The Hypergeometric Approach to Integral Transforms and Convolutions

Yakubovich, Semyon; Luchko, Yurii F.

doi:10.1007/978-94-011-1196-6

Cited by 213 publications

(285 citation statements)

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“…The same function, called "vector-indexed M-L function" happened to be introduced also in some works by Luchko et al, for example in the book Yakubovich-Luchko [48], the paper [31], etc. as a tool to represent explicitly the solutions of some fractional order differential equations.…”

Section: Multi-index Mittag-leffler Functionsmentioning

confidence: 99%

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

Kiryakova

2014

Fract Calc Appl Anal

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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.MSC . Definitions of classical FCThe classical fractional calculus can be thought as an upgrade of the Calculus, originated as early as in 1695 (the famous l'Hospital's letter to Leibnitz). Since then, many known mathematicians and applied scientists contributed to the development of this "strange" calculus, but the first book and the first conference dedicated specially to that topic appeared 279 years (1974−1695 = 279) after the mentioned correspondence. And this year we are marking 40 years (2014−1974 = 40) of these two events, when there are published more than 100 monographs and topical selections on the area of FC and its applications. The detailed history, theory and its various applications, by the years of 1987-1993 was presented in the "FC Encyclopedia" [42], and currently -in several recent surveys as [47], and the posters at http://www.math.bas.bg/∼fcaa.The classical FC is based on several (equivalent or alternative) definitions for the operators of integration and differentiation of arbitrary (including real fractional or complex) order, as continuations of the classical integration and differentiation operators and their integer order powers (n ∈ N), namely -the n-fold integration

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Section: Multi-index Mittag-leffler Functionsmentioning

confidence: 99%

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

Kiryakova

2014

Fract Calc Appl Anal

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“…The theory of the H-function, its properties, and applications are presented in a number of textbooks and papers (see, e.g., [23][24][25][26][27][28]); thus, here we do not discuss this subject in detail and prefer to directly deduce the properties of the fundamental solution G α,β,n from its Mellin-Barnes representation (Equation (23)). Starting with this representation and using simple linear variables' substitutions, we can easily derive some other forms of this representation that will be useful for further discussions.…”

Section: Mellin-barnes Representations Of the Fundamental Solutionmentioning

confidence: 99%

On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation

Luchko

2017

Mathematics

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Abstract:In this paper, some new properties of the fundamental solution to the multi-dimensional space-and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to obtain two new representations of the fundamental solution in the form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed-form formulas for particular cases of the fundamental solution are derived. In particular, we solve the open problem of the representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

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“…The background and the basic knowledge on the results in this paper are contained as essence in Luchko, Yakubobich and Luchko [11,24]. In this section, the stuff is taken from Hilfer-Luchko-Tomovski [6], unless specially indicated.…”

Section: Preliminaries and Operational Calculus For Generalized Fractmentioning

confidence: 99%

“…Operational calculus techniques have been alreday applied successfully to ordinary differential equations and partial differential equations (both of integer or fractional order), integral equations, and to the theory of special functions (see, for example [11,24,9,10], etc.). For example, in [12] Mikusinski's operational calculus has been applied to solve a Cauchy boundary-value problem for a certain linear equation involving RiemannLiouville fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives

Kim¹,

Ri²,

Hyong-Chol³

2013

fcaa

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This paper provides results on the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski's type. We prove that the initial value problem has the solution if and only if some initial values are zero.MSC 2010 : Primary 34A08, 34A25, 44A40; Secondary 26A33

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The Hypergeometric Approach to Integral Transforms and Convolutions

Cited by 213 publications

References 0 publications

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

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

On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation

Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives

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