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Cut-Off Distributions And Probabilistic Interpretations Of The General Fractional Derivatives With Memory Effect

Author

Listed:
  • 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)

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.

Suggested Citation

  • Dazhi Zhao & Yan Yu & Liang Pu, 2022. "Cut-Off Distributions And Probabilistic Interpretations Of The General Fractional Derivatives With Memory Effect," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(04), pages 1-8, June.
    • Handle: RePEc:wsi:fracta:v:30:y:2022:i:04:n:s0218348x2250089x
    DOI: 10.1142/S0218348X2250089X
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