EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6774
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Efficient Treatment of Some Important
Fractal-Fractional Models: Theoretical and Numerical

Study
M. Adel1,∗, M. M. Khader2,3, Dragan Pamucar4, Hijaz Ahmad5,6,7

1 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah,
42351, Saudi Arabia
2 Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud
Islamic University (IMSIU), Riyadh, Saudi Arabia
3 Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
4 Széchenyi István University, Győr, Hungary
5 Operational Research Center in Healthcare, Near East University, Nicosia/TRNC, 99138
Mersin 10, Turkey
6 VIZJA University, Okopowa 59, 01-043 Warsaw, Poland
7 Department of Mathematics, College of Science, Korea University, Seoul 02841,
Republic of Korea

Abstract. In this study, we investigate two fundamental fractal-fractional (FF) models: the com-
petitive dynamics among Egyptian banks and the Brusselator system. For the banking model,
optimal control strategies are proposed to mitigate profit downturns during crises, such as the
COVID-19 pandemic, through a system of four fractional differential equations. Recognizing the
slow convergence of traditional numerical methods, an efficient integration technique is developed
to simulate both models with enhanced accuracy and computational efficiency. The simulation
results reveal the dynamic behaviors of the studied systems for various FF-operator values, con-
firming the robustness and precision of the proposed approach when compared with the classical
fourth-order Runge–Kutta method (RK4M). The presented technique offers a simple yet powerful
framework for modeling and analyzing complex FF-based dynamical systems.
2020 Mathematics Subject Classifications: 34A12, 41A30, 47H10, 65N20
Key Words and Phrases: Banks’ competition model, optimal control, Brusselator system,
fractal-fractional derivative, numerical integration, RK4M

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6774

Email addresses: adel@sci.cu.edu.eg (M. Adel),
mmkhader@imamu.edu.sa (M. M. Khader), pamucar.dragan@sze.hu (D. Pamucar),
hijazahmad@korea.ac.kr (H. Ahmad)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 2 of 16

1. Introduction

Mathematical models are not only instrumental in addressing scientific challenges but
are also gaining increasing prominence in sectors that drive economic growth, such as
banking and finance. A mathematical competition model, which is typically represented
by a system of ordinary differential equations (ODEs), is commonly employed to simu-
late competitive interactions among banks ([1], [2]). The Lotka�Volterra system is fre-
quently utilized in the financial domain to compare and analyze the profit dynamics of
banks ([3], [4]). Previous studies, such as [5] and [6], have extended this framework by
incorporating various definitions of fractional derivatives to capture the complex competi-
tive dynamics between Indonesian commercial and rural banks. According to the Central
Bank of Egypt (CBE), the Egyptian banking system comprises 33 banks that are catego-
rized into four distinct groups [7], and the regulations issued by the CBE apply uniformly
to all these institutions.

The CBE performs the dual roles of national monetary authority and central bank for
the nation. Like all other central banks, its primary function is to oversee both domestic
and international institutions. Additionally, its responsibilities include creating and imple-
menting Egypt’s financial laws, printing banknotes, overseeing foreign exchange reserves,
regulating the usage of the country’s currency, and overseeing the management of both
private and public loans. Since there is no obvious distinction in their business sectors,
which will be our emphasis in this study, there may be competition amongst the four bank
groups previously described in [8].

Several scholars have lately examined the fractional Brusselator system ([9], [10]).
In [9], authors studied the stability of this model, where in [10], the authors demonstrated
the existence of the system’s solutions numerically.

Since most of the fractional differential equations lack exact solutions, one must resort
to approximate and numerical methods ([11],[12]). In the case of the Egyptian banks model
and Brusselator system, it was shown that by offering a great deal more alternatives for
the optimal data fitting setting [13].

A lot of researchers have been interested in the FF-derivatives because it lets them look
at the fractal dimension while still analyzing the phenomenon in fractional items. This
category of FF derivatives has been employed in various significant investigations, publica-
tions, and scientific reports; these can be found there and in the references therein ([14],[15]).
Also, the The operational research ([16],[17]) is a very important tool to make the best
decision.

The outline of the paper is as follows: Section 2 presents some preliminaries concerning
the fractional calculus. Section 3 gives the description of the two fractal-fractional models:
the banks’ competition model and the Brusselator system. Section 4 investigates the
numerical implementation of the proposed method for solving the models under study.
Section 5 outlines the numerical simulation for these models. Finally, Section 6 ends the
paper with conclusions and future work.



