Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions

Fernandez, Arran; Băleanu, Dumitru; Srivástava, H. M.

doi:10.1016/j.cnsns.2018.07.035

Cited by 123 publications

(86 citation statements)

References 30 publications

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“…Newer investigations include themes as chaos and statistic[3], reaction-diffusion systems[39], time-fractional variable-order telegraph equation[17], Lévy Flights[13], viscoelastic response[23], etc. In fact, in this scenario, the Mittag-Leffler kernels have been a mathematical tool that has been more understood day by day[44,14]. Thereby, this work extends 1 Or Riemann-Liouville, depending on how the Prabhakar derivative is defined.…”

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confidence: 87%

Mittag-Leffler functions in superstatistics

Santos

2020

Chaos, Solitons & Fractals

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Nowadays, there is a series of complexities in biophysics that require a suitable approach to determine the measurable quantity. In this way, the superstatistics has been an important tool to investigate dynamic aspects of particles, organisms and substances immersed in systems with non-homogeneous temperatures (or diffusivity). The superstatistics admits a general Boltzmann factor that depends on the distribution of intensive parameters β = 1 D (inverse-diffusivity). Each value of β is associated with a local equilibrium in the system. In this work, we investigate the consequences of Mittag-Leffler function on the definition of f (β)-distribution of a complex system. Thus, using the techniques belonging to the fractional calculus with non-singular kernels, we constructed a distribution to β using the Mittag-Leffler function. This function implies distributions with power-law behaviour to high energy values in the context of Cohen-Beck superstatistics. This work aims to present the generalised probabilities distribution in statistical mechanics under a new perspective of the Mittag-Leffler function inspired in Atangana-Baleanu and Prabhakar forms.

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confidence: 87%

Mittag-Leffler functions in superstatistics

Santos

2020

Chaos, Solitons & Fractals

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“…Remark 2.2. It has been demonstrated [33,34] that the Prabhakar kernel is general enough to include several other kernel functions of fractional calculus, including the AB one, as special cases. We now demonstrate that all of the fractional models mentioned above can be viewed as special cases of our new generalised model.…”

Section: Definition and Basic Properties 21 Fractional Integralsmentioning

confidence: 99%

“…This is the same domain of definition that works for the RL [3], AB [16], and Prabhakar [18] operators. Theorem 2.6 establishes a way of expressing the general operators in terms of only the classical Riemann-Liouville fractional integrals, following the method used in [16,34]. This is the reason for our assumption that A was analytic, i.e.…”

Section: This Proves Thatmentioning

confidence: 99%

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On fractional calculus with general analytic kernels

Fernandez

Özarslan

Băleanu

2019

Applied Mathematics and Computation

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117

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Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann-Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators.

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“…These equations have many applications in many fields of science. [2][3][4][5][6][7] Let Ω ⊂ R d (d ≥ 1) be a bounded domain with the sufficiently smooth boundary Ω. We consider the following nonlinear fractional diffusion equation:…”

Section: Introductionmentioning

confidence: 99%

On a final value problem for fractional reaction‐diffusion equation with Riemann‐Liouville fractional derivative

Tran

Zhou

et al. 2019

Math Methods in App Sciences

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In this paper, we study a backward problem for a fractional diffusion equation with nonlinear source in a bounded domain. By applying the properties of Mittag‐Leffler functions and Banach fixed point theorem, we establish some results above the existence, uniqueness, and regularity of the mild solutions of the proposed problem in some suitable space. Moreover, we also show the ill‐posedness of our problem in the sense of Hadamard. The regularized solution is given, and the convergence rate between the regularized solution and the exact solution is also obtained.

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Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions

Cited by 123 publications

References 30 publications

Mittag-Leffler functions in superstatistics

Mittag-Leffler functions in superstatistics

On fractional calculus with general analytic kernels

On a final value problem for fractional reaction‐diffusion equation with Riemann‐Liouville fractional derivative

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