Skip to main content
SpringerLink
Log in

Fractional Integration and Fractals

  • Chapter
Fractional Dynamics

Part of the book series: Nonlinear Physical Science ((NPS,volume 0))

Abstract

The theory of integration and differentiation of non-integer order has a long history from 30 September 1695, when the derivatives of order α = 1/2 was described by Leibniz in a letter to L’Hospital (Oldham and Spanier, 1974; Samko et al., 1993; Ross, 1975). The earliest theory of integrals and derivatives of non-integer order goes back to Liouville and Riemann (Ross, 1975). There are many interesting books about fractional calculus and fractional differential equations (Oldham and Spanier, 1974; Samko et al., 1993; Miller and Ross, 1993; Podlubny, 1999; Kilbas et al., 2006; Nahushev, 2003; Pshu, 2005); see also (McBride, 1976, 1986; Srivastava and Owa, 1989; Nishimoto, 1989; Kiryakova, 1994; Rubin, 1996). Derivatives and integrals of non-integer order, and fractional integro-differential equations have found many applications in recent studies in physics (for example, books (West et al., 2003; Zaslavsky, 2005; Uchaikin, 2008; Mainardi, 2010), edited volumes (Carpinteri and Mainardi, 1997; Hilfer, 2000; Sabatier et al., 2007), and reviews (Metzler and Klafter, 2000; Zaslavsky, 2002; Montroll and Shlesinger, 1984; Metzler and Klafter, 2004)). Now we can state that fractional dynamics form a new paradigm in science.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • F. Ben Adda, 1997, Geometric interpretation of the fractional derivative, Journal of Fractional Calculus, 11, 21–52.

    MathSciNet  MATH  Google Scholar 

  • A.S. Besicovitch, 1929, On linear sets of points of fractional dimensions, Mathematische Annalen, 101, 161–193.

    Article  MathSciNet  MATH  Google Scholar 

  • R.C. Blei, 2003, Analysis in Integer and Fractional Dimensions, Cambridge University Press, Cambridge.

    Google Scholar 

  • V.L. Bonch-Bruevich, I.P. Zvyagin, R. Kaiper, A.G. Mironov, R. Eiderlain, B. Esser, 1981, Electron Theory of Disordered Semiconductors, Nauka, Moscow, Section 1.5, 1.6, 2.5. In Russian.

    Google Scholar 

  • G. Calcagni, 2010, Quantum Field Theory, Gravity and Cosmology in a Fractal Universe, E-print: arXiv: 1001.0571.

    Google Scholar 

  • A. Carpinteri, F. Mainardi (Eds.), 1997, Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York.

    MATH  Google Scholar 

  • R.M. Christensen, 2005, Mechanics of Composite Materials, Dover, New York.

    Google Scholar 

  • J.C. Collins, 1984, Renormalization, Cambridge University Press, Cambridge. Section 4.

    Book  MATH  Google Scholar 

  • G.A. Edgar, 1990, Measure, Topology, and Fractal Geometry, Springer, New York.

    MATH  Google Scholar 

  • G. Eyink, 1989, Quantum field theory models on fractal space-time. I: Introduction and overview, Communication in Mathematical Physics, 125, 613–636.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • K.F. Falconer, 1985, The Geometry of Fractal Sets, Cambridge University Press, Cambridge.

    Book  MATH  Google Scholar 

  • K.F. Falconer, 1990, Fractal Geometry. Mathematical Foundations and Applications, Wiley, Chichester, New York.

    MATH  Google Scholar 

  • J. Feder, 1988, Fractals, Plenum Press, New York, London.

    MATH  Google Scholar 

  • H. Federer, 1969, Geometric Measure Theory, Springer, Berlin.

    MATH  Google Scholar 

  • M. Frame, B. Mandelbrot, N. Neger, 2006, Fractal Geometry, http://classes.yale.edu/fractals

    Google Scholar 

  • R. Gorenflo, 1998, Afterthoughts on interpretation of fractional derivatives and integrals, in: P. Rusev, I. Dimovski, V. Kiryakova, (Eds.), Transform Methods and Special Functions, Varna, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, 589–591.

    Google Scholar 

  • F. Hausdorff, 1919, Dimension und ausseres Mass, Mathematische Annalen, 79, 157–179. In German.

    Article  MathSciNet  Google Scholar 

  • Xing-Fei He, 1990, Fractional dimensionality and fractional derivative spectra of interband optical transitions, Physical Review B, 42, 11751–11756.

    Article  ADS  Google Scholar 

  • Xing-Fei He, 1991, Excitons in anisotropic solids: The model of fractional-dimensional space, Physical Review B, 43, 2063–2069.

    Article  ADS  Google Scholar 

  • N. Heymans, I. Podlubny, 2006, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheo-logica Acta, 45, 765–771. E-print: math-ph/0512028.

    Article  Google Scholar 

  • R. Hilfer (Ed.), 2000, Applications of Fractional Calculus in Physics, World Scientific Publishing, Singapore.

