From the hyper-Bessel operators of Dimovski to the generalized fractional calculus

By Virginia S. Kiryakova. 2014

56 citations recorded by the source corpus.

In 1966 Ivan Dimovski introduced and started detailed studies on the Bessel type differential operators B of arbitrary (integer) order m ≥ 1. He also suggested a variant of the Obrechkoff integral transform (arising in a paper of 1958 by another Bulgarian mathematician Nikola Obrechkoff) as a Laplace-type transform basis of a corresponding operational calculus for B and for its linear right inverse integral operator L. Later, the developments on these linear singular differential operators appearing in many problems of mathematical physics, have been continued by the author of this survey who called them hyper-Bessel differential operators, in relation to the notion of hyper-Bessel functions of Delerue (1953), shown to form a fundamental system of solutions of the IVPs for By(t) = λy(t). We have been able to extend Dimovski’s results on the hyper-Bessel operators and on the Obrechkoff transform due to the happy hint to attract the tools of the special functions as Meijer’s G-function and Fox’s H-function to handle successfully these matters. These author’s studies have lead to the introduction and development of a theory of generalized fractional calculus (GFC) in her monograph (1994) and subsequent papers, and to various applications of this GFC in other topics of analysis, differential equations, special functions and integral transforms.Here we try briefly to expose the ideas leading to this GFC, its basic facts and some of the mentioned applications.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/f44df187697975a80a2f68341dbebef9fd9b43db

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In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

Models of dielectric relaxation based on completely monotone functions (2016, methodology): “This is the case, for instance, of the model proposed in 1999 by Raicu [103], which is not treated in this survey since it can be reverted in a special case of the HN model, or the model based on fractional order operators of hyper-Bessel type (for details one can see in [69]), and still having CM r”