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Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


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Online
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1314-2224
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Band 18, Heft 1

Hefte

Several Results of Fractional Derivatives in Dʹ(R+)

Chenkuan Li
Online erschienen: 10.02.2015 | DOI: https://doi.org/10.1515/fca-2015-0013

Abstract

In this paper, we define fractional derivative of arbitrary complex order of the distributions concentrated on R+, based on convolutions of generalized functions with the supports bounded on the same side. Using distributional derivatives, which are generalizations of classical derivatives, we present a few interesting results of fractional derivatives in D'(R+), as well as the symbolic solution for the following differential equation by Babenko's method

y(x)+λΓ(-α)0xy(σ)(x-σ)α+1dσ=δ(x),

where Re α>0

MSC 2010: 46F10; 26A33

Key Words and Phrases: distribution; convolution; Dirac delta function; Abel's equation; Gamma function; Caputo derivative and Riemann-Liouville derivative

References

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  • [2] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier, New York (2006).Google Scholar

  • [3] S. B. Yuste and L. Acedo, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42 (2005), 1862-1874.Google Scholar

  • [4] C. P. Li and Z. G. Zhao, Introduction to fractional integrability and differentiability. Euro. Phys. J. - Special Topics 193, (2011), 5-26.Google Scholar

  • [5] I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. I. Academic Press, New York (1964).Google Scholar

  • [6] C. K. Li, Asymptotic expressions of several dsitributions on the sphere. British J. of Mathematics and Computer Science 3 (2013), 73-85.CrossrefGoogle Scholar

  • [7] C. K. Li and M. A. Aguirre, The distributional products on spheres and Pizzetti's formula. J. Comput. Appl. Math. 235 (2011), 1482-1489.Web of ScienceGoogle Scholar

  • [8] M. Aguirre and C. K. Li, The distributional products of particular distributions. Appl. Math. Comput 187 (2007), 20-26.Web of ScienceGoogle Scholar

  • [9] I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).Google Scholar

  • [10] H. M. Srivastava and R. G. Buschman, Theory and Applications of Convolution Integral Equations. Kluwer Academic Publishers, Dordrecht-Boston-London (1992).Google Scholar

Artikelinformationen

Erhalten: 18.06.2014

Online erschienen: 10.02.2015


Quellenangabe: Fractional Calculus and Applied Analysis, Band 18, Heft 1, Seiten 192–207, ISSN (Online) 1314-2224, DOI: https://doi.org/10.1515/fca-2015-0013.

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