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 3 of 16

2. Preliminaries

Fractional models define the fractional derivative using power law memory kernels
(PLMKs). This makes it easier for the system to describe memory and global correlation.
This is the paramount definition employed in the formulation of fractional calculus the-
ory ([18]-[19]). Thus, we will quickly go over some fundamental definitions of fractional
calculus with the PLMKs and fractal notions in this brief part.

Definition 1. [20]

Let Ω(t) belong to C(0, b) and be fractal differentiable on (0, b) with order %. The
fractal-fractional derivative of order ξ, with Riemann-Liouville’s derivative including the
PLMK, is given as follows:

FFPDξ,%Ω(t) =
1

Γ(n− ξ)

d

dt%

∫ t

0
(t− y)n−ξ−1Ω(y) dy,

where n− 1 < ξ, % ≤ n, n ∈ N, and dΩ(y)
d y% = limt→y

Ω(t)−Ω(y)
t%−y% .

Definition 2. [20]

Let Υ(t) belongs to C(0, b) and fractal differentiable on (0, b) with order %. The fractal-
fractional integral of order ξ, with including the PLMK is given as follows:

FFP Iξ,%Υ(t) =
%

Γ(ξ)

∫ t

0
(t− y)ξ−1 y%−1Υ(y) dy.

3. Description of the fractal-fractional models

Here, we will try to describe and introduce the fractal-fractional form of the studied
models: The FF banks’ competition model and the FF Brusselator system.

3.1. Fractal-fractional banks’ competition model

We suppose that, at any given period t, the profits gained by each of the four classes
that make up Egypt’s banking system�public, private, Arab, foreign, and investment col-
laboration banks are P (t), A(t), F (t), and I(t). Here, we will use the symbols
[P (t), A(t), F (t), I(t)] = [Υs(t), s = 1, 2, 3, 4]. As a result, the suggested competition
model between them is given as follows:

dΥs(t)

dt
= αsΥs(t)

(
1− Υs(t)

βs

)
− δs N̄[Υ1, Υ2, Υ3, Υ4], Υs(0) =

ˆ̄Υs, s = 1, 2, 3, 4.

(1)

The nonlinear term N̄[Υ1, Υ2, Υ3, Υ4] = Υ1(t)Υ2(t)Υ3(t)Υ4(t) in the system (1) is the
area of the market where the four banks engage with one another [21]. Where



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 4 of 16

(i) αk, k = 1, 2, 3, 4 are the rates of growth of banks that collaborate on investments,
international, Arabic, and private sectors, respectively.

(ii) βk, k = 1, 2, 3, 4 are the highest earnings recorded by the four categories mentioned
above.

(iii) δk, k = 1, 2, 3, 4 are the competition parameters.

Let us now reinterpret, in the Caputo sense, the banks’ competitiveness model under
fractal-fractional derivative which is formulated as follows:

FFPDξ,%Υs(t) = αsΥs(t)

(
1− Υs(t)

βs

)
− δs N̄[Υ1, Υ2, Υ3, Υ4]. (2)

To examine numerically the system with the FF derivative (2), we can rewrite (2) as:

RLDξΥs(t) = % t%−1

[
αsΥs(t)

(
1− Υs(t)

βs

)
− δs N̄[Υ1, Υ2, Υ3, Υ4]

]
. (3)

Upon substituting the derivative RLD with CD and utilizing the fractional integral, the
following is the obtained solution:

Ῡ(t) = Ῡ(0) +
%

Γ(ξ)

∫ t

0
y%−1(t− y)ξ−1F(Ῡ(y), y) dy, (4)

the vector function Ῡ(t) is given by Ῡ(t) = [Υ1(t),Υ2(t),Υ3(t),Υ4(t)]
T and

F(Ῡ(t), t) = (f1(Υ1,Υ2,Υ3,Υ4, t), f2(Υ1,Υ2,Υ3,Υ4, t), f3(Υ1,Υ2,Υ3,Υ4, t), f4(Υ1,Υ2,Υ3,Υ4, t))
T ,

(5)
where

fs(Υ1,Υ2,Υ3,Υ4, t) = αsΥs(t)

(
1− Υs(t)

βs

)
− δsΥ1(t)Υ2(t)Υ3(t)Υ4(t).

