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.
Similar content being viewed by others
References
Yang X J. General Fractional Derivatives: Theory, Methods and Applications. New York: CRC Press, 2019
Kumar S, Ranbir K, Carlo C, Bessem S. Chaotic behaviour of fractional predator-prey dynamical system. Chaos, Solitons & Fractals, 2020, 135: 109811
Kumar S, Kumar A, Samet B, Dutta H. A study on fractional host-parasitoid population dynamical model to describe insect species. Numer Meth Part Diff Equ, 2021, 37(2): 1673–1692
Zhao Y W, Xia J W, Lü X. The variable separation solution, fractal and chaos in an extended coupled (2+1)-dimensional Burgers system. Nonl Dyn, 2022, 108(4): 4195–4205
Liu J G, Zhang Y F, Wang J J. Investigation of the time fractional generalized (2+1)-dimensional Zakharov-Kuznetsov equation with single-power law nonlinearity. Fractals, 2023, 31(5): 2350033
Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Prog Fract Diff Appl, 2015, 1: 73–85
Atangana A, Baleanu D. New fractional derivative without nonlocal and nonsingular kernel: Theory and application to heat transfer model. Therm Sci, 2016, 20: 763–769
Sun H, Hao X, Zang Y, Baleanu D. Relaxation and diffusion models with non-singular kernel. Phys A, 2017, 468: 590–596
Teodoro G S. Derivadas Fracionarias: Tipos e Criterios de Validade[D]. Campinas: Imecc-Unicamp, 2019
Yang X J. A new integral transform operator for solving the heat-diffusion problem. Appl Math Lett, 2017, 64: 193–197
Yang X J, Tenreiro Machado J A, Baleanu D. Anomalous diffusion models with general fractional derivatives within the kernels of the extended Mittag-Leffler type functions. Rom Rep Phys, 2017, 69: S1–S9
Khalil R, Horani M A, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math, 2014, 264: 65–70
Vanterler da C Sousa J, Capelas de Oliveira E. On the ψ-Hilfer fractional derivative. Commun Nonl Sci Numer Simul, 2018, 60: 72–91
Saigo M, Saxena R K, Kilbas A A. Generalized Mittag-Leffler function and generalized fractional calculus operators. Inte Tran Spec Fun, 2004, 15(1): 31–49
Zhao D, Luo M. Representations of acting processes and memory effects: general fractional derivatives and its application to theory of heat conduction with finite wave speeds. Appl Math Comput, 2019, 346: 531–544
Kumar S, Chauhan R P, Momani S, Hadid S. Numerical investigations on COVID-19 model through singular and non-singular fractional operators. Numer Meth Part Diff Equ, 2024, 40(1): e22707
Muhammad Altaf K, Ullah S, Kumar S. A robust study on 2019-nCOV outbreaks through non-singular derivative. Eur Phys J Plus, 2021, 136: 1–20
Kumar S, Kumar R, Osman M S, Samet B. A wavelet based numerical scheme for fractional order SEIR epidemic of measles by using Genocchi polynomials. Numer Meth Part Diff Equ, 2021, 37(2): 1250–1268
Liu J G, Yang X J, Feng Y Y, Geng L L. A new fractional derivative for solving time fractional diffusion wave equation. Math Meth Appl Sci, 2022, 46(1): 267–272
Yang X J, Gao F, Ju Y. General Fractional Derivatives with Applications in Viscoelasticity. New York: Academic Press, 2020
Hakimeh M, Kumar S, Rezapour S, Etemad S. A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control. Chaos, Solitons & Fractals, 2021, 144: 110668
Liu J G, Yang X J, Feng Y Y, Geng L L. On the generalized weighted Caputo-type differential operator. Fractals, 2022, 31(1): 2250032
Mittag-Leffler G M. Sur la nouvelle fonction eα(x). CR Acad Sci Paris, 1903, 137: 554–558
Wiman A. Uber den fundamental satz in der theorie der funcktionen, Eα(x). Acta Math, 1905, 29: 191–201
Prabhakar T R. A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yoko Math J, 1971, 19: 171–183
Dorrego G A, Cerutti R A. The k-Mittag-Leer function. Int J Contemp Math Sci, 2012, 7(1): 705–716
Díaz R, Eddy P. On hypergeometric functions and Pochhammer k-symbol. Divulg Mat, 2007, 15(2): 179–192
Dorrego G A. Generalized Riemann-Liouville fractional operators associated with a generalization of the Prabhakar integral operator. Int J Prog Fract Diff Appl, 2016, 2(2): 131–140
Mehmet Z S, Aysel K. On the k-Riemann-Liouville fractional integral and applications. Int J Stat Math, 2014, 1(3): 33–43
Liu J G, Yang X J, Wang J J. A new perspective to discuss Korteweg-de Vries-like equation. Phys Lett A, 2022, 451: 128429
Yin Y H, Luü X, Ma W X. Baücklund transformation, exact solutions and diverse interaction phenomena to a (3+1)-dimensional nonlinear evolution equation. Nonl Dyn, 2022, 108(4): 4181–4194
Wen L L, Fan E G, Chen Y. The Sasa-Satsuma equation on a non-zero background: the inverse scattering transform and multi-soliton solutions. Acta Math Sci, 2023, 43(3): 1045–1080
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
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).
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-024-0405-7
Key words
- k-Prabhakar fractional operator
- Caputo-Fabrizio operator
- Atangana-Baleanu operator
- Sun-Hao-Zhang-Baleanu operator
- generalized Caputo type operator