Skip to main content
SpringerLink
Log in

New relationships connecting a class of fractal objects and fractional integrals in space

  • Survey Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

Many specialists working in the field of the fractional calculus and its applications simply replace the integer differentiation and integration operators by their non-integer generalizations and do not give any serious justifications for this replacement. What kind of “Physics” lies in this mathematical replacement? Is it possible to justify this replacement or not for the given type of fractal and find the proper physical meaning? These or other similar questions are not discussed properly in the current papers related to this subject. In this paper new approach that relates to the procedure of the averaging of smooth functions on a fractal set with fractional integrals is suggested. This approach contains the previous one as a partial case and gives new solutions when the microscopic function entering into the structural-factor does not have finite value at N ≫ 1 (N is number of self-similar objects). The approach was tested on the spatial Cantor set having M bars with different symmetry. There are cases when the averaging procedure leads to the power-law exponent that does not coincide with the fractal dimension of the self-similar object averaged. These new results will help researches to understand more clearly the meaning of the fractional integral. The limits of applicability of this approach and class of fractal are specified.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods. Ser. on Complexity, Nonlinearity and Chaos, World Scientific (2012).

    MATH  Google Scholar 

  2. A.A. Khamzin, R.R. Nigmatullin, I.I. Popov, B.A. Murzaliev, Microscopis model of dielectric α-relaxation in disordered media. Fract. Calc. Appl. Anal. 16, No 1 (2013), 158–170; DOI: 10.2478/s13540-013-0011-1; http://link.springer.com/article/10.2478/s13540-013-0011-1.

    MathSciNet  Google Scholar 

  3. J.A.T. Machado, V. Kiryakova, F. Mainardi, A poster about old history of fractional calculus. Fract. Calc. Appl. Anal. 13, No 4 (2010), 447–454.

    MathSciNet  MATH  Google Scholar 

  4. J.A.T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus. Commun. in Nonlinear Sci. and Numer. Simulation 16 (2011), 1140–1153.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Le Mehaute, R.R. Nigmatullin, L. Nivanen, Fleches du Temps et Geometrie Fractale. Paris, Editions Hermes (1998), In French.

    MATH  Google Scholar 

  6. R.R. Nigmatullin, A. Le Mehaute, Is there a geometrical/physical meaning of the fractional integral with complex exponent? J. Non-Cryst. Sol. 351 (2005), 2888–2899.

    Article  Google Scholar 

  7. R.R. Nigmatullin, Theory of dielectric relaxation in non-crystalline solids: From a set of micromotions to the averaged collective motion in the mesoscale region. Physica B: Physics of Condensed Matter 358 (2005), 201–215.

    Article  Google Scholar 

  8. R.R. Nigmatullin, Fractional kinetic equations and universal decoupling of a memory function in mesoscale region. Phys. A 363 (2006), 282–298.

    Article  Google Scholar 

  9. R.R. Nigmatullin, Strongly correlated variables and existence of the universal disctribution function for relative fluctuations. Phys. Wave Phen. 16, No 2 (2008), 119–145.

    Article  Google Scholar 

  10. R.R. Nigmatullin, Universal distribution function for the stronglycorrelated fluctuations: General way for description of random sequences. Commun. Nonlin. Sci. 15 (2010), 637–647.

    Article  MathSciNet  MATH  Google Scholar 

  11. R.R. Nigmatullin, D. Baleanu, The derivation of the generalized functional equations describing self-similar processes. Fract. Calc. Appl. Anal. 15, No 4 (2012), 718–740; DOI: 10.2478/s13540-012-0049-5; http://link.springer.com/article/10.2478/s13540-012-0049-5.

    MathSciNet  Google Scholar 

  12. L. Pietronero, E. Tosatti (Eds.), Fractals in Physics. Proc. 6-th Trieste International Symposium on Fractals in Physics, ICTP, Trieste, Italy, July 9–12, 1985, North-Holland (1986).

    Google Scholar 

  13. I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).

    MATH  Google Scholar 

  14. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach Sci. Publ., London — N. York (1993).

    MATH  Google Scholar 

  15. V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Ser.: Nonlinear Phys. Sci., Springer (2013).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretical Physics Department, Institute of Physics, Kazan Federal University, Kremlevskaya str. 18, 420008, Kazan, Russia

    Raoul R. Nigmatullin

  2. Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara, Turkey

    Dumitru Baleanu

  3. Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia

    Dumitru Baleanu

  4. Institute of Space Sciences, Magurele — Bucharest, Romania

    Dumitru Baleanu

Authors
  1. Raoul R. Nigmatullin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dumitru Baleanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raoul R. Nigmatullin.

About this article

Cite this article

Nigmatullin, R.R., Baleanu, D. New relationships connecting a class of fractal objects and fractional integrals in space. fcaa 16, 911–936 (2013). https://doi.org/10.2478/s13540-013-0056-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s13540-013-0056-1

MSC 2010

Key Words and Phrases

Search

Navigation