Some theoretical results on fractional-order continuous information measures

Benmahmoud, Slimane

doi:10.2478/aupcsm-2024-0001

Abstract

By rewriting the differential entropy in a form of a differ-integral function’s limit, and deforming the ordinary derivative to a fractional-order one, we derive in this paper a novel generalized fractional-order differential entropy along with its related information measures. When the order of fractional differentiation α → 1, the ordinary Shannon’s differential entropy is recovered, which corresponds to the results from first-order ordinary differentiation.

References

Boltzmann, Ludwig. “Lectures on the Principles of Mechanics.” In Theoretical Physics and Philosophical Problems editeg by B. McGuinness. Dordrecht: Springer, 1974. Cited on 5.
Search in Google Scholar Back to article
Clausius, Rudolf and Thomas Archer Hirst. The Mechanical Theory of Heat: With its Applications to the Steam-Engine and to the Physical Properties of Bodies. London: J. Van Voorst, 1867. Cited on 5.
Search in Google Scholar Back to article
Cover, Thomas M., and Joy A. Thomas. Elements of Information Theory. New-York: John Wiley & Sons, INC., 1991. Cited on 7.
Search in Google Scholar Back to article
Hilfer, Rudolf. Applications Of Fractional Calculus In Physics. Universität Mainz & Universität Stuttgart, 2000. Cited on 6.
Search in Google Scholar Back to article
Jumarie, Guy. Relative Information: Theories and Applications. Vol. 47 of Springer Series in Synergetics. Heidelberg: Springer Berlin, 2011. Cited on 6.
Search in Google Scholar Back to article
Kilbas, Anatoly A., and Hari Mohan Srivastava, and Juan J. Trujillo. Theory and Applications of Fractional Differential Equations. Oxford: Elsevier LTD, 2006. Cited on 7.
Search in Google Scholar Back to article
Leibniz G.W. “Letter from Hanover; Germany; to G.F.A. L’Hopital, September 30; 1695.” In: Mathematische Schriften, 301-302. Hidesheim: Olms Verlag, 1962. Cited on 6.
Search in Google Scholar Back to article
Liouville, Joseph. Mémoire sur le calcul des différentielles à indices quelconques. 1832. Cited on 6.
Search in Google Scholar Back to article
Olver, Frank W.J. et al. NIST Handbook of Mathematical Functions. Cambridge University Press, 2010. Cited on 8.
Search in Google Scholar Back to article
Riemann, Bernhard. “Versuch Einer Allgemeinen Auffassung Der Integration Und Differentiation. (1847.).” In Bernard Riemann’s Gesammelte Mathematische Werke Und Wissenschaftlicher Nachlass, edited by Richard Dedekind and Heinrich Martin Weber, 331-344. Cambridge: Cambridge University Press, 2013. Cited on 6.
Search in Google Scholar Back to article
Ubriaco, Marcelo R. “Entropies based on fractional calculus.” Physics Letters A 373 (2009): 2516-2519. Cited on 6.
Search in Google Scholar Back to article
Shannon, Claude E. “A mathematical theory of communication.” Bell Syst. Tech. J. 27 (1948): 379-423. Cited on 5.
Search in Google Scholar Back to article

Some theoretical results on fractional-order continuous information measures

Abstract

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