New Fractional Operators Theory and Applications

Document Type : Research Paper


Department of Mathematics, College of Science, AL-Mustansiriyah University, Baghdad, Iraq


In this article, we present a new fractional integral with a non-singular kernel and by using Laplace transform, we derived the corresponding fractional derivative. By composition between our fractional integration operator with classical Caputo and Riemann-Liouville fractional operators, we establish a new fractional derivative which is interpolated between the generalized fractional derivatives in a sense Riemann-Liouville and Caputo-Fabrizio with non-singular kernels. Additionally, we introduce the fundamental properties of these fractional operators with applications and simulations. Finally, a model of Coronavirus (COVID-19) transmission is presented as an application.


[1] Mouaouine A, Boukhouima A, Hattaf K, Yousfi N. A fractional order SIR epidemic model with nonlinear incidence
rate. Adv Differ Equations. 2018; 2018(1):1–9.
[2] Boukhouima A, Hattaf K, Yousfi N. Dynamics of a fractional order HIV infection model with specific functional
response and cure rate. Int J Differ Equations. 2017; 2017(1).
[3] Magin RL. Fractional calculus in bioengineering. Vol. 2. Begell House Redding; 2006.
[4] Acay B, Bas E, Abdeljawad T. Fractional economic models based on market equilibrium in the frame of different
type kernels. Chaos, Solitons and Fractals. 2020; 130:109438.
[5] U¸car S. Existence and uniqueness results for a smoking model with determination and education in the frame of
non-singular derivatives. Discret Contin Dyn Syst - S. 2021; 14(7):2571-2589.
[6] Moze M, Sabatier J, Oustaloup A. Theoretical developments and applications in physics and engineering. Adv
Fract Calc. 2007;
[7] Atangana A, Koca I. On the new fractional derivative and application to nonlinear Baggs and freedman model.
J Nonlinear Sci Appl. 2016; 9(5):2467–2480.
[8] Caponetto R, Dongola G, Fortuna L, Petras I. Fractional order systems. Modeling and control applications. Vol.
Series A, World Scientific Series on Nonlinear Science. 2010.
[9] Dos Santos JPC, Monteiro E, Vieira GB. Global stability of fractional SIR epidemic model. 2017; 5:1–7.
[10] Fabrizio M. Fractional rheological models for thermomechanical systems. Dissipation and free energies. Fract Calc
Appl Anal. 2014; 17(1):206–223.
[11] Hilfer R. Fractional time evolution. Applications of fractional calculus in physics. 2000. 87–130 p.
[12] Diethelm K. The analysis of fractional differential equations: An application-oriented exposition using differential
operators of Caputo type. Vol. 2004, Lecture Notes in Mathematics. Springer Science & Business Media; 2010.
1–262 p.
[13] Kilbas A, Srivastava H, Trujillo J. Theory and applications of fractional differential equations. Vol. 129, Journal
of the Electrochemical Society. Elsevier; 2006. 2865 p.
[14] Podlubny I. Fractional differential equations, Volume 198: An introduction to fractional derivatives, fractional
differential equations, to methods of their. Elsevier; 1998.
[15] Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Prog Fract Differ Appl.
2015; 1(2):73–85.
[16] Caputo M, Fabrizio M. Applications of new time and spatial fractional derivatives with exponential kernels. Prog
Fract Differ Appl. 2016; 2(1):1–11.
[17] Atangana A, Baleanu D. New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model. Therm Sci. 2016; 20(2):763–769.
[18] Al-Refai M. On weighted Atangana–Baleanu fractional operators. Adv Differ Equations. 2020; 2020(1).
[19] Hattaf K. A new generalized definition of fractional derivative with non-singular kernel. Computation. 2020;
[20] Hristov J. On the Atangana – Baleanu derivative and its relation to the fading memory concept: The Diffusion
Equation. :175–193.
[21] Hristov J. Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising
analysis with emphasis on diffusion models. 2018; 1:269–341.
[22] Syam MI, Al-Refai M. Fractional differential equations with Atangana–Baleanu fractional derivative: Analysis
and applications. Chaos, Solitons Fractals X. 2019; 2:3–7.
[23] Tuan NH, Mohammadi H, Rezapour S. A mathematical model for COVID-19 transmission by using the Caputo
fractional derivative. Chaos, Solitons and Fractals. 2020; 140.
[24] Baleanu D, Mohammadi H, Rezapour S. A fractional differential equation model for the COVID-19 transmission
by using the Caputo–Fabrizio derivative. Adv Differ Equations. 2020; 2020(1).
Volume 12, Special Issue
December 2021
Pages 825-845
  • Receive Date: 14 February 2021
  • Revise Date: 09 July 2021
  • Accept Date: 01 August 2021