Anatoly A. Kilbas

H-Transforms : Theory and Applications

Definition, Representations and Expansions of the H-Function. Properties of the H-Function H-Transform on the Space Ln,2. H-Transform on the Space L n,t Modified H-Transforms on the Space L n,t. G-Transform and Modified G-Transforms on the Space L n,t. Hypergeometric Type Integral Transforms on the Space L n,t. Bessel Type Integral Transforms on the Space L n,t. Bibliography Subject Index Author Index Symbol Index.

Mittag-Leffler Functions, Related Topics and Applications

As a result of researchers and scientists increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have recently caught the interest of the scientific community. Focusing on the theory of the Mittag-Leffler functions, the present volume offers a self-contained, comprehensive treatment, ranging from rather elementary matters to the latest research results. In addition to the theory the authors devote some sections of the work to the applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular the Mittag-Leffler functions allow us to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and its successors. The book is intended for a broad audience, comprising graduate students, university instructors and scientists in the field of pure and applied mathematics, as well as researchers in applied sciences like mathematical physics, theoretical chemistry, bio-mathematics, theory of control and several other related areas.

GENERALIZED MITTAG-LEER FUNCTION AND GENERALIZED FRACTIONAL CALCULUS OPERATORS

The paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mittag-Leffler function E γ ρ,μ(z) in the kernels. Similar assertions are presented for the integral operators containing the Mittag-Leffler and Kummer functions, E ρ,μ(z) and Φ(γ, μ; z), in the kernels, and applications are given to obtain solutions in closed form of the integral equations of the first kind.

Mellin transform analysis and integration by parts for Hadamard-type fractional integrals

Abstract This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard, in the space Xpc of Lebesgue measurable functions f on R + =(0,∞) such that ∫ 0 ∞ u c f(u) p du u ess sup u>0 u c f(u) for c∈ R =(−∞,∞) , in particular in the space Lp(0,∞) (1⩽p⩽∞). Formulas for the Mellin transforms of the four Hadamard-type fractional integral operators are established as well as relations of fractional integration by parts for them.

Fractional calculus in the Mellin setting and Hadamard-type fractional integrals

Abstract The purpose of this paper and some to follow is to present a new approach to fractional integration and differentiation on the half-axis R + =(0,∞) in terms of Mellin analysis. The natural operator of fractional integration in this setting is not the classical Liouville fractional integral Iα0+f but J α 0+,c f (x):= 1 Γ(α) ∫ 0 x u x c log x u α−1 f(u) du u (x>0) for α>0, c∈ R . The Mellin transform of this operator is simply (c−s) −α M [f](s) , for s=c+it, c,t∈ R . The Mellin transform of the associated fractional differentiation operator D α 0+,c f is similar: (c−s) α M [f](s) . The operator D α 0+,c f may even be represented as a series in terms of xkf(k)(x), k∈ N 0 , the coefficients being certain generalized Stirling functions Sc(α,k) of second kind. It turns out that the new fractional integral J α 0+,c f and three further related ones are not the classical fractional integrals of Hadamard (J. Mat. Pure Appl. Ser. 4, 8 (1892) 101–186) but far reaching generalizations and modifications of these. These four new integral operators are first studied in detail in this paper. More specifically, conditions will be given for these four operators to be bounded in the space Xcp of Lebesgue measurable functions f on (0,∞), for c∈(−∞,∞), such that ∫ ∞ 0 |u c f(u)| p du/u for 1⩽p ess sup u>0 [u c |f(u)|] for p=∞, in particular in the space Lp(0,∞) for 1⩽p⩽∞. Connections of these operators with the Liouville fractional integration operators are discussed. The Mellin convolution product in the above spaces plays an important role.

Compositions of Hadamard-type fractional integration operators and the semigroup property

Abstract This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard in the space Xcp of functions f on R + =(0,∞) such that ∫ 0 ∞ u c f(u) p du u ess sup u>0 u c |f(u)| for c∈ R =(−∞,∞) , in particular in the space Lp(0,∞) (1⩽p⩽∞). The semigroup property and its generalizations are established for the generalized Hadamard-type fractional integration operators under consideration. Conditions are also given for the boundedness in Xcp of these operators; they involve Kummer confluent hypergeometric functions as kernels.

On the generalized mittag-leffler type functions

The paper is devoted to the study of the properties of the special functions generalizing the Mittag-Leffler type functions. The order and type of such entire functions are evaluated and recurrence relations are given. Connections with hypergeometric functions are discussed and differentiation formulae are proved.

On Mittag-Leffler type function and applications

This is the continuation of the paper [4] which was devoted to introduce a special entire function named a Mittag -Leffler type function E α,m,l (z) to discuss its connections with the Riemnn Loiuville fractional integral and derivatives and to solve the linear Abel-Volterra integral equation. Here we construct explicit soliutions of special differential equations of fractional order and as their consequences, of ordinary differential equations. Examples are also exhibited.

Multi-parametric mittag-leffler functions and their extension

This paper surveys one of the last contributions by the late Professor Anatoly Kilbas (1948–2010) and research made under his advisorship. We briefly describe the historical development of the theory of the discussed multi-parametric Mittag-Leffler functions as a class of the Wright generalized hypergeometric functions. The method of the Mellin-Barnes integral representations allows us to extend the considered functions to the case of arbitrary values of parameters. Thus, the extended Mittag-Leffler-type functions appear. The properties of these special functions and their relations to the fractional calculus are considered. Our results are based mainly on the properties of the Fox H-functions, as one of the widest class of special functions.

Hadamard-type integrals as G-transforms

This paper is devoted to the study of four integral operators, which are generalizations and modifications of integrals of Hadamard, in the space X c p of Lebesgue measurable functions f on R + = (0, ∞) such that for c ∈ R = (−∞, ∞) [Formula: See Text] Representations for the operators are given in the form of integral transforms involving the Meijer G-function in the kernels. The mapping properties such as the boundedness, representation and range are established.