    W. Heisenberg, Z. Phys. 43, 172 (1927)ADSCrossRefGoogle Scholar
  35. 35.
    L.A. Rozema, A. Darabi, D.H. Mahler, A. Hayat, Y. Soudagar, A.M. Steinberg, Phys. Rev. Lett. 109, 100404 (2012)ADSCrossRefGoogle Scholar
  36. 36.
    D.J. Gross, F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973)ADSCrossRefGoogle Scholar
  37. 37.
    R. Gorenflo, J. Loutchko, Y. Luchko, Fract. Calc. Appl. Anal. 5, 491 (2002)MathSciNetGoogle Scholar
  38. 38.
    R.N. Pillai, Ann. Inst. Stat. Math. 42, 157 (1990)CrossRefGoogle Scholar
  39. 39.
    C.S. Kumar, B.U. Nair, J. Stat. Appl. 6, 23 (2011)Google Scholar
  40. 40.
    C.S. Kumar, B.U. Nair, OPSEARCH 52, 86 (2015)MathSciNetCrossRefGoogle Scholar
  41. 41.
    A.M. Mathai, P. Moschopoulos, J. Stat. Appl. Prob. 1, 15 (2012)CrossRefGoogle Scholar
  42. 42.
    J. Hristov, Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, in Frontiers in Fractional Calculus (Bentham Science Publishers, 2017) pp. 235--295Google Scholar
  43. 43.
    J. Hristov, Electrical Circuits of Non-integer Order: Introduction to an Emerging Interdisciplinary Area with Examples, in Analysis and Simulation of Electrical and Computer Systems (Springer, 2018) pp. 251--273Google Scholar
  44. 44.
    A. Atangana, I. Koca, Chaos, Solitons Fractals 89, 447 (2016)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    J. Hristov, Therm. Sci. 21, 827 (2016)CrossRefGoogle Scholar
  46. 46.
    B.S.T. Alkahtani, Chaos, Solitons Fractals 89, 547 (2016)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    V.F. Morales-Delgado, J.F. Gómez-Aguilar, M.A. Taneco-Hernández, Eur. Phys. J. Plus 132, 527 (2017)CrossRefGoogle Scholar
  48. 48.
    J.F. Gómez-Aguilar, J. Math. Sociol. 41, 172 (2017)MathSciNetCrossRefGoogle Scholar
  49. 49.
    J.F. Gómez-Aguilar, Chaos, Solitons Fractals 95, 179 (2017)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    A.A. Tateishi, H.V. Ribeiro, E.K. Lenzi, Front. Phys. 5, 52 (2017)CrossRefGoogle Scholar
  51. 51.
    I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Academic Press, New York, 1998)Google Scholar
  52. 52.
    L. Changpin, T. Chunxing, Comput. Math. Appl. 58, 1573 (2009)MathSciNetCrossRefGoogle Scholar
  53. 53.
    K. Diethelm, N.J. Ford, A.D. Freed, Numer. Algorithms 36, 31 (2004)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    M. Caputo, M. Fabricio, Progr. Fract. Differ. Appl. 1, 73 (2015)Google Scholar
  55. 55.
    A. Atangana, J.J. Nieto, Adv. Mech. Eng. 7, 1 (2015)Google Scholar
  56. 56.
    J.F. Gómez-Aguilar, Numer. Methods Part. Differ. Equ. (2017)  https://doi.org/10.1002/num.22219
  57. 57.
    A.M.A. El-Sayed, A.E.M. El-Mesiry, H.A.A. El-Saka, Appl. Math. Lett. 20, 817 (2007)MathSciNetCrossRefGoogle Scholar
  58. 58.
    J.G. Lu, Chaos, Solitons Fractals 26, 1125 (2005)ADSCrossRefGoogle Scholar
  59. 59.
    D.R. Willé, C.T. Baker, Appl. Numer. Math. 9, 223 (1992)CrossRefGoogle Scholar
  60. 60.
    V. Daftardar-Gejji, Y. Sukale, S. Bhalekar, Fract. Calc. Appl. Anal. 18, 400 (2015)MathSciNetCrossRefGoogle Scholar
  61. 61.
    A. Atangana, J.F. Gómez-Aguilar, Numer. Methods Part. Differ. Equ. (2017)  https://doi.org/10.1002/num.22195
  62. 62.
    A. Atangana, E.F. Doungmo-Goufo, Therm. Sci. 19, 231 (2015)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Groundwater Studies, Faculty of Natural and Agricultural SciencesUniversity of the Free StateBloemfonteinSouth Africa
  2. 2.CONACyT-Tecnológico Nacional de México/CENIDETCuernavacaMexico

Personalised recommendations