Skip to main content

A Fruitful Interplay: From Nonlocality to Fractional Calculus

  • Conference paper
Nonlinear Waves: Classical and Quantum Aspects

Part of the book series: NATO Science Series II: Mathematics, Physics and Chemistry ((NAII,volume 153))

Abstract

The Fractional Calculus represents a natural instrument to model nonlocal phenomena either in space or time. From Physics and Chemistry to Biology, there are many processes that involve different space/time scales. In many problems of the above context the dynamics of the system can be formulated by fractional differential equations which include the nonlocal effects. We give a panoramic view of the problem and show some examples.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

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

  1. S.S. Schweber, An Introduction to Relativistic Quantum Field Theory. Ed. Harper and Row (1964).

    Google Scholar 

  2. P. I. Naumkin and I.A. Shishmarev, Nonlinear Nonlocal Equations in the Theory of Waves. A.M.S. Vol. 133 (1994).

    Google Scholar 

  3. L. Vázquez, W.A.B. Evans and G. Rickayzen, Phys. Lett. A 189, 454–459 (1994)

    Article  ADS  Google Scholar 

  4. M.D. Cunha, V.V. Konotop and L. Vázquez, Phys. Lett. A 221, 317 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. G.L. Alfimov and V.P. Silin, J. Exp. Theor. Phys. 79, 369 (1994).

    ADS  Google Scholar 

  6. G.L. Alfimov and V.P. Silin, J. Exp. Theor. Phys. 81, 915 (1995).

    ADS  Google Scholar 

  7. G.L. Alfimov, D. Usero and L. Vázquez, J. Phys. A: Math. Gen. 33, 6707 (2000).

    Article  ADS  MATH  Google Scholar 

  8. B. Ross (Ed.) Fractional Calculus and its Applications. Lecture Notes in Mathematics 457. Springer-Verlag (1975).

    Google Scholar 

  9. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach (1993).

    Google Scholar 

  10. F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics. Eds. A. Carpinteri and F. Mainardi, CISM Courses and Lectures 378, Springer-Verlag, 291–348 (1997).

    Google Scholar 

  11. G. Turchetti, D. Usero and L. Vázquez, Tamsui Oxford J. Math. Sciences 18, 31 (2002).

    MATH  Google Scholar 

  12. D. Usero and L. Vázquez, Localization and Energy Transfer in Nonlinear Systems. Eds. L. Vázquez, R.S. MacKay and M.P. Zorzano, World Scientific, 296–303 (2003).

    Google Scholar 

  13. R.T. Baillie and M.L. King (Eds.), Fractional Differencing and Long Memory Processes. J. Econometrics 73, 1 (1996)

    Google Scholar 

  14. D. Helbing, H.J. Herrmann, M. Schreckenberg and D.E. Wolf (Eds.), Traffic and Granular Flow 99, Springer Verlag (2000).

    Google Scholar 

  15. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific (2000).

    Google Scholar 

  16. G. Horneck, Christa Baumtark-Khan (Eds.), Astrobiology: The Quest for the Conditions of Life, Springer Verlag (2002).

    Google Scholar 

  17. K. Morinaga and T. Nono, J. Sci. Hiroshima Univ.(A) 16, 13–41 (1952).

    MathSciNet  MATH  Google Scholar 

  18. L. Vázquez, J. Comp. Math. 21, 491 (2003)

    MATH  Google Scholar 

  19. L. Vázquez and R. Vilela-Mendes, Applied Math. and Comp. 141, 125 (2003)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Kluwer Academic Publishers

About this paper

Cite this paper

Vázquez, L. (2004). A Fruitful Interplay: From Nonlocality to Fractional Calculus. In: Abdullaev, F.K., Konotop, V.V. (eds) Nonlinear Waves: Classical and Quantum Aspects. NATO Science Series II: Mathematics, Physics and Chemistry, vol 153. Springer, Dordrecht. https://doi.org/10.1007/1-4020-2190-9_10

Download citation

Publish with us

Policies and ethics