Mittag-Leffler Functions, Related Topics and Applications

Gorenflo, Rudolf; Kilbas, Anatoly A.; Mainardi, Francesco; Rogosin, Sergei

doi:10.1007/978-3-662-43930-2

Cited by 992 publications

(914 citation statements)

References 135 publications

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“…Note that the above results well agree with the corresponding ones for the order and type of the Mittag-Leffler function (1.2) and its multi-index extension (1.4) (see in [7] and [1,14,15,11]). …”

Section: Integral Representations Of the Le Roy Type-functionsupporting

confidence: 84%

“…[7], [24]). Below we establish two integral representations for our function E (γ) α,β which use the technique similar to that in the Mellin-Barnes formulas.…”

Section: Integral Representations Of the Le Roy Type-functionmentioning

confidence: 99%

“…where E α,β and E α,β;α,β are respectively the 2-parameter and 4-parameter Mittag-Leffler functions in the sense of (1.4) (see [7,Chs. 4,6] …”

Section: Integral Representations Of the Le Roy Type-functionmentioning

confidence: 99%

“…., [1,14,15,11]). The function (1.2) is named after the great Swedish mathematician Gösta Magnus Mittag-Leffler (1846-1927 who defined it in 1-parameter case (E α (z) with β = 1) by a power series and studied its properties in 1902-1905 (see detailed description in [7]). As a matter of fact, MittagLeffler introduced the function E α (z) for the purposes of his method for summation od divergent series.…”

Section: Introductionmentioning

confidence: 99%

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On a generalized three-parameter wright function of Le Roy type

Garrappa

Rogosin

Mainardi

2017

FCAA

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Recently S. Gerhold and R. Garra -F. Polito independently introduced a new function related to the special functions of the Mittag-Leffler family. This function is a generalization of the function studied byÉ. Le Roy in the period 1895-1905 in connection with the problem of analytic continuation of power series with a finite radius of convergence. In our note we obtain two integral representations of this special function, calculate its Laplace transform, determine an asymptotic expansion of this function on the negative semi-axis (in the case of an integer third parameter γ) and provide its continuation to the case of a negative first parameter α. An asymptotic result is illustrated by numerical calculations. Discussion on possible further studies and open questions are also presented.MSC 2010 : Primary 33E12; Secondary 30D10, 30F15, 35R11

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Section: Integral Representations Of the Le Roy Type-functionsupporting

confidence: 84%

“…[7], [24]). Below we establish two integral representations for our function E (γ) α,β which use the technique similar to that in the Mellin-Barnes formulas.…”

Section: Integral Representations Of the Le Roy Type-functionmentioning

confidence: 99%

“…where E α,β and E α,β;α,β are respectively the 2-parameter and 4-parameter Mittag-Leffler functions in the sense of (1.4) (see [7,Chs. 4,6] …”

Section: Integral Representations Of the Le Roy Type-functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On a generalized three-parameter wright function of Le Roy type

Garrappa

Rogosin

Mainardi

2017

FCAA

Self Cite

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“…if α ∈ (0, 1), for continuous functions y for which lim t→0 t −α (v(t) − v(0)) exists, is finite, and z j Γ(αj + β) (1.4) and the one-parameter Mittag-Leffler functions E α : C → C, α > 0, defined by E α := E α,1 (see e.g. [13]). The structure of the paper is as follows.…”

Section: (T − S) β−1 Y(s) Dsmentioning

confidence: 99%

Asymptotic behavior of solutions of linear multi-order fractional differential systems

Diethelm

Siegmund

Tuan

2017

FCAA

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In this paper, we investigate some aspects of the qualitative theory for multi-order fractional differential equation systems. First, we obtain a fundamental result on the existence and uniqueness for multi-order fractional differential equation systems. Next, a representation of solutions of homogeneous linear multi-order fractional differential equation systems in series form is provided. Finally, we give characteristics regarding the asymptotic behavior of solutions to some classes of linear multi-order fractional differential equation systems.

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Composite learning fuzzy synchronization for incommensurate fractional‐order chaotic systems with time‐varying delays

Zhou

Liu

Cao

et al. 2019

Adaptive Control & Signal

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This paper presents a composite learning fuzzy control to synchronize two different uncertain incommensurate fractional-order time-varying delayed chaotic systems with unknown external disturbances and mismatched parametric uncertainties via the Takagi-Sugeno fuzzy method. An adaptive controller together with fractional-order composite learning laws is designed based on both a parallel distributed compensation technology and a fractional Lyapunov criterion. The boundedness of all variables in the closed-loop system and the Mittag-Leffler stability of tracking error can be guaranteed. T-S fuzzy systems are provided to tackle unknown nonlinear functions. The distinctive features of the proposed approach consist in the following: (1) a supervisory control law is designed to compensate the lumped disturbances; (2) both the prediction error and the tracking error are used to estimate the unknown fuzzy system parameters; (3) parameter convergence can be ensured by an interval excitation condition. Finally, the feasibility of the proposed control strategy is demonstrated throughout an illustrative example.

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Mittag-Leffler Functions, Related Topics and Applications

Cited by 992 publications

References 135 publications

On a generalized three-parameter wright function of Le Roy type

On a generalized three-parameter wright function of Le Roy type

Asymptotic behavior of solutions of linear multi-order fractional differential systems

Composite learning fuzzy synchronization for incommensurate fractional‐order chaotic systems with time‐varying delays

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