Fractional Calculus and Applied Analysis

, Volume 17, Issue 2, pp 361–370 | Cite as

Towards a geometric interpretation of generalized fractional integrals — Erdélyi-Kober type integrals on R N , as an example

  • Richard Herrmann
Discussion Survey


A family of generalized Erdélyi-Kober type fractional integrals is interpreted geometrically as a distortion of the rotationally invariant integral kernel of the Riesz fractional integral in terms of generalized Cassini ovaloids on R N . Based on this geometric point of view, several extensions are discussed.

Key Words and Phrases

fractional calculus Riesz fractional integrals Erdélyi-Kober fractional integrals generalized fractional calculus Cassini ovaloids 


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  1. [1]
    A. Erdélyi, On fractional integration and its application to the theory of Hankel transforms. The Quarterly J. of Mathematics (Oxford), Second Ser., 11 (1940), 293–303.CrossRefGoogle Scholar
  2. [2]
    W. Feller, On a generalization of Marcel Riesz’ potentials and the semigroups generated by them. Comm. Sem. Mathem. Universite de Lund (1952), 72–81.Google Scholar
  3. [3]
    R. Herrmann, Fractional Calculus — An Introduction for Physicists. World Scientific Publishing, Singapore (2011).CrossRefzbMATHGoogle Scholar
  4. [4]
    V.S. Kiryakova, Generalized Fractional Calculus and Applications. Longman (Pitman Res. Notes in Math. Ser. 301), Harlow; Co-publ.: John Wiley and Sons, New York (1994).zbMATHGoogle Scholar
  5. [5]
    V. Kiryakova, A long standing conjecture failed? In: Transform Methods & Special Functions’, Varna’ 96 (Proc. 2nd Internat. Workshop), Inst. Math. Inform. — Bulg. Acad. Sci., Sofia (1998), 584–593.Google Scholar
  6. [6]
    H. Kober, On fractional integrals and derivatives. Quarterly J. of Mathematics (Oxford Ser.) 11, No 1 (1940), 193–211.CrossRefMathSciNetGoogle Scholar
  7. [7]
    J. Liouville, Sur le calcul des differentielles á indices quelconques. J. École Polytechn. 13 (1832), 1–162.Google Scholar
  8. [8]
    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press & World Sci., London - Singapore (2010).CrossRefzbMATHGoogle Scholar
  9. [9]
    J.C. Maxwell, On the description of oval curves, and those having a plurality of foci (focus geometry). (Proc.) Royal Society of Edinburgh 2 (1846); Reprinted in: The Scientific Letters and Papers of James Clerk Maxwell: 1846–1862, Cambridge University Press, UK (1990).Google Scholar
  10. [10]
    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; Scholar
  11. [11]
    G. Pagnini, Erdelyi-Kober fractional diffusion. Fract. Calc. Appl. Anal. 15, No 1 (2012), 117–127; DOI: 10.2478/s13540-012-0008-1; Scholar
  12. [12]
    V.V. Pashkevich, On the asymmetric deformation of fissioning nuclei. Nucl. Phys. 169 (1971), 275–293; doi:10.1016/0375-9474(71)90884-0.CrossRefGoogle Scholar
  13. [13]
    I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5, No 4 (2002), 367–386;; and Corrections to Figure 4 in: Fract. Calc. Appl. Anal. 6, No 1 (2003), 109–110.zbMATHMathSciNetGoogle Scholar
  14. [14]
    M. Riesz, L’integrale de Riemann-Liouville et le probléme de Cauchy. Acta Math. 81 (1949), 1–223.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    S.G. Samko, Fractional integration and differentiation of variable order. Anal. Math. 21 (1995), 213–236.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    I.N. Sneddon, Mixed Boundary Value Problems in Potential Theory. North-Holland Publ. Co., Amsterdam (1966).zbMATHGoogle Scholar
  17. [17]
    I.N. Sneddon, The use in mathematical analysis of Erdélyi-Kober operators and some of their applications. In: Lecture Notes in Math. 457 (1975), 37–79 (Proc. Intern. Conf. on Fractional Calculus Held in New Haven, 1974), Springer-Verlag, N. York.CrossRefMathSciNetGoogle Scholar

Copyright information

© Versita Warsaw and Springer-Verlag Wien 2014

Authors and Affiliations

  1. 1.GigaHedronDreieichGermany

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