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.
[1] Yang X J.General Fractional Derivatives: Theory, Methods and Applications. New York: CRC Press, 2019
[2] Kumar S, Ranbir K, Carlo C, Bessem S. Chaotic behaviour of fractional predator-prey dynamical system. Chaos, Solitons & Fractals, 2020, 135: 109811
[3] 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
[4] 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
[5] 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
[6] Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Prog Fract Diff Appl, 2015, 1: 73-85
[7] 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
[8] Sun H, Hao X, Zang Y, Baleanu D. Relaxation and diffusion models with non-singular kernel. Phys A, 2017, 468: 590-596
[9] Teodoro G S.Derivadas Fracionarias: Tipos e Criterios de Validade[D]. Campinas: Imecc-Unicamp, 2019
[10] Yang X J. A new integral transform operator for solving the heat-diffusion problem. Appl Math Lett, 2017, 64: 193-197
[11] 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
[12] Khalil R, Horani M A, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math, 2014, 264: 65-70
[13] Vanterler da C Sousa J, Capelas de Oliveira E. On the $\psi$-Hilfer fractional derivative. Commun Nonl Sci Numer Simul, 2018, 60: 72-91
[14] 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
[15] 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
[16] 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
[17] Muhammad Altaf K, Ullah S, Kumar S. A robust study on2019-nCOV outbreaks through non-singular derivative. Eur Phys J Plus, 2021, 136: 1-20
[18] 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
[19] 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
[20] Yang X J, Gao F, Ju Y.General Fractional Derivatives with Applications in Viscoelasticity. New York: Academic Press, 2020
[21] 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
[22] 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
[23] Mittag-Leffler G M. Sur la nouvelle fonction ${e_\alpha }(x)$. CR Acad Sci Paris, 1903, 137: 554-558
[24] Wiman A. Über den fundamental satz in der theorie der funcktionen, ${E_\alpha }(x)$. Acta Math, 1905, 29: 191-201
[25] Prabhakar T R. A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yoko Math J, 1971, 19: 171-183
[26] Dorrego G A, Cerutti R A. The $k$-Mittag-Leer function. Int J Contemp Math Sci, 2012, 7(1): 705-716
[27] Díaz R, Eddy P. On hypergeometric functions and Pochhammer $k$-symbol. Divulg Mat, 2007, 15(2): 179-192
[28] 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
[29] Mehmet Z S, Aysel K. On the $k$-Riemann-Liouville fractional integral and applications. Int J Stat Math, 2014, 1(3): 33-43
[30] 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
[31] Yin Y H, Lü X, Ma W X. Bäcklund transformation, exact solutions and diverse interaction phenomena to a (3+1)-dimensional nonlinear evolution equation. Nonl Dyn, 2022, 108(4): 4181-4194
[32] 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, 43B(3): 1045-1080
