Fractional Calculus: D’où Venons-Nous? Que Sommes-Nous? Où Allons-Nous?

Machado, J. A. Tenreiro; Mainardi, Francesco; Kiryakova, Virginia; Atanacković, Teodor M.

doi:10.1515/fca-2016-0059

Cited by 31 publications

(13 citation statements)

References 50 publications

(47 reference statements)

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“…The reason is that the differential and integral equations and dynamical systems of fractional order can model mathematically the phenomena of Nature and Society more adequately than these restricted to integer order. The evolution of FC was analyzed in several survey papers and round tables discussions [103,104,105,101,102,107,108] revealing a considerable and sustained progress. Therefore, it is important to collect the most recent information not only to keep a precise record, but also to monitor the present day development.…”

Section: Introductionmentioning

confidence: 99%

The Chronicles of Fractional Calculus

Machado

Kiryakova

2017

FCAA

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130

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Since the 60s of last century Fractional Calculus exhibited a remarkable progress and presently it is recognized to be an important topic in the scientific arena. This survey analyzes and measures the evolution that occurred during the last five decades in the light of books, journals and conferences dedicated to the theory and applications of this mathematical tool, dealing with operations of integration and differentiation of arbitrary (fractional) order and their generalizations.MSC 2010 : Primary 26A33; 01A60, 01A61, 01A67; Secondary 34A08, 35R11, 60G22Key Words and Phrases: fractional calculus, development, fractional order differential equations, fractional order mathematical models, applications 1. IntroductionFractional Calculus (FC) started with the ideas of Gottfried Leibniz by the end of the XVII century and had been developed progressively up to now. During the recent decades FC, as an extension of the classical Calculus, attracted the attention of many researchers in several areas, namely mathematics, physics, engineering, biology, finance, economy, chemistry and social sciences. The reason is that the differential and integral equations and dynamical systems of fractional order can model mathematically the phenomena of Nature and Society more adequately than these restricted to integer order.

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Section: Introductionmentioning

confidence: 99%

The Chronicles of Fractional Calculus

Machado

Kiryakova

2017

FCAA

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130

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“…generalising the causal expression (19) to real orders. For the anti-causal case, we get the general expression:…”

Section: Second Unificationmentioning

confidence: 99%

“…Several of these questions and others were discussed in round tables in the scope of several meetings held by the fractional calculus community [18,19]. Nevertheless, present day state of affairs reveals that they were not sufficient to stimulate all researchers for a systematic definition of the fundamental concepts.…”

Section: Introductionmentioning

confidence: 99%

Fractional Derivatives: The Perspective of System Theory

Ortigueira

Machado

2019

Mathematics

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“…Notwithstanding this development a singular situation exists, since there is no definition of "fractional definite integral" [5,6]. Even more strangely, it seems that no-one has the goal to propose such a adefinition [7,8]. In fact, the term FI is used for an indefinite integral, which, as we will see, stands for a primitivation operation.…”

Section: Introductionmentioning

confidence: 99%