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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Feng, Jonathan

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Volume 11, Issue 10 (Oct 2013)

Issues

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators

Virginia Kiryakova
  • Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, “Acad. G. Bontchev” Str., Block 8, 1113, Sofia, Bulgaria
  • Email:
/ Yuri Luchko
  • Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences, Luxemburger Str. 10, D - 13353, Berlin, Germany
  • Email:
Published Online: 2013-12-19 | DOI: https://doi.org/10.2478/s11534-013-0217-1

Abstract

In this paper some generalized operators of Fractional Calculus (FC) are investigated that are useful in modeling various phenomena and systems in the natural and human sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (FO) differential equations. We start, as a background, with an overview of the Riemann-Liouville and Caputo derivatives and the Erdélyi-Kober operators. Then the multiple Erdélyi-Kober fractional integrals and derivatives of R-L type of multi-order (δ 1,…,δ m) are introduced as their generalizations. Further, we define and investigate in detail the Caputotype multiple Erdélyi-Kober derivatives. Several examples and both known and new applications of the FC operators introduced in this paper are discussed. In particular, the hyper-Bessel differential operators of arbitrary order m > 1 are shown as their cases of integer multi-order. The role of the so-called special functions of FC is emphasized both as kernel-functions and solutions of related FO differential equations.

Keywords: fractional calculus; operators of Riemann-Liouville and Caputo type; Erdélyi-Kober operators; special functions; integral transforms; Cauchy problems

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About the article

Published Online: 2013-12-19

Published in Print: 2013-10-01



Citation Information: Open Physics, ISSN (Online) 2391-5471, DOI: https://doi.org/10.2478/s11534-013-0217-1. Export Citation

© 2013 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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