We can implement the Banach fixed point (BFP) theorem to prove adequate con-
straints on the existence of a unique solution of the model (1). Let B = R4 be a Banach
space with the norm:

‖Ῡ(t)‖ = max
t∈I

‖Υ1(t) + Υ2(t) + Υ3(t) + Υ4(t)‖,

and I = (t0 − ω, t0 + ω). To accomplish our aim, we define the mapping T : B → B by:

T(Ῡ(t)) = Ῡ(t) = Ῡ(0) +
%

Γ(ξ)

∫ t

0
y%−1 (t− y)ξ−1F(Ῡ(y), y) dy. (6)

We can follow the same manner as in [22] to prove the existence by showing that T is a
contraction mapping.



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 5 of 16

3.2. Fractal-fractional Brusselator system

Here, we will examine the Brusselator system, which is expressed as follows ([9], [10]):

θ̇1(t) = δ − (γ + 1) θ1 + θ21θ2, (7)

θ̇2(t) = γ θ1 − θ21θ2, (8)

θk(0) = θk,0, k = 1, 2. (9)

The parameters δ > 0, γ > 0, and θk,0, k = 1, 2 are constants [18].
Now, we formulate the fractal-fractional Brusselator system in the following form:

FFPDξ,%θ1(t) = δ − (γ + 1) θ1 + θ21(t)θ2,
FFPDξ,%θ2(t) = γ θ1 − θ21θ2.

(10)

The system (10) can be written as:

RLDξθ1(t) = % t%−1 (δ − (γ + 1) θ1 + θ21θ2),
RLDξθ2(t) = % t%−1 (γ θ1 − θ21θ2).

(11)

Upon substituting the derivative RLD with CD and utilizing the fractional integral, the
following is the obtained solution:

θ̄(t) = θ̄(0) +
%

Γ(ξ)

∫ t

0
(t− s)ξ−1s%−1F(θ̄(s), s) ds, (12)

where

θ̄(t) = [θ1(t), θ2(t)]
T , θ̄(0) = [θ1(0), θ2(0)]

T ,

F(θ̄(t), t) =
(

f1(θ1, θ2, t)
f2(θ1, θ2, t)

)
=

(
δ − (γ + 1) θ1 + θ21θ2

γ θ1 − θ21θ2

)
.

(13)

Using the BFP theorem, we can now establish adequate constraints on the existence and
uniqueness of the solution of the system (11). Let B = R2 with the norm:

‖θ̄(t)‖ = max
t∈I

‖θ1(t) + θ2(t)‖.

After that, we proceed as we did in the preceding section.

4. Numerical implementation on the proposed models

To demonstrate the numerical simulation of the models under study, we will set up
numerical schemes for them in their fractional form of the FF type in this section.



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 6 of 16

4.1. The Banks’ competition model

To do this, let us modify the model found in (3) as follows:

CDξΥi(t) = % t%−1
[
αiΥi(t)

(
1− β−1

i Υi(t)
)
− δi N̄[Υ1, Υ2, Υ3, Υ4]

]
, i = 1, 2, 3, 4.

(14)
By utilizing the integral on the system (14), we obtain:

Υi(t) = Υi(0) +
%

Γ(ξ)

∫ t

0
(t− y)ξ−1 y%−1 fi(Υ1(y),Υ2(y),Υ3(y),Υ4(y), y) dy, i = 1, 2, 3, 4,

(15)
where fi(Υ1(t),Υ2(t),Υ3(t),Υ4(t), t), are given in (5). Thus, at t = tn+1

Υn+1
i = ˆ̄Υi +

%

Γ(ξ)

∫ tn+1

0
(tn+1 − y)ξ−1 y%−1 fi(Υ1(y),Υ2(y),Υ3(y),Υ4(y), y) dy. (16)

Let us approximate y%−1 fi(Υ1,Υ2,Υ3,Υ4, y), as follows:

Lj(y) ≈ y%−1 fi(Υ1,Υ2,Υ3,Υ4, y)

=
y − tj−1

tj − tj−1
t%−1
j fi (Υ1,Υ2,Υ3,Υ4, tj)−

y − tj
tj − tj−1

t%−1
j−1 fi (Υ1,Υ2,Υ3,Υ4, tj−1) .

