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The fractional integral related to the CFFD and proposed by Losada and Nieto reads as
As a response to the CFFD and being aware of the conflicting situations that exist between the classical Riemann-Liouville and Caputo derivatives, the classical Riemann-Liouville definition was modified [9, 10] in order to propose another definition known as the new Riemann-Liouville fractional derivative (NRLFD) without singular kernel and expressed for [gamma] [member of] [0, 1] as
This paper however uses the CFFD, so for more details about those recent definitions, please feel free to consult the articles and works mentioned above and also the references mentioned therein.
In this section we prove the existence and uniqueness results for the seventh order Korteweg-de Vries equation (KdV) with one perturbation level, expressed with the CFFD and given by
where [zeta] is the perturbation parameter and [sup.cf][D.sup.[gamma].sub.t] is the Caputo-Fabrizio fractional order derivative (CFFD) given in (5).
If the condition 1 > ((2(1-[gamma])/(2-[gamma])M([gamma]))K+ 2tK[gamma]/(2 - [gamma])M([gamma])) holds, then, there exists a unique and continuous solution to the seventh order Korteweg-de Vries equation with one perturbation level expressed with the CFFD given in (5):
Let us now come back to the full model (48) with the nonsingular kernel derivative CFFD (given in (5)) and with no higher order perturbation parameter [zeta], given as
The fractional integral (anti-derivative) associated to the CFFD was proposed as well by Losada and Nieto and proved to be:
The Laplace transform of the CaputoFabrizio fractional derivative (CFFD) is given by
where v and [mu] are the perturbation parameters, [sup.cf] [D.sup.[alpha]].sub.t] is the CFFD given in (2.1) with initial condition
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