    MATH  Google Scholar 

  • A.J. Katz, A.H. Thompson, 1985, Fractal sandstone pores: implications for conductivity and pore formation, Physical Review Lettetrs, 54, 1325–1328.

    Article  ADS  Google Scholar 

  • S. Kempfle, I. Schafer, 2000, Fractional differential equations and initial conditions, Fractional Calculus and Applied Analysis, 3, 387–400.

    MathSciNet  MATH  Google Scholar 

  • A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.

    MATH  Google Scholar 

  • V. Kiryakova, 1994, Generalized Fractional Calculus and Applications, Longman, Harlow and Wiley, New York.

    MATH  Google Scholar 

  • C. Kittel, 2004, Introduction to Solid State Physics, 8th ed., Wiley, New York.

    Google Scholar 

  • H. Kröger, 2000, Fractal geometry in quantum mechanics, field theory and spin systems, Physics Reports, 323, 81–181.

    Article  MathSciNet  Google Scholar 

  • J. Kugami, 2001, Analysis on Fractals, Cambridge University Press, Cambridge.

    Book  Google Scholar 

  • M.I. Kulak, 2002, Fractal Mechanics of Materials, Visheishaya Shkola, Minsk. In Russian.

    Google Scholar 

  • J.A.T. Machado, 2003, A probabilistic interpretation of the fractional-order differentiation, Fractional Calculus and Applied Analysis, 6, 73–80.

    MathSciNet  MATH  Google Scholar 

  • F. Mainardi, 2010, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific Publishing, Singapore.

    Book  MATH  Google Scholar 

  • F. Mainardi, 1998, Considerations on fractional calculus: interpretations and applications, in: P. Rusev, I. Dimovski, V. Kiryakova (Eds.), Transform Methods and Special Functions, Varna 1996, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, 594–597.

    Google Scholar 

  • B. Mandelbrot, 1983, The Fractal Geometry of Nature, Freeman, New York.

    Google Scholar 

  • A.C. McBride, 1979, Fractional Calculus and Integral Transforms of Generalized Functions, Pitman Press, San Francisco.

    MATH  Google Scholar 

  • A.C. McBride, 1986, Fractional Calculus, Halsted Press, New York.

    Google Scholar 

  • R. Metzler, J. Klafter, 2000, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339, 1–77.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • R. Metzler, J. Klafter, 2004, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Journal of Physics A, 37, R161–R208.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • K.S. Miller, B. Ross, 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.

    MATH  Google Scholar 

  • M. Monsrefi-Torbati, J.K. Hammond, 1998, Physical and geometrical interpretation of fractional operators, Journal of The Franklin Institute B, 335, 1077–1086.

    Article  Google Scholar 

  • E.W. Montroll, M.F. Shlesinger, 1984, The wonderful world of random walks, In: Studies in Statistical Mechanics, Vol.11, J. Lebowitz, E. Montroll (Eds.), North-Holland, Amsterdam, 1–121.

    Google Scholar 

  • A.M. Nahushev, 2003, Fractional Calculus ans its Appication, Nauka, Moscow. In Russian.

    Google Scholar 

  • G.R. Newkome, P. Wang, C.N. Moorefield, T.J. Cho, P.P. Mohapatra, S. Li, S.H. Hwang, O. Lukoyanova, L. Echegoyen, J.A. Palagallo, V. Iancu, S.W. Hla, 2006, Nanoassembly of a fractal polymer: A molecular Sierpinski hexagonal gasket, Science, 312, 1782–1785.

    Article  MathSciNet  ADS  Google Scholar 

  • R.R. Nigmatullin, 1992, Fractional integral and its physical interpretation, Theoretical and Mathematical Physics, 90, 242–251.

    Article  MathSciNet  ADS  Google Scholar 

  • K. Nishimoto, 1989, Fractional Calculus: Integrations and Differentiations of Arbitrary Order, University of New Haven Press, New Haven.

    MATH  Google Scholar 

  • K.B. Oldham, J. Spanier, 1974, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York.

    MATH  Google Scholar 

  • C. Palmer, P.N. Stavrinou, 2004, Equations of motion in a non-integer-dimensional space, Journal of Physics A, 37, 6987–7003.

    Article  MathSciNet  MATH  Google Scholar 

  • I. Podlubny, 1996, Fractional Differential Equations, Academic Press, New York.

    Google Scholar 

  • I. Podlubny, 2002, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis, 5, 367–386.

    MathSciNet  MATH  Google Scholar 

  • A.A. Potapov, 2005, Fractals in Radiophysics and Radiolocation, 2nd ed., Universitetskaya Kniga, Moscow. In Russian.

    Google Scholar 

  • A.V. Pshu, 2005, Equations with Partial Derivatives of Fractional Order, Nauka, Moscow. In Russian.

