# br In Table we present the

In Table 1, we present the description and baseline values of parameters in system given by system (1).

System (4) can be more realistic as the breast cancer competi-tion model as a function of time should not follow the same frac-tional order dynamics. For this reason, we introducing five differ-ent orders of the fractional operators α1, α2, α3, α4, α5 ∈ (0, 1]. The system (4) is called commensurate when α1 = α2 = α3 = α4 = α5 = α, otherwise is called non-commensurate (in this case, the total order of the system is then changed from 5 to the sum of each particular order).

Using the Laplace transform given by Eq. (3) and the inverse Laplace transform on both sides of the Eq. (4), have the following iterative formula
Cn (t ) = C(0) + L −

Hn

sγ

sγ

L

sγ

Considering the limit when n tend to infinity, we obtain the approximate solution

Stability analysis of iteration method.

Theorem 1. We demonstrate that the recursive method given by Eq. (5) is stable.

Proof. We assume the following. It is possible to find five positive constants x1, x2, x3, x4 and x5 such that for all

and consider the following operator

s

μ

τ

Then

k

n

a

n

I
t

E

υ

d

d

where

q

t

t

a

H

k

M

a

T
t

T

q

From the results obtained in Eqs. (11) and (12), we conclude that the used iterative method is stable. This completes the proof.

Now, we propose a numerical solution of the breast cancer Midostaurin (PKC412) model using the numerical scheme proposed by Atangana and Toufik [30].

We consider a fractional differential equation of the form

C0Dtα g(t ) = f (t, g(t )),
the above equation can be converted to a fractional integral equation if we apply the fundamental theorem of fractional calculus, i.e.

t

j

tm

Using the two-step Lagrange polynomial interpolation, the function f(θ , y(θ )) can be approximate within the interval [tm, tm+1]
Pm (θ )

h

h

f

h

j

For simplicity, we define the following expressions

A

Considering Eq. (18), the approximate solution of (17) is given by

j

α(α
m

α

Using the suggested numerical scheme, we obtain the numerical solution of Eq. (4) as follows

j

j

j

j

j

where

Breast cancer competition model with Caputo–Fabrizio–Caputo fractional derivative.

The Caputo–Fabrizio fractional derivative without singular kernel in Liouville–Caputo sense (CFC) [31] is given by
CFC0
Dtα

α

n
−
α
dtn
n
−
α

where

The fractional breast cancer competition model (1) in Caputo–Fabrizio–Caputo sense is given by

The model is subject to initial conditions

Existence of the coupled solutions.

We investigate the numerical results predicted by the fractional model given by the system (26). Firstly start to investigate the existence and uniqueness of the solutions. By using the fixed-point theorem, we define the existence of the solution. First, transform system (26) into an integral equation as follows

CF
α

d

C

k

T

k

n

ω

ν

E

d

Now, we consider the following kernels

Now, we prove that the kernels 1, 2, 3, 4 and 5 satisfy the Lipschitz condition. To achieve we first prove this condition for each kernel proposed. Using the Cauchy’s inequality, then we assess the following