(17)

When we use the approximation (17) in the first equation of (16), respectively, we get:

Υn+1
1 = ˆ̄Υ1 +

%

Γ(ξ)

n∑
j=0

∫ tj+1

tj

(tn+1 − y)ξ−1 y%−1 f1(Υ1,Υ2,Υ3,Υ4, y) dy

= ˆ̄Υ1 +
%

Γ(ξ)

n∑
j=0

∫ tj+1

tj

(tn+1 − y)ξ−1
[ y − tj−1

tj − tj−1
t%−1
j f1 (Υ1,Υ2,Υ3,Υ4, tj)

− y − tj
tj − tj−1

t%−1
j−1 f1 (Υ1,Υ2,Υ3,Υ4, tj−1)

]
dy

= ˆ̄Υ1 +
%

Γ(ξ)

n∑
j=0

∫ tj+1

tj

(tn+1 − y)ξ−1

[
y − tj−1

tj − tj−1
t%−1
j f1 (Υ1,Υ2,Υ3,Υ4, tj)

]
dy

− %

Γ(ξ)

n∑
j=0

∫ tj+1

tj

(tn+1 − y)ξ−1

[
y − tj

tj − tj−1
t%−1
j−1 f1 (Υ1,Υ2,Υ3,Υ4, tj−1)

]
dy,

(18)



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 7 of 16

and after some arrangements, we can write:

Υn+1
1 =ˆ̄Υ1 +

%

Γ(ξ)

n∑
j=0

∫ tj+1

tj

(tn+1 − y)ξ−1 (y − tj−1)

tj − tj−1

[
t%−1
j f1 (Υ1,Υ2,Υ3,Υ4, tj)

]
dy

− %

Γ(ξ)

n∑
j=0

∫ tj+1

tj

(tn+1 − y)ξ−1 (y − tj)

tj − tj−1

[
t%−1
j−1 f1 (Υ1,Υ2,Υ3,Υ4, tj−1)

]
dy

= ˆ̄Υ1 +
%

Γ(ξ)

n∑
j=0

I1 −
%

Γ(ξ)

n∑
j=0

I2,

(19)
where

I1 =

∫ tj+1

tj

(tn+1 − y)ξ−1 (y − tj−1)

tj − tj−1

[
t%−1
j f1 (Υ1,Υ2,Υ3,Υ4, tj)

]
dy,

=
t%−1
j hξ

ξ(ξ + 1)
f1 (Υ1,Υ2,Υ3,Υ4, tj)

{
(n− j + 1)ξ(ξ + n− j + 2)− (n− j)ξ(2ξ + n− j + 2)

}
,

I2 =

∫ tj+1

tj

(tn+1 − y)ξ−1 (y − tj)

tj − tj−1

[
t%−1
j−1 f1 (Υ1,Υ2,Υ3,Υ4, tj−1)

]
dy

=
t%−1
j−1 h

ξ

ξ(ξ + 1)
f1 (Υ1,Υ2,Υ3,Υ4, tj−1)

{
(n− j + 1)ξ+1 − (n− j)ξ(ξ + n− j + 1)

}
.

(20)
The details of the derivation the formula given in (20) can be found in [22].
Now, using I1 and I2 in (19), and general for all components of the system (Υn+1

i , i =
1, 2, 3, 4), the model defined in (16) converts to the following system of algebraic equations
(for i = 1, 2, 3, 4):

Υn+1
i =

% hξ

Γ(ξ + 2)

n∑
j=0

t%−1
j fi (Υ1,Υ2,Υ3,Υ4, tj)

{
(n− j + 1)ξ(ξ + n− j + 2)− (n− j)ξ(2ξ + n− j + 2)

}
− % hξ

Γ(ξ + 2)

n∑
j=0

t%−1
j−1 fi (Υ1,Υ2,Υ3,Υ4, tj−1)

{
(n− j + 1)ξ+1 − (n− j)ξ(ξ + n− j + 1)

}
+ ˆ̄Υi.

