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An explanation on four new definitions of fractional operators

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Abstract

Fractional calculus has drawn more attentions of mathematicians and engineers in recent years. A lot of new fractional operators were used to handle various practical problems. In this article, we mainly study four new fractional operators, namely the Caputo-Fabrizio operator, the Atangana-Baleanu operator, the Sun-Hao-Zhang-Baleanu operator and the generalized Caputo type operator under the frame of the k-Prabhakar fractional integral operator. Usually, the theory of the k-Prabhakar fractional integral is regarded as a much broader than classical fractional operator. Here, we firstly give a series expansion of the k-Prabhakar fractional integral by means of the k-Riemann-Liouville integral. Then, a connection between the k-Prabhakar fractional integral and the four new fractional operators of the above mentioned was shown, respectively. In terms of the above analysis, we can obtain this a basic fact that it only needs to consider the k-Prabhakar fractional integral to cover these results from the four new fractional operators.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Changshu Institute of Technology, Changshu, 215500, China

    Jiangen Liu

  2. Qin Institute of Mathematics, Shanghai Hanjing Centre for Science and Technology, Shanghai, 201609, China

    Jiangen Liu & Fazhan Geng

Authors
  1. Jiangen Liu
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  2. Fazhan Geng
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Correspondence to Jiangen Liu.

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Conflict of Interest The authors declare no conflict of interest.

Additional information

Liu’s research was supported by NSFC (11971475), the Natural Science Foundation of Jiangsu Province (BK20230708) and the Natural Science Foundation for the Universities in Jiangsu Province (23KJB110003); Geng’s research was supported by the NSFC (11201041) and the China Postdoctoral Science Foundation (2019M651765).

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Liu, J., Geng, F. An explanation on four new definitions of fractional operators. Acta Math Sci 44, 1271–1279 (2024). https://doi.org/10.1007/s10473-024-0405-7

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  • DOI: https://doi.org/10.1007/s10473-024-0405-7

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