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Open AccessArticle

An Entropy Formulation Based on the Generalized Liouville Fractional Derivative

Grupo Física-Matemática, Faculdade de Ciências, Universidade de Lisboa, Avenida Professor Gama Pinto, 2, 1649-003 Lisboa, Portugal
Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, R. Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Entropy 2019, 21(7), 638;
Received: 15 June 2019 / Revised: 21 June 2019 / Accepted: 25 June 2019 / Published: 28 June 2019
(This article belongs to the Special Issue The Fractional View of Complexity)
PDF [368 KB, uploaded 28 June 2019]


This paper presents a new formula for the entropy of a distribution, that is conceived having in mind the Liouville fractional derivative. For illustrating the new concept, the proposed definition is applied to the Dow Jones Industrial Average. Moreover, the Jensen-Shannon divergence is also generalized and its variation with the fractional order is tested for the time series. View Full-Text
Keywords: fractional derivatives; fractional calculus; entropy fractional derivatives; fractional calculus; entropy

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Ferreira, R.A.C.; Tenreiro Machado, J. An Entropy Formulation Based on the Generalized Liouville Fractional Derivative. Entropy 2019, 21, 638.

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