A Practical Guide to Prabhakar Fractional Calculus

Giusti, Andrea; Colombaro, Ivano; Garra, Roberto; Garrappa, Roberto; Polito, Federico; Popolizio, Marina; Mainardi, Francesco

doi:10.1515/fca-2020-0002

Cited by 148 publications

(102 citation statements)

References 163 publications

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“…which is upper triangular circulant with a Poisson distribution in the non-vanishing entries and we directly verify the general properties (18)- (20) of Laplacian matrix exponentials in digraphs. This relation underlines the utmost importance of the Poisson distribution in strictly increasing walks.…”

Section: Circulant Transition Matrices and Strictly Increasing Walksmentioning

confidence: 59%

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Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

Michelitsch¹,

Polito²,

Riascos³

2020

Preprint

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We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps subordinated to a (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time fractional Mittag-Leffler process. The approach of construction of good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

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Section: Circulant Transition Matrices and Strictly Increasing Walksmentioning

confidence: 59%

“…In the meantime several generalizations of fractional calculus have been proposed. One of the most pertinent generalizations seems to be the so called Prabhakar generalization [18][19][20]. This generalization involves the Prabhakar function as a generalization of the Mittag-Leffler function and was first introduced by Prabhakar [21].…”

Section: Introductionmentioning

confidence: 99%

Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

Michelitsch¹,

Polito²,

Riascos³

2020

Preprint

Self Cite

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“…The Fox-Wright function that is used in Eqs. (38) and (39) can be represented (Kilbas et al 2006, p. 45) through the three-parameter Mittag-Leffler function (Gorenflo et al 2014), which is also called the Prabhakar function (Garra and Garrappa (2018); Giusti et al (2020), by the equation…”

Section: Solution For Equation Of Cagan Model With Memorymentioning

confidence: 99%

Cagan model of inflation with power-law memory effects

Tarasov

2020

Comp. Appl. Math.

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This paper considers a generalization of the model that has been proposed by Phillip D. Cagan to describe the dynamics of the actual inflation. In this generalization, the memory effects and memory fading are taken into account. In the standard Cagan model, the indicator of nervousness of economic agents, which characterizes the speed of revising the expectations, is represented as a constant parameter. In general, the speed of revising the expectations of inflation can depend on the history of changes in the difference between the real inflation rate and the rate expected by economic agents. We assume that the nervousness of economic agents can be caused not only by the current state of the process, but also by the history of its changes. The use of the memory function instead of the indicator of nervousness allows us to take into account the memory effects in the Cagan model. We consider the fractional dynamics of the inflation that takes into account memory with power-law fading. The fractional differential equation, which describes the proposed economic model with memory, and the expression of its exact solution are suggested.

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“…The latter especially has discovered many applications, since various real-life processes have behaviour which is better described by a Mittag-Leffler law than a power law Kumar et al 2019;Yusuf et al 2018). From the mathematical point of view, Prabhakar's is perhaps the most relevant univariate Mittag-Leffler function to the current work, with several recent studies and surveys on Prabhakar fractional calculus (Garra et al 2014;Garrappa 2016;Garra and Garrappa 2018;Giusti et al 2020) and also some applications (Garrappa et al 2016;Sandev 2017;D'Ovidio and Polito 2018;Zhao and Sun 2019). Recently, bivariate Mittag-Leffler functions have also been used to define fractional-calculus operators (Özarslan and Kürt 2019;Kürt et al 2020).…”

Section: Introductionmentioning

confidence: 97%

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

2020

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We define an analogue of the classical Mittag-Leffler function which is applied to two variables, and establish its basic properties. Using a corresponding single-variable function with fractional powers, we define an associated fractional integral operator which has many interesting properties. The motivation for these definitions is twofold: firstly, their link with some fundamental fractional differential equations involving two independent fractional orders, and secondly, the fact that they emerge naturally from certain applications in bioengineering. Keywords Mittag-Leffler functions • Fractional integrals • Fractional derivatives • Fractional differential equations • Bivariate Mittag-Leffler functions ρ α,β (x) and E α,β,γ (x), the "two-parameter" and "three-parameter" Mittag-Leffler functions, being defined by power series similar to the Communicated by Roberto Garrappa.

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A Practical Guide to Prabhakar Fractional Calculus

Cited by 148 publications

References 163 publications

Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

Cagan model of inflation with power-law memory effects

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

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