A Practical Guide to Prabhakar Fractional Calculus

The Mittag–Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key...

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Published inFractional calculus & applied analysis Vol. 23; no. 1; pp. 9 - 54
Main Authors Giusti, Andrea, Colombaro, Ivano, Garra, Roberto, Garrappa, Roberto, Polito, Federico, Popolizio, Marina, Mainardi, Francesco
Format Journal Article
LanguageEnglish
Published Warsaw Versita 01.02.2020
De Gruyter
Nature Publishing Group
Subjects
26A33
34K37
60G22
Abstract Harmonic Analysis
Analysis
anomalous physical phenomena
Calculus
Fractional calculus
Functional Analysis
Integral Transforms
Mathematics
Mittag-Leffler type functions
numerical methods
Operational Calculus
Prabhakar fractional calculus
Prabhakar function
Primary: 33E12
Secondary: 65R10
stochastic processes
Survey Paper
Prabhakar function
numerical methods
Primary: 33E12, 26A33
Prabhakar fractional calculus
Mittag-Leffler type functions
stochastic processes
Secondary: 65R10, 34K37, 60G22
anomalous physical phenomena
Online AccessGet full text
ISSN1311-0454
1314-2224
DOI10.1515/fca-2020-0002

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Abstract The Mittag–Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.
AbstractList The Mittag–Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.
Author Popolizio, Marina
Polito, Federico
Garra, Roberto
Giusti, Andrea
Colombaro, Ivano
Mainardi, Francesco
Garrappa, Roberto
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  surname: Giusti
  fullname: Giusti, Andrea
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  givenname: Ivano
  surname: Colombaro
  fullname: Colombaro, Ivano
  organization: Department of Information and Communication Technologies, Universitat Pompeu Fabra
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  surname: Garra
  fullname: Garra, Roberto
  organization: Department of Statistical Sciences, Sapienza University of Rome
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  surname: Garrappa
  fullname: Garrappa, Roberto
  organization: Department of Mathematics, University of Bari, INdAM Research group GNCS
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  givenname: Federico
  surname: Polito
  fullname: Polito, Federico
  organization: Department of Mathematics, University of Torino
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  surname: Popolizio
  fullname: Popolizio, Marina
  organization: Department of Electrical and Information Engineering, Polytechnic University of Bari
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  givenname: Francesco
  surname: Mainardi
  fullname: Mainardi, Francesco
  organization: Department of Physics and Astronomy, University of Bologna
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Issue 1
Keywords Prabhakar function
numerical methods
Primary: 33E12, 26A33
Prabhakar fractional calculus
Mittag-Leffler type functions
stochastic processes
Secondary: 65R10, 34K37, 60G22
anomalous physical phenomena
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PublicationSubtitle An International Journal for Theory and Applications
PublicationTitle Fractional calculus & applied analysis
PublicationTitleAbbrev Fract Calc Appl Anal
PublicationYear 2020
Publisher Versita
De Gruyter
Nature Publishing Group
Publisher_xml – name: Versita
– name: De Gruyter
– name: Nature Publishing Group
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Snippet The Mittag–Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and...
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StartPage 9
SubjectTerms 26A33
34K37
60G22
Abstract Harmonic Analysis
Analysis
anomalous physical phenomena
Calculus
Fractional calculus
Functional Analysis
Integral Transforms
Mathematics
Mittag-Leffler type functions
numerical methods
Operational Calculus
Prabhakar fractional calculus
Prabhakar function
Primary: 33E12
Secondary: 65R10
stochastic processes
Survey Paper
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Title A Practical Guide to Prabhakar Fractional Calculus
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