Virginia Kiryakova

Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus

The classical Mittag{Leer (M{L) functions have already proved their eciency as solutions of fractional-order differential and integral equations and thus have become important elements of the fractional calculus’ theory and applications. In this paper we introduce analogues of these functions, depending on two sets of multiple (m-tuple, m> 2i s an integer) indices. The hint for this comes from a paper by Dzrbashjan (Izv. AN Arm. SSR 13 (3) (1960) 21{63) related to the case m= 2. We study the basic properties and the relations of the multiindex M{L functions with the operators of the generalized fractional calculus. Corresponding generalized operators of integration and dierention of the so-called Gelfond{Leontiev-type, as well as Borel{Laplace-type integral transforms, are also introduced and studied. c 2000 Elsevier Science B.V. All rights reserved. MSC: 26A33; 33C60; 33E30; 44A15; 44A30

The special functions of fractional calculus as generalized fractional calculus operators of some basic functions

Abstract We propose a unified approach to the so-called Special Functions of Fractional Calculus (SFs of FC), recently enjoying increasing interest from both theoretical mathematicians and applied scientists. This is due to their role as solutions of fractional order differential and integral equations, as the better mathematical models of phenomena of various physical, engineering, automatization, chemical, biological, Earth science, economics etc. nature. Our approach is based on the use of Generalized Fractional Calculus (GFC) operators. Namely, we show that all the Wright generalized hypergeometric functions (W.ghf-s) Ψ q p ( z ) can be represented as generalized fractional integrals, derivatives or differ-integrals of three basic simpler functions as cos q − p ( z ) , exp ( z ) and Ψ 0 1 ( z ) (reducible in particular to the elementary function z α ( 1 − z ) β , the Beta-distribution), depending on whether p q , p = q or p = q + 1 and on the values of their indices and parameters. In this way, the SFs of FC can be separated into three classes with similar behaviour, and also new integral and differential formulas can be derived, useful for computational procedures.

Some pioneers of the applications of fractional calculus

In the last decades fractional calculus (FC) became an area of intensive research and development. This paper goes back and recalls important pioneers that started to apply FC to scientific and engineering problems during the nineteenth and twentieth centuries. Those we present are, in alphabetical order: Niels Abel, Kenneth and Robert Cole, Andrew Gemant, Andrey N. Gerasimov, Oliver Heaviside, Paul Lévy, Rashid Sh. Nigmatullin, Yuri N. Rabotnov, George Scott Blair.

Fractional Calculus: Quo Vadimus? (Where Are We Going?)

Abstract Discussions under this title were held during a special session in frames of the International Conference “Fractional Differentiation and Applications” (ICFDA ’14) held in Catania (Italy), 23-25 June 2014, see details at http://www.icfda14.dieei.unict.it/. Along with the presentations made during this session, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend. The intention of this special session was to continue the useful traditions from the first conferences on the Fractional Calculus (FC) topics, to pose open problems, challenging hypotheses and questions “where to go”, to discuss them and try to find ways to resolve.

The mellin integral transform in fractional calculus

In Fractional Calculus (FC), the Laplace and the Fourier integral transforms are traditionally employed for solving different problems. In this paper, we demonstrate the role of the Mellin integral transform in FC. We note that the Laplace integral transform, the sin- and cos-Fourier transforms, and the FC operators can all be represented as Mellin convolution type integral transforms. Moreover, the special functions of FC are all particular cases of the Fox H-function that is defined as an inverse Mellin transform of a quotient of some products of the Gamma functions.In this paper, several known and some new applications of the Mellin integral transform to different problems in FC are exemplarily presented. The Mellin integral transform is employed to derive the inversion formulas for the FC operators and to evaluate some FC related integrals and in particular, the Laplace transforms and the sin- and cos-Fourier transforms of some special functions of FC. We show how to use the Mellin integral transform to prove the Post-Widder formula (and to obtain its new modi-fication), to derive some new Leibniz type rules for the FC operators, and to get new completely monotone functions from the known ones.

Explicit solutions of fractional integral and differential equations involving Erdélyi-Kober operators

By means of fractional calculus techniques we find explicit solutions of Volterra integral equations of second kind and fractional differential equations, involving Erdelyi-Kober fractional integrals or derivatives. We use the transmutation method to reduce the solutions of these equations to known solutions of simpler (Riemann-Liouville) equations of the same type. Some examples are given.

Further results on a family of generalized radiation integrals

In this paper we continue the investigation by Kalla et al. (1991) of the family of generalized radiation integrals defined by Ia, b, p, λ, μα, β γ=σα4π∫0bxλ(x2+p)−α1−x2b2μ2F1α, β, γ;−a2x2+pdx , where Re(γ) >Re(β) > 0; −1 −1; μ > −1; p, a, b > 0; 0 < a ⩽ b < ∞. Several recurr ence relations are presented. By differentiation of these integrals with respect to the parameters λ and μ we obtain also various integrals that include the logarithmic function in the integrand. Finally, we propose an algorithm for numerical evaluation of the generalized radiation integrals and illustrate it by tables of their values computed for selected values of the parameters.

The Chronicles of Fractional Calculus

Abstract Since the 60s of last century Fractional Calculus exhibited a remarkable progress and presently it is recognized to be an important topic in the scientific arena. This survey analyzes and measures the evolution that occurred during the last five decades in the light of books, journals and conferences dedicated to the theory and applications of this mathematical tool, dealing with operations of integration and differentiation of arbitrary (fractional) order and their generalizations.

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

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.

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators

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 (δ1,…,δm) 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.