(21)

4.1.1. Optimal control of banks’ profits

Recently, a wide variety of issues have been successfully addressed by the optimal control
theory in practical applications [23]. An intriguing question that arises in our present BCM
is how to affect the profits of the given competition model (2) by selecting an effective
mechanism to guide the model from (Υ1(0), Υ2(0), Υ3(0), Υ4(0)) to a desired final state
(Υ1(Tf ), Υ2(Tf ), Υ3(Tf ), Υ4(Tf )) in time Tf . This prefix objective is profit maximization.
To do this, let’s look at the system below:

FFPDξ,%Υi(t) = αiΥi(t)
(
1− β−1

i Υi(t)
)
− δi N̄[Υ1, Υ2, Υ3, Υ4]− εiOc(t)Υi(t). (22)



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 8 of 16

with the same previous initial conditions. Where εi = 1, i = 1, 2, 3, 4 indicates that there
are no investment limits because the constants εk, k = 1, 2, 3, 4 reflect the maximum level
of investment by the four bank categories. For that matter, Oc(t) ∈ L2[0, Tf ] is the
control parameter that indicates how much profit from each population is either retained
by the bank for market reinvestment. The best choice of this function can be derived and
obtained through Theorem 4 in [8]. For this type of problem, the majority of the current
numerical approaches converge slowly, leading to imprecise approximations [24].

4.2. The Brusselator system

In the same manner as in the previous subsection, we can construct the numerical
scheme for solving the Brusselator system (10). For this purpose, let us recall the model
given in (10):

CDξθ1(t) = % t%−1(δ − (γ + 1) θ1(t) + θ21(t)θ2(t)),
CDξθ2(t) = % t%−1(γ θ1(t)− θ21(t)θ2(t)).

(23)

By utilizing the integrals on the system (23), we obtain:

θ1(t) = θ1(0) +
%

Γ(ξ)

∫ t

0
(t− s)ξ−1 s%−1 f1(θ1(s), θ2(s), s) ds,

θ2(t) = θ2(0) +
%

Γ(ξ)

∫ t

0
(t− s)ξ−1 s%−1 f2(θ1(s), θ2(s), s) ds,

(24)

where the functions fk(θ1(t), θ2(t), t), k = 1, 2, are given in (13). Thus, at time t = tn+1

θn+1
1 = θ1(0) +

%

Γ(ξ)

∫ tn+1

0
(tn+1 − s)ξ−1 s%−1 f1(θ1(s), θ2(s), s) ds,

θn+1
2 = θ2(0) +

%

Γ(ξ)

∫ tn+1

0
(tn+1 − s)ξ−1 s%−1 f2(θ1(s), θ2(s), s) ds.

(25)

Thus, the following algebraic equation system is transformed by the system provided in
(25) as follows (k = 1, 2):

θn+1
k = θk,0 +

% hξ

Γ(ξ + 2)
.

n∑
j=0

[
t%−1
j

{
(n− j + 1)ξ(ξ + n− j + 2)− (n− j)ξ(2ξ + n− j + 2)

}
fk (θ1, θ2, tj)

− t%−1
j−1

{
(n− j + 1)ξ+1 − (n− j)ξ(ξ + n− j + 1)

}
fk (θ1, θ2, tj−1)

]
.

(26)



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 9 of 16

5. Numerical simulation

5.1. The Banks’ competition model

We will show a numerical simulation in [0, 9], for the model (2) and the modified system
(22) with optimal control, with different values of %, ξ, n, h, to show the precision and
caliber of the proposed scheme. We utilize the same values for the parameters in [8] in all
figures:

(α1, α2, α3, α4) = (0.7, 0.5, 0.45, 0.3), (β1, β2, β3, β4) = (33696, 31162, 11679, 3710),

(δ1, δ2, δ3, δ4) = (1.9, 2.3, 1.02, 5.0)×10−18, (ε1, ε2, ε3, ε4) = (0.392, 0.327, 0.245, 0.295).

Use the following initial conditions [8]:

ˆ̄Υ1 = 2.25, ˆ̄Υ2 = 2.3, ˆ̄Υ3 = 0.74, ˆ̄Υ4 = 0.6250.

We also give a comparison of the outcomes obtained by the suggested technique with those
acquired by employing the RK4M.

Figures 1-4 give the approximate solution that was achieved for the system under
investigation by using the suggested technique.

(i) The behavior of the numerical solution is given in Figure 1 for n = 50, using different
values of % = ξ = 1.0, 0.95, 0.9, 0.85.

(ii) In Figure 2, we show a comparison between the results using the RK4M at (% = ξ =
1) with n = 90 and the results using the suggested technique.

(iii) In Figure 3, the approximate solution of the modified system with the optimal control
Oc(t) = −2 e−2t by using the proposed method at (% = ξ = 0.95) and RK4M
(% = ξ = 1), at n = 90.

