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Journal of Mathematical Sciences

, Volume 89, Issue 5, pp 1570–1575 | Cite as

The fractal nature of the asymptotic solution of an integral equation with stable (in Lévy's sense) kernel

  • V. A. Slobodenyuk
  • V. V. Uchaikin
Article

Abstract

We consider the asymptotic behavior of the solution\(n (\vec r)\) of a nonhomogeneous integral equation for a threedimensional density with a right-hand side and kernel qp(α), where\((\vec r)\) is the density of a stable (in Lévy's sense) distribution. We demonstrate that the solution is of an asymptotically fractal nature, i.e.
$$n (\vec r) \sim Br^{D - 3} , 0< D< 3,$$
for q=1, α<2, and the fractal dimensionality D coincides with the characteristic of the stable law α.

Keywords

Integral Equation Asymptotic Behavior Asymptotic Solution Fractal Nature Fractal Dimensionality 
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.
    Ph. J. E. Peebles,The Large-Scale Structure of the Universe, Princeton Univ. Press, Princeton (1980).Google Scholar
  2. 2.
    V. M. Zolotarev,One-Dimensional Stable Distributions, AMS, Providence (1986).Google Scholar
  3. 3.
    A. F. Nikiforov and V. B. Uvarov,Special Functions of Mathematical Physics [in Russian], Nauka, Moscow (1978).Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • V. A. Slobodenyuk
    • 1
  • V. V. Uchaikin
    • 1
  1. 1.Ulyanovsk State UniversityUlyanovskRussia

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