数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (4): 1271-1279.doi: 10.1007/s10473-024-0405-7

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AN EXPLANATION ON FOUR NEW DEFINITIONS OF FRACTIONAL OPERATORS

Jiangen Liu1,2,*, Fazhan Geng3   

  1. 1. School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China;
    2. Qin Institute of Mathematics, Shanghai Hanjing Centre for Science and Technology, Shanghai 201609, China;
    3. School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China
  • 收稿日期:2022-11-23 修回日期:2023-10-06 出版日期:2024-08-25 发布日期:2024-08-30

AN EXPLANATION ON FOUR NEW DEFINITIONS OF FRACTIONAL OPERATORS

Jiangen Liu1,2,*, Fazhan Geng3   

  1. 1. School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China;
    2. Qin Institute of Mathematics, Shanghai Hanjing Centre for Science and Technology, Shanghai 201609, China;
    3. School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China
  • Received:2022-11-23 Revised:2023-10-06 Online:2024-08-25 Published:2024-08-30
  • Contact: *E-mail: ljgzr557@126.com
  • About author:E-mail: gengfazhan@sina.com
  • Supported by:
    Liu's research was supported by the 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).

摘要: 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.

关键词: $k$-Prabhakar fractional operator, Caputo-Fabrizio operator, Atangana-Baleanu operator, Sun-Hao-Zhang-Baleanu operator, generalized Caputo type operator

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.

Key words: $k$-Prabhakar fractional operator, Caputo-Fabrizio operator, Atangana-Baleanu operator, Sun-Hao-Zhang-Baleanu operator, generalized Caputo type operator

中图分类号: 

  • 33E12