CUT-OFF DISTRIBUTIONS AND PROBABILISTIC INTERPRETATIONS OF THE GENERAL FRACTIONAL DERIVATIVES WITH MEMORY EFFECT

Dazhi Zhao, Yan Yu () and Liang Pu ()
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Dazhi Zhao: School of Sciences, Southwest Petroleum University, Chengdu 610500, P. R. China†Institute for Artificial Intelligence, Southwest Petroleum University, Chengdu 610500, P. R. China‡State Key Laboratory of Oil and Gas Reservoir Geology and Exploration, Southwest Petroleum University, Chengdu 610500, P. R. China
Yan Yu: School of Sciences, Southwest Petroleum University, Chengdu 610500, P. R. China
Liang Pu: School of Sciences, Southwest Petroleum University, Chengdu 610500, P. R. China

FRACTALS (fractals), 2022, vol. 30, issue 04, 1-8

Abstract: The physical/probabilistic meanings of fractional derivatives or integrals are very basic and important problems in fractional calculus. In this paper, we investigate the probabilistic interpretations of the general fractional derivatives with memory effect and their connections with cut-off distributions. Some examples like the cut-off Rayleigh distribution and the Atangana–Gómez fractional derivative are especially discussed. This paper shows that the kernels of the general fractional derivatives with memory effect may be not the same as the density functions of the physical processes described. For example, the kernel of Atangana–Gómez fractional derivative is a Gaussian function but its corresponding probabilistic distribution is a cut-off Rayleigh distribution.

Keywords: Probabilistic Interpretation; General Fractional Derivative; Cut-Off Distribution; Memory Effect (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X2250089X

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