Fractional Powers of a Class of Ordinary Differentilal Operators

McBride, Adam C.

doi:10.1112/plms/s3-45.3.519

Cited by 69 publications

(62 citation statements)

References 9 publications

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“…This representation of the fractional powers of the hyper-Bessel operators is first published in our paper joint with Dimovski [9] and coincides with the results proposed by McBride [35] found in completely different way.…”

Section: M Wheresupporting

confidence: 69%

“…Several authors, like Love [32], Saxena [43], Kalla and Saxena [18], Saigo [39,40], McBride [35], also Tricomi, Sprinkhuizen-Kuiper, Koornwinder, etc., have studied and used different modifications (mainly in 60's-70's) of the so-called hypergeometric operators of fractional integration 6) involving the Gauss hypergeometric function. An example of fractional integration operators involving other special functions, is given by the operators of Lowndes [33]:…”

Section: Attempts For Generalized Fractional Calculimentioning

confidence: 99%

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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: M Wheresupporting

confidence: 69%

Section: Attempts For Generalized Fractional Calculimentioning

confidence: 99%

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

Kiryakova

2014

Fract Calc Appl Anal

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“…3.2] and [31]). We also want to mention the works of W. Lamb [29,30] on fractional powers of operators on Frechét spaces following that of A.C. McBride [36].…”

Section: Representation Formulae For Fractional Powersmentioning

confidence: 99%

Representation of complex powers of C-sectorial operators

Chuang

Kostić

2014

Fract Calc Appl Anal

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“…Refs. in [11], [19]) and appearing very often in differential equations of mathematical physics, specially in the theory of potential, viscoelasticity, etc.…”

Section: Appendix: On the Theory Of Fractional Powers Of Hyper-besselmentioning

confidence: 99%

“…The linear singular differential operators with variable coefficients of arbitrary integer order n = 1, 2, 3, ... of the form [19], and called later by the name hyper-Bessel operators in the works by Kiryakova as [11]. The reason is because the hyper-Bessel functions of Delerue (variants of the generalized hypergeometric function 0 F n−1 , reducing to the Bessel functions 0 F 1 for n = 2) are the solutions (forming f.s.s.…”

Section: Appendix: On the Theory Of Fractional Powers Of Hyper-besselmentioning

confidence: 99%

Fractional relaxation with time-varying coefficient

Garra

Giusti

Mainardi

et al. 2014

fcaa

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From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument due to the presence of Erdélyi-Kober fractional integrals in our operator. We present solutions, both singular and regular in the time origin, that are locally integrable and completely monotone functions in order to be consistent with the physical phenomena described by non-negative relaxation spectral distributions.MSC 2010 : Primary 26A33; Secondary 33E12, 34A08, 76A10

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Fractional Powers of a Class of Ordinary Differentilal Operators

Cited by 69 publications

References 9 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

Representation of complex powers of C-sectorial operators

Fractional relaxation with time-varying coefficient

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