EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 1128-1139
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A New Scheme for Solving a Fractional Differential
Equation and a Chaotic System

Ahmad Qazza1,∗, Mohamed A.Abdoon2,3, Rania Saadeh1, Mohammed Berir3,4

1 Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan
2 Department of Basic Sciences (Mathematics) Deanship of Common First Year,
Shaqra University, Riyadh 15342, Saudi Arabia
3 Department of Mathematics, Faculty of Science, Bakht Al-Ruda University, Duwaym,
Sudan
4 Department of Mathematics, Faculty of Science and Arts, Al-Baha university, Baljurashi
1988, Saudi Arabia

Abstract. The subject of this study is the solution of a fractional Bernoulli equation and a chaotic
system by using a novel scheme for the fractional derivative and comparison of approximate and
exact solutions. It is found that the suggested method produces solutions that are identical to
the exact solution. We can therefore generalize the strategy to different systems to get more
accurate results. We think that the novel fractional derivative scheme that has been offered and
the algorithm that has been suggested will be utilized in the future to construct and simulate
a variety of fractional models that can be used to solve more difficult physics and engineering
challenges.

2020 Mathematics Subject Classifications: 34C28, 34A08, 65P20

Key Words and Phrases: Numerical solutions, numerical scheme for ABC operator, analytical
solutions, Laplace decomposition method, chaos

1. Introduction

Due to the modeling of diffusion, control, and viscoelasticity in fractional calculus,
applied mathematics has grown in popularity over the past few decades. In physics and
engineering research, fractional differential equations are used [29, 32]. There are several
techniques for resolving fractional differential equations, see [11, 17]. The body of research
on modeling chaotic and hyperchaotic systems has been exploded recently with several ap-
plications in disciplines as diverse as electrical circuits, biology, and physics [12, 18, 31, 34].
Electrical circuit modeling, which is described in multiple works, is one of the most well-
known applications of chaos. It is justified to employ chaotic models given how difficult

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4769

Email addresses: aqazza@zu.edu.jo (A. Qazza), moh.abdoon@gmail.com (M. A. Abdoon),
rsaadeh@zu.edu.jo (R. Saadeh), mabberer@gmail.com (M. Berir)

https://www.ejpam.com 1128 © 2023 EJPAM All rights reserved.



A. Qazza et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1128-1139 1129

it is to predict many real-world situations. Asymptotic stability, which identifies the pre-
cise nature of the chaos, is only one of the many unique techniques for analyzing chaotic
systems that have emerged in recent years. The mathematical and scientific domains of
fractional calculus [2, 16, 25, 40] are extremely diverse: mathematics, biology, and other
domains[19, 39, 44] are rapidly expanding cutting-edge applications in the area of frac-
tional calculus [15, 21, 22, 42].
This research is significant since fractional operators have many different meanings. Deriva-
tives with exponential and Mittag-Leffler kernels are examples of singular-free derivatives
[20]. The fractional derivatives are helpful since they account for the influence of long-term
memory [24]. Recent research has shown that there are several convincing grounds for us-
ing fractional derivatives in practical contexts [41]. Chaotic systems violently respond to
both initial conditions and small changes in their parameters, as is well known [30].
The main goal of this research is to introduce a new approach for solving fractional dif-
ferential equations [4, 5, 8, 23, 27, 38, 45]. Also, we study the Chaotic behavior of the
studied problems.
Moreover, we sketch some figures to illustrate the efficiency of the proposed method.
This article is organized as follows, in the next section, we introduce the basic definitions
and properties. In Section 3, we introduce the numerical scheme of the ABC operator. In
Section 4, we introduce some applications and finally, we present the conclusion section.

2. Basic Principles

Definition 1. The Riemann Liouville integral (RLI) order of 0 < α < 1 and v(τ) is
provided by [33]:

Dαv (t) =
1

Γ(n− α)

∫ t

0
(t− τ)n−α−1vn (τ) dτ = In−αvn (t) , t > 0. (1)

Definition 2. The Riemann-Liouville fractional integral of order α > 0, given by [1]:

Iαa+f(t) =
1

Γ(α)

∫ t

a
(t− s)α−1f(s)ds, t > a. (2)

Definition 3. For a function y (τ) Caputo derivative of order 0 < α < 1 is given by [10]:

Iαy (t) =
1

Γ (α)

∫ t

0
(t− τ)α−1 y (τ) dτ , t > 0. (3)

Definition 4. The Mittag Leffler function can be expressed as follows [6]:

Eα (t) =

∞∑
k=0

tk

Γ(αk + 1)
. (4)



A. Qazza et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1128-1139 1130

Definition 5. (The Lagrange’s polynomial interpolation)
The Lagrange’s polynomial interpolation which defined by [8]

Pn(x) =
n∑

i=0

f (xi)Li(x),

where

Li (x) =
n∏

j=0,j ̸=i

x− xi
xi − xj

.

