On the theory of the operational calculus for the Bessel equation

Ditkin, V. A.; Prudnikov, A. P.

doi:10.1016/0041-5553(63)90022-3

Cited by 11 publications

(8 citation statements)

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“…In all these mentioned cases, the kernel functions have the form of (8) as also studied earlier by Erdélyi [40]. We conclude here the list of cases of the Obrechkoff transform with emphasize on the works by Ditkin-Prudnikov (as [50]) on operational calculi for (hyper-Bessel) operators of the form…”

Section: Use Of G-and H-functions As Kernels Of Laplace Type Integralsupporting

confidence: 51%

“…For m = 2, the corresponding integral transform is a variant of the Meijer transform (with ν = 0), and in the general case m > 1, Ditkin and Prudnikov [50] made use of an integral transform of the form…”

Section: Use Of G-and H-functions As Kernels Of Laplace Type Integralmentioning

confidence: 99%

“…. It remains to combine the results (48) and (50) to verify the fact that the multi-index M-L functions that generate the G-L operators (47) appear as their eigenfunctions:…”

Section: Some Basic Properties Of the Multi-index Mittag-leffler Funcmentioning

confidence: 99%

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A Guide to Special Functions in Fractional Calculus

Kiryakova

2021

Mathematics

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Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.

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Section: Use Of G-and H-functions As Kernels Of Laplace Type Integralsupporting

confidence: 51%

Section: Use Of G-and H-functions As Kernels Of Laplace Type Integralmentioning

confidence: 99%

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A Guide to Special Functions in Fractional Calculus

Kiryakova

2021

Mathematics

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“…The Mikusiński operational calculus was successfully used in ordinary differential equations, integral equations, partial differential equations and in the theory of special functions. It is worth mentioning that the Mikusiński scheme was extended by several mathematicians to develop operational calculi for differential operators with variable coefficients (see, for example, [8], [9], [32]), too. These operators are all particular cases of the so called hyper-Bessel differential operator…”

Section: Introductionmentioning

confidence: 99%

Operational Calculus for the General Fractional Derivative and Its Applications

Luchko

2021

Fract Calc Appl Anal

103

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In this paper, we first address the general fractional integrals and derivatives with the Sonine kernels that possess the integrable singularities of power function type at the point zero. Both particular cases and compositions of these operators are discussed. Then we proceed with a construction of an operational calculus of the Mikusiński type for the general fractional derivatives with the Sonine kernels. This operational calculus is applied for analytical treatment of some initial value problems for the fractional differential equations with the general fractional derivatives. The solutions are expressed in form of the convolution series that generalize the power series for the exponential and the Mittag-Leffler functions.

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“…The operator L generalizes the 2nd order Bessel differential operator, the n-th order hyper-Bessel operator of Ditkin and Prudnikov [6] …”

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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On the theory of the operational calculus for the Bessel equation

Cited by 11 publications

References 1 publication

A Guide to Special Functions in Fractional Calculus

A Guide to Special Functions in Fractional Calculus

Operational Calculus for the General Fractional Derivative and Its Applications

Fractional relaxation with time-varying coefficient

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