Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
W. Feller, On a generalization of Marcel Riesz’ potentials and the semigroups generated by them. Comm. Sem. Mathem. Universite de Lund (1952), 72–81.
R. Herrmann, Fractional Calculus — An Introduction for Physicists. World Scientific Publishing, Singapore (2011).
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).
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.
H. Kober, On fractional integrals and derivatives. Quarterly J. of Mathematics (Oxford Ser.) 11, No 1 (1940), 193–211.
J. Liouville, Sur le calcul des differentielles á indices quelconques. J. École Polytechn. 13 (1832), 1–162.
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press & World Sci., London - Singapore (2010).
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).
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; http://link.springer.com/article/10.2478/s13540-012-0049-5.
G. Pagnini, Erdelyi-Kober fractional diffusion. Fract. Calc. Appl. Anal. 15, No 1 (2012), 117–127; DOI: 10.2478/s13540-012-0008-1; http://link.springer.com/article/10.2478/s13540-012-0008-1.
V.V. Pashkevich, On the asymmetric deformation of fissioning nuclei. Nucl. Phys. 169 (1971), 275–293; doi:10.1016/0375-9474(71)90884-0.
I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5, No 4 (2002), 367–386; http://www.math.bas.bg/~fcaa/; and Corrections to Figure 4 in: Fract. Calc. Appl. Anal. 6, No 1 (2003), 109–110.
M. Riesz, L’integrale de Riemann-Liouville et le probléme de Cauchy. Acta Math. 81 (1949), 1–223.
S.G. Samko, Fractional integration and differentiation of variable order. Anal. Math. 21 (1995), 213–236.
I.N. Sneddon, Mixed Boundary Value Problems in Potential Theory. North-Holland Publ. Co., Amsterdam (1966).
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.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Herrmann, R. Towards a geometric interpretation of generalized fractional integrals — Erdélyi-Kober type integrals on R N, as an example. fcaa 17, 361–370 (2014). https://doi.org/10.2478/s13540-014-0174-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s13540-014-0174-4