Definition 6. The ABC operator, y (t) in the RLI is given by [7]:

ABC
0 Dα

t y (t) =
B (α)

1− α

d

dt

∫ t

0
y (τ)Eα

(
α

1− α
(t− τ)α

)
dτ, 0 < α < 1. (5)

Where B (α) satisfies the condition B (1) = B (0) = 1.

3. Numerical scheme for ABC

The goal of this section is to investigate chaotic models in the sense of the ABC
fractional derivative, of the form:

ABC
0 Dα

0 v (t) = g (t, v (t)) ,

v (0) = v0.
(6)

A fractional integral equation can be derived from the equation above

v(t)− v(0) =
(1− α)g(t, v(t))

ABC(α)
+

α

Γ(α+ 1)×ABC(α)

∫ t

0
g(τ, v(τ))(t− τ)α−1dτ, (7)

where n = 0, 1, 2, 3 . . ., reformulated as

v(tn+1)− v(0) =
(1− α)g(tn, v(tn))

ABC(α)

+
α

ABC(α)× Γ(α+ 1)

∫ tn+1

0
g (τ, v (τ)) (tn+1 − τ)α−1 dτ

=
(1− α)g(tn, v(tn))

ABC(α)

+
α

ABC(α)× Γ(α)

n∑
k=0

∫ tk+1

tk

g(τ, v(τ)) (tn+1 − τ)α−1 dτ.

(8)

The following can be approximated using two-step Lagrange polynomial interpolation:

Pk (τ) =
(τ − tk−1)g (tk, v (tk))

tk − tk−1
− (τ − tk)g (tk−1, v (tk−1))

tk − tk−1

=
g (tk, v (tk)) (τ − tk−1)

h
− g (tk−1, v (tk−1)) (τ − tk)

h

≃ g (tk, vk) (τ − tk−1)

h
− g (tk−1, vk−1) (τ − tk)

h
,

(9)



A. Qazza et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1128-1139 1131

vn+1 = v0 +
(1− α)

ABC(α)
g(tn, v(tn))

+
α

ABC(α)× Γ(α)

n∑
k=0

(
g(tk, vk)

h

∫ tk+1

tk

(τ + tk−1t)(tn+1 − τ)α−1dτ

− g(tk−1, vk−1)

h

∫ tk+1

tk

(τ − tk)(tn+1 − τ)α−1dτ

)
.

(10)

For simplicity

Aα,k,1 =

∫ tk+1

tk

(τ − tk−1)(tn+1 − τ)α−1dτ, (11)

Aα,k,2 =

∫ tk+1

tk

(τ − tk)(tn+2 − τ)α−1dτ

Aα,k,1 = hα+1 (n+ 1− k)α(n− k + 2 + α)− (n− k)α(n− k + 2 + 2α)

α(α+ 1)

Aα,k,2 = (hα+1)
(n+ 1− k)α+1 − (n− k)α(n− k + 1 + α)

α(α+ 1)
.

(12)

By combining equations (11) and (12) and substituting in (10),

vn+1 = v(1) +
(1− α)

ABC(α)
g (tn, v (tn)) +

α

ABC(α)
n∑

j=0

(
hαg (tk, vk)

Γ(α+ 1)
((1 + n− j)α (2 + α+ n− k) + (j − n)α(2 + n− k + 2α))

− hαg (tj−1, vj−1)

Γ(1 + α)

(
(n− j + 1)α+1 + (j − n)α(n− j + 1 + α)

))
.

(13)

4. Applications

In this part, we explore the usefulness of the novel scheme for ABC fractional derivative
for solving an initial value problem (IVP) numerically.

Problem 1. We start with the Bernoulli equation [7]:

ABC
0 Dα

t y(t) = 2y(t)− 4y2(t), (14)

where 0 < α ≤ 1 and y(0) = 1, ABC
0 Dα

t is ABC operator, given in Eq. (7). The exact
solution of the Bernoulli equation is

y(t) =
−1

e−2t − 1
, (15)

under y(0) = 1, where ABC
0 Dα

t is defined by Eq. (5) with the parameter α, when α = 1,
the Bernoulli Equation (14) has an exact solution according to the proposed the numerical



A. Qazza et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1128-1139 1132

scheme for ABC fractional derivative, and we show this results in the Table 1 and Table
2.

Table 1: The numerical solutions of Eq. (14) when α = 1.

h t = 10 t = 12 t = 14

1/10 0.500000000938279 0.500000000018475 0.500000000000364
1/20 0.500000000627998 0.500000000011706 0.500000000000218
1/40 0.500000000554842 0.500000000010206 0.500000000000188
1/80 0.500000000531467 0.500000000009744 0.500000000000179
1/160 0.500000000522537 0.500000000009573 0.500000000000175
1/320 0.500000000518709 0.50000000000950 0.500000000000174
1/640 0.500000000516948 0.500000000009468 0.500000000000173
yExact 0.500000000515288 0.500000000009438 0.500000000000173

Table 2: The numerical solutions of Eq. (14) when α = 0.99.

h t = 10 t = 12 t = 14

1/10 0.500000003717573 0.500000002289498 0.500000001921669
1/20 0.500000003391435 0.500000002272609 0.500000001916067
1/40 0.500000003312222 0.500000002268483 0.500000001909773
1/80 0.500000003291813 0.500000002273635 0.500000001940562
1/160 0.500000003340172 0.500000002329872 0.500000001934836
1/320 0.500000003462054 0.500000002529303 0.500000002330563
1/640 0.500000004155025 0.500000003461087 0.500000004209577

In Table 1, we provide numerical results from our novel scheme for ABC fractional
derivative to fractional Bernoulli equation Eq. (14) when α = 1 at t = 10, t = 12 and
t = 14, and when α = 0.99 at t = 10, t = 12 and t = 14 in Table 2. The numerical answers
we provided matched the exact solution perfectly, and the step size h is small enough.



A. Qazza et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1128-1139 1133

 

Figure 1: A comparison between the exact and approximate solutions of Eq.

(14).

Problem 2. We discuss the Chen system:
ABC
0 Dα

t u (t) = a(v (t)− u(t)),
ABC
0 Dα

t v (t) = (c− a)u (t)− u (t)w (t) + cv(t),
ABC
0 Dα

t w (t) = u (t) v (t)− bw (t) .

(16)

With u (0) = −5, v (0) = −1 and w (0) = −1, where a, b, c ∈ R, t > 0, and ABC
0 Dα

t is
the ABC operator, the parameters a = 7.5, b = 1.0 and c = 5. We show this results in the
Table 3 and Table 4.

Table 3: The numerical solutions of Eq. (16) at t = 14 and α = 1.

h x y z

1/10 0.996232907806605 1.581099450806809 0.495241231468361
1/20 1.043578882387420 1.651420448120412 0.544251070796891
1/40 1.058429831301523 1.675181183975680 0.560064850780525
1/80 1.065523942097168 1.687112985888534 0.567652302039966
1/160 1.069190461371803 1.693385027057710 0.571579495559338
1/320 1.071078400638952 1.696632550666957 0.573603308346396
1/640 1.072039151580891 1.698288510008531 0.574633680132214



A. Qazza et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1128-1139 1134

Table 4: The numerical solutions of Eq. (16) at t = 14 and α = 0.99.

h x y z

1/10 1.621482381001720 1.976822616207164 1.315702543995343
1/20 1.614809101418269 1.966952972142942 1.304706684275194
1/40 1.615906131167622 1.964093205028113 1.306415886448812
1/80 1.617681111000321 1.962647251183329 1.309263163719868
1/160 1.618884483012851 1.961844891929927 1.311201894320728
1/320 1.619565828368396 1.961413183136856 1.312301530738447
1/640 1.619926462286785 1.961188279022434 1.312884047918658

In Table 3 and Table 4 provide numerical results from the novel scheme for ABC
fractional derivative to fractional Chen system Eq. (16), when α = 1 and t = 14 in Table
3, and when α = 0.99 and t = 14 in Table 4.

Figure 2: Chaotic attractor of Eq. (16), when α = 1.



A. Qazza et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 1128-1139 1135

Figure 3: Chaotic attractor of Eq. (16), when α =
0.99.

In Figure 2 and Figure 3, we plot the numerical solutions of Eq. (16) at the values
(a, b, c) = (7.5, 1, 5), and (x0, y0, z0) = (−5,−1,−1). In these figures, we display the Eq.
(16) attractors obtained using the novel scheme for the fractional derivative when α = 0.99
and α = 1. This phenomena is known as the chaos and it is characterized by complex
non-linear behaviors such as a periodic long-term behavior, erratic responses [9, 26, 28, 43].

5. Conclusions

A unique numerical approach was developed to solve Bernoulli equation and Chen sys-
tem based on ABC operator. The shortcomings of the well-known predictor-corrector ap-
proach are addressed by a numerical methodology. It is based on the Lagrange polynomial
and the fundamental theorem of fractional calculus. Rapid convergence, high efficiency
and accuracy, and user-friendliness are the distinguishing features of this approach. The
method was used to solve a fractional equation and system for which there exist solutions
as well as a nonlinear system. We advise wider use of the method to address physics and
engineering challenges that are becoming ever more complicated. In the future, we intend



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