(iv) In Figure 4, we study the influence of the approximation order, n, on the approximate
solution of the proposed model with and without optimal control.

These findings indicate that the presented technique is suitable for solving the proposed
system, as the characteristics of the numerical solution derived from this technique are
contingent upon the values of %, ξ, n. Furthermore, we established a reinvestment-control
mechanism as a proposal to implement a pre-reinvestment-control system that can absorb
this unexpected fall in profits during crises, using the optimal function to recommend
remedies. Simulations were run again after taking the controlling function into account,
and the outcomes demonstrate how the banks’ earnings are managed.



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 10 of 16

Fig. 1: The approximate solution with various values of %, ξ.

5.2. The Brusselator model

We will show a numerical simulation on a test case of the system, to show the quality
and correctness of the given numerical method (26) in the interval [0, 4], h = 0.01 where
fractional order % ∈ (0, 1], and fractal dimension ξ ∈ (0, 1], with different values of δ, γ,
with initial conditions θ1,0 = θ2,0 = 1. Figures 5-8 give the numerical solutions for the
model under consideration by implementing the given method.

(i) The behavior of the approximate solution is shown in Figure 5 using different values
of % = ξ = 1, 0.9, 0.8, 0.7, with δ = 0.25, γ = 1, and n = 50.

(ii) In Figure 6, we compare the outcomes obtained by the proposed strategy with the
outcomes produced by utilizing the RK4M at (% = ξ = 1), and δ = 0.0, γ = 1 with
n = 90.

(iii) In Figure 7, we demonstrate the approximate solution’s behavior using several values
of γ = 0.5, 1.0, 1.5, and h = 0.1, δ = 0.2 at (% = ξ = 0.95) with n = 90.

(iv) In Figure 8, the approximation solution’s behavior using various values δ = 0.0, 0.5, 1.0,
and γ = 0.2 is presented with n = 75.



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 11 of 16

Fig. 2: The numerical solution by present method and RK4M at % = ξ = 1.

The results shown in Figures 5-8 demonstrate that the behavior of the approximate
solution derived from the specified technique is contingent upon the values of γ, δ, frac-
tional order %, and fractal dimension ξ. This signifies that the given scheme is suitable for
addressing the studied system.

Fig. 5: The solution θ1(t), θ2(t) against distinct values of ξ = %.



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 12 of 16

Fig. 3: The numerical solution with optimal control by present method (% = 0.95) and RK4M
(% = 1).

Fig. 6: The solution θ1(t), θ2(t) by the present method and the RK4M.



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 13 of 16

Fig. 4: The approximate solution with and without optimal control against distinct values of n.

Fig. 7: The numerical solution θ1(t), θ2(t) against different values of γ.



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 14 of 16

Fig. 8: The numerical solution θ1(t), θ2(t) against different values of δ.

6. Conclusions

This study delved into two significant fractal-fractional models: the competition among
Egyptian banks and the Brusselator system employing the fractal-fractional derivative op-
erator and fractional calculus methodologies. We investigated several choices of fractional
order (%), fractal dimension (ξ), and step size (h) to address these models. Our method
proves remarkably successful in modeling these systems, with precision regulated by re-
ducing h. Notably, numerical simulations using the FF operator outperform the RK4
approach for the models considered. We proposed a reinvestment-control function to
manage profit loss during crises, demonstrating its efficacy through simulation outcomes.
Numerical simulations are conducted using the Mathematica software package. Future
endeavors may involve exploring alternative fractional derivatives. Extending the model
to two-dimensional systems and using fractional variable orders. Exploring real experi-
mental data validation monitoring stations. Developing efficient numerical solvers tailored
for non-singular fractional kernels.

Acknowledgements

The researchers wish to extend their sincere gratitude to the Deanship of Scientific
Research at the Islamic University of Madinah for the support provided to the Post-
Publishing Program.

References

[1] C. Michalakelis, C. Christodoulos, D. Varoutas, and T. Sphicopoulos. Dynamic esti-
mation of markets exhibiting a prey-predator behavior. Expert Systems with Appli-
cations, 39:7690–7700, 2012.



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 15 of 16

[2] S. Lakka, C. Michalakelis, D. Varoutas, and D. Martakos. Competitive dynamics
in the operating systems market: modeling and policy implications. Technological
Forecasting and Social Change, 80:88–105, 2013.