    Google Scholar 

  • F.Y Ren, J.R. Liang, X.T. Wang, W.Y Qiu, 2003, Integrals and derivatives on net fractals, Chaos, Solitons and Fractals, 16, 107–117.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • F.-Y. Ren, Z.-G. Yu, F. Su, 1996, Fractional integral associated to the self-similar set or the generalized self-similar set and its physical interpretation, Physics Letters A, 219, 59–68.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • F.Y. Ren, Z.-G. Yu, J. Zhou, A. Le Mehaute, R.R. Nigmatullin, 1997, The relationship between the fractional integral and the fractal structure of a memory set, Physica A, 246, 419–429.

    Article  ADS  Google Scholar 

  • C.A. Rogers, 1998, Hausdorff Measures, 2nd ed., Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • B. Ross, 1975, A brief history and exposition of the fundamental theory of fractional calculus, Lecture Notes in Mathematics, 457, 1–36.

    Google Scholar 

  • B. Rubin, 1996, Fractional Integrals and Potentials, Longman, Harlow.

    MATH  Google Scholar 

  • R.S. Rutman, 1994, On the paper by R.R. Nigmatullin Fractional integral and its physical interpretation, Theoretical and Mathematical Physics, 100, 1154–1156.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • R.S. Rutman, 1995, On physical interpretations of fractional integration and differentiation, Theoretical and Mathematical Physics, 105, 1509–1519.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • J. Sabatier, O.R Agrawal, J.A. Tenreiro Machado, (Eds.), 2007, Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht.

    MATH  Google Scholar 

  • S.G. Samko, A.A. Kilbas, O.I. Marichev, 1993, Integrals and Derivatives of Fractional Order and Applications, Nauka i Tehnika, Minsk, 1987, in Russian; and Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York, 1993.

    Google Scholar 

  • J. Schweizer, B. Frank, 2002, Calculus on linear Cantor sets, Archiv der Mathematik, 79, 46–50.

    Article  MathSciNet  MATH  Google Scholar 

  • H.M. Srivastava, S. Owa, (Eds.), 1989, Univalent Functions, Fractional Calculus and Their Applications, Prentice Hall, New Jersey.

    MATH  Google Scholar 

  • A.A. Stanislavsky, 2004, Probability interpretation of the integral of fractional order, Theoretical and Mathematical Physics, 138, 418–431.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • A.A. Stanislavsky, K. Weron, 2002, Exact solution of averaging procedure over the Cantor set, Physica A, 303, 57–66.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • F.H. Stillinger, 1977, Axiomatic basis for spaces with noninteger dimensions, Journal of Mathematical Physics, 18, 1224–1234.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • R.S. Strichartz, 1999, Analysis on fractals, Notices of the American Mathematical Society, 46, 1199–1208.

    MathSciNet  MATH  Google Scholar 

  • R.S. Strichartz, 2006, Differential Equations on Fractals, Princeton Press, Princeton.

    MATH  Google Scholar 

  • K. Svozil, 1987, Quantum field theory on fractal spacetime: a new regularization method, Journal of Physics A, 20, 3861–3875.

    Article  MathSciNet  Google Scholar 

  • V.E. Tarasov, 2004, Fractional generalization of Liouville equations, Chaos, 14, 123–127.

    Article  ADS  Google Scholar 

  • V.E. Tarasov, 2005a, Continuous medium model for fractal media, Physics Letters A, 336, 161–114.

    Article  ADS  Google Scholar 

  • V.E. Tarasov, 2005b, Fractional hydrodynamic equations for fractal media, Annals of Physics, 318, 286–307.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • F.B. Tatom, 1995, The relationship between fractional calculus and fractals, Fractals, 3, 217–229.

    Article  MathSciNet  MATH  Google Scholar 

  • V.V. Uchaikin, 2008, Method of Fractional Derivatives, Artishok, Ulyanovsk. In Russian.

    Google Scholar 

  • P.S. Uryson, 1951, Works on Topology and Other Fields of Mathematics Vol. 1–2, Gostehizdat, Moscow. In Russian.

    Google Scholar 

  • B.J. West, M. Bologna, P. Grigolini, 2003, Physics of Fractal Operators, Springer, New York.

    Google Scholar 

  • Z.-G. Yu, F.-Y Ren, J. Zhou, 1997, Fractional integral associated to generalized cookie-cutter set and its physical interpretation, Journal of Physics A, 30, 5569–5577.

    Article  MathSciNet  MATH  Google Scholar 

  • G.M. Zaslavsky, 2002, Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371, 461–580.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • G.M. Zaslavsky, 2005, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford.

    MATH  Google Scholar 

  • V.V. Zverev, 1996, On conditions of existence of fractal domain integrals, Theoretical and Mathematical Physics, 107, 419–426.

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992, Moscow, Russia

    Vasily E. Tarasov

Authors
  1. Vasily E. Tarasov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Tarasov, V.E. (2010). Fractional Integration and Fractals. In: Fractional Dynamics. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14003-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation