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Fractals pp 171-181 | Cite as

Local Fractional Calculus: a Calculus for Fractal Space-Time

  • Kiran M. Kolwankar
  • Anil D. Gangal

Abstract

Recently, new notions such as local fractional derivatives and local fractional differential equations were introduced. Here we argue that these developments provide a possible calculus to deal with phenomena in fractal space-time. We show how the usual calculus is generalized to deal with non Lipschitz functions. We also indicate how a definition of a fractal measure arises from these developments much the same way as the Lebesgue measure from ordinary calculus.

Keywords

Fractional Order Fractional Derivative Fractional Calculus Fractional Differential Equation Hausdorff Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    B.B. Mandelbrot (1977): The Fractal Geometry of Nature. Freeman, New York.Google Scholar
  2. 2.
    A. Bunde and S. Havlin, Eds (1995): Fractals in science. Springer.Google Scholar
  3. 3.
    S. Baldo, F. Normant and C. Tricot, Eds. (1994): Fractals in Engineering. World Scientific, Singapore.MATHGoogle Scholar
  4. 4.
    J. Lévy-vehel, E. Lutton and C. Tricot, Eds. (1997): Fractals in Engineering. Springer.MATHGoogle Scholar
  5. 5.
    A.-L. Barabási and H. E. Stanley (1995): Fractal concepts in surface growth. Cambridge University Press.MATHCrossRefGoogle Scholar
  6. 6.
    T. Vicsek (1989):Fractal growth phenomenon. World Scientific.Google Scholar
  7. 7.
    H. Peitgen, H. Jurgens and D. Saupe (1992): Chaos and fractals: New frontiers of science. Springer, New York.Google Scholar
  8. 8.
    L. Nottale (1996): Chaos, Solitons & Fractals. 7, 877.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    K. Falconer (1990): Fractal Geometry. John Wiley, New York.MATHGoogle Scholar
  10. 10.
    C. Tricot (1993): Curves and fractal dimension. Springer, New York.MATHGoogle Scholar
  11. 11.
    J. L. Kaplan, J. Malet-Peret and J. A. Yorke (1994). Ergodic Th. and Dyn. Syst. 4, 261.Google Scholar
  12. 12.
    R. P. Feynmann and A. R. Hibbs (1965): Quantum Mechanics and Path Integrals. McGraw-Hill, New York.Google Scholar
  13. 13.
    P. Constantin, I. Procaccia and K. R. Sreenivasan (1991). Phys. Rev. Lett. 67, 1739.CrossRefGoogle Scholar
  14. 14.
    P. Constantin and I. Procaccia (1994). Nonlinearity 7, 1045.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    L. F. Abott and M. B. Wise (1981): Am. J. Phys. 49, 37.CrossRefGoogle Scholar
  16. 16.
    K. M. Kolwankar (1997). Ph. D. thesis, University of Pune.Google Scholar
  17. 17.
    K. S. Miller and B. Ross (1993): An Introduction to the Fractional. Calculus and Fractional Differential Equations. John Wiley, New York.Google Scholar
  18. 18.
    K. B. Oldham and J. Spanier (1974): The Fractional Calculus. Academic Press, New York.MATHGoogle Scholar
  19. 19.
    S. G. Samko, A. A. Kilbas and O. I. Marichev (1993): Fractional integrals and derivatives, theory and applications. Gordon & Breach.MATHGoogle Scholar
  20. 20.
    A. Carpinteri and F. Mainardi, eds. (1997): Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York.MATHGoogle Scholar
  21. 21.
    R. Hilfer, ed. (1998): Applications of Fractional Calculus in Physics. World Scientific, Singapore.Google Scholar
  22. 22.
    T. J. Osier (1971). SIAM J. Math. Anal. 2, 37–48.MathSciNetCrossRefGoogle Scholar
  23. 23.
    M. Caputo and F. Mainardi (1971). Pure Appl. Geophys., 91, 134–147.CrossRefGoogle Scholar
  24. 24.
    Nigmatullin R. (1986). Phys. Stat. Sol. B133, 425–430.Google Scholar
  25. 25.
    T. F. Nonnenmacher and G. W. Glockle (1991). Phil. Mag. Lett. B64, 89–93.CrossRefGoogle Scholar
  26. 26.
    H. Schiessel and A. Blumen (1993). J. Phys. A: Math. Gen. 26, 5057–5069.CrossRefGoogle Scholar
  27. 27.
    F. Mainardi (1996). Chaos, Solitons and Fractals 7, 1461–1477.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    A. Compte and M. O. Cáceres (1998). Phys. Rev. Lett, 81, 3140–3143.CrossRefGoogle Scholar
  29. 29.
    M. Giona and H. E. Roman (1992). J. Phys. A: Math Gen. 25, 2093.MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    H. E. Roman and M. Giona (1992). J. Phys. A: Math. Gen. 25, 2107.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    G. M. Zaslavsky (1994). Physica D 76, 110.MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    G. M. Zaslavsky (1994). Chaos, 4, 25.MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    T. F. Nonnenmacher (1990). J. Phys. A: Math. gen. 23, L697-L700.MathSciNetCrossRefGoogle Scholar
  34. 34.
    J. P. Bouchaud and A. Georges (1990). Phys. Rep. 195, 127.MathSciNetCrossRefGoogle Scholar
  35. 35.
    B. Souillard (1993). In Chance and Matter edited by J. Souletie, J. Vannimenusans R. Stora, North Holland, Amsterdam.Google Scholar
  36. 36.
    K. Sarkar and C. Meneveau (1993). Phys. Rev. E 47 957.CrossRefGoogle Scholar
  37. 37.
    B. B. Mandelbrot and J. W. Van Ness (1968). SIAM Rev. 10, 422.MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    K. L. Sebastian (1995). J. Phys. A28, 4305.MathSciNetMATHGoogle Scholar
  39. 39.
    K. M. Kolwankar and A. D. Gangal (1996). Chaos 6, 505.MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    H. Risken (1984): The Fokker-Planck Equation. Springer-Verlag, Berlin.MATHGoogle Scholar
  41. 41.
    W. Feller (1968): An Introduction to Probability Theory and its Applications. Wiley, New York, Vol 2.MATHGoogle Scholar
  42. 42.
    K. M. Kolwankar and A. D. Gangal (1998). Phys. Rev. Lett. 80 214.MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    K. M. Kolwankar and A. D. Gangal. Unpublished.Google Scholar
  44. 44.
    K. M. Kolwankar and A. D. Gangal (1997). Pramana-J. Phys. 48, 49.CrossRefGoogle Scholar
  45. 45.
    K. M. Kolwankar and A. D. Gangal (1997). In proceedings of ‘Fractals in Engineering’, Arcachon, France.Google Scholar
  46. 46.
    R. Courant and F. John (1965): Introduction to calculus and analysis. John Wiley, Vol 1.MATHGoogle Scholar
  47. 47.
    H. L. Royden, Real Analysis 3e (Macmillan, New York, 1988).Google Scholar
  48. 48.
    K. J. Falconer, The Geometry of Fractal Sets (Cambridge University Press, Cambridge, 1986).Google Scholar
  49. 49.
    P. R. Haimos, Measure Theory (Springer, New York, 1986).Google Scholar
  50. 50.
    Feder J., Fractals, Pergamon, 1988MATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 1999

Authors and Affiliations

  • Kiran M. Kolwankar
    • 1
  • Anil D. Gangal
    • 2
  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.Department of PhysicsUniversity of PunePuneIndia

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