[3] C. Zhu and G. Yin. On competitive Lotka-Volterra model in random environments.
Journal of Mathematical Analysis and Applications, 357:154–170, 2009.

[4] C. A. Comes. Banking system: three-level Lotka-Volterra model. Procedia Economics
and Finance, 3:251–255, 2012.

[5] W. Wang and M. A. Khan. Analysis and numerical simulation of the fractional
model of bank data with fractal-fractional Atangana-Baleanu derivative. Journal of
Computational and Applied Mathematics, 369:112646, 2019.

[6] X. Gong and F. M. A. Khan. A new numerical solution of the competition model
among bank data in Caputo-Fabrizio derivative. Alexandria Engineering Journal,
59:2251–2259, 2020.

[7] Central Bank of Egypt (CBE). Central Bank of Egypt Official Website, 2023. Ac-
cessed: 10 February 2023.

[8] O. A. M. Omar, H. M. Ahmed, and W. Hamdy. Investigation of Egyptian banks’
competition through a Riesz-Caputo fractional model. Fractal and Fractional,
7(473):1–21, 2023.

[9] V. Gafiychuk and B. Datsko. Stability analysis and limit cycle in a fractional system
with Brusselator nonlinearities. Physics Letters A, 372(29):4902–4904, 2008.

[10] Y. Wang and C. Li. Does the fractional Brusselator with efficient dimension less than
1 have a limit cycle? Physics Letters A, 363(5):414–419, 2007.

[11] N. H. Sweilam, M. M. Khader, and M. Adel. Numerical simulation of fractional Cable
equation of spiny neuronal dendrites. Journal of Advanced Research, 5(2):253–259,
2014.

[12] M. M. Khader, N. H. Sweilam, and A. M. S. Mahdy. Two computational algorithms
for the numerical solution for system of fractional differential equations. Arab Journal
of Mathematical Sciences, 21(1):39–52, 2015.

[13] S. Qureshi and A. Yusuf. Fractional derivatives applied to MSEIR problems: Com-
parative study with real-world data. The European Physical Journal Plus, 134:171,
2019.

[14] M. P. Yadav and R. Agarwal. Numerical investigation of fractional-fractal Boussinesq
equation. Chaos, 29(1):0131, 2019.

[15] K. M. Owolabi, A. Atangana, and A. Akgül. Modelling and analysis of fractal-
fractional partial differential equations: application to reaction-diffusion model.
Alexandria Engineering Journal, 59(4):2477–2490, 2020.

[16] D. Mittal. A Study for Application of Decision-Making Model in a Public Organiza-
tion. Spectrum of Operational Research, 3(41):183–192, 2026.

[17] A. Biswas, K. H. Gazi, S. P. Mondal, and A. Ghosh. A Decision-Making Framework
for Sustainable Highway Restaurant Site Selection: AHP-TOPSIS Approach based
on the Fuzzy Numbers. Spectrum of Operational Research, 2(1):1–26, 2025.

[18] H. Jafari, Abdelouahab Kadem, and D. Baleanu. Variational iteration method for
a fractional-order Brusselator system. Abstract and Applied Analysis, page 496323,



M. Adel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6774 16 of 16

2014.
[19] M. Adel, H. M. Srivastava, and M. M. Khader. Modeling and numerical simulation

for covering the fractional Covid-19 model using spectral collocation-optimization
algorithms. Fractal and Fractional, 6:1–19, 2022.

[20] A. Atangana, A. Akgül, and K. M. Owolabi. Analysis of fractal fractional differential
equations. Alexandria Engineering Journal, 59(3):1117–1134, 2020.

[21] Z. Li, Z. Liu, and M. A. Khan. Fractional investigation of bank data with fractal-
fractional Caputo derivative. Chaos, Solitons & Fractals, 131:109528, 2020.

[22] M. Adel and M. M. Khader. Theoretical and numerical treatment for the fractal-
fractional model of pollution for a system of lakes using an efficient numerical tech-
nique. Alexandria Engineering Journal, 82:415–425, 2023.

[23] R. S. Sharp and H. Peng. Vehicle dynamics applications of optimal control theory.
Vehicle System Dynamics, 49:1073–1111, 2011.

[24] M. Adel, H. M. Srivastava, and M. M. Khader. Implementation of an accurate method
for the analysis and simulation of electrical R-L circuits. Mathematical Methods in
the Applied Sciences, 12:1–10, 2022.