Numerical Comparisons Between Some Recent Modifications of
Fractional Euler Methods
Received: October 22, 2023. Revised: May 21, 2024. Accepted: July 11, 2024. Published: September 2, 2024.
1 Introduction
Fractional calculus allows for procedures such as the
half-derivative by extending the ideas of integrals and
derivatives to non-integer orders. In situations where
classical calculus is inadequate, this area of math-
ematical analysis offers methods for modeling and
problem-solving in a variety of domains, such as fi-
nance, engineering, and physics. Fractional calcu-
lus provides a more comprehensive framework that
allows for the capture of complicated dynamics and
memory effects in systems, leading to more precise
descriptions and forecasts, see [1, 2, 3, 4, 5, 6].
Fractional differential equations are regarded very
helpful tools used to formulate memory-like or fractal
features in many physical models. These equations
have progressively been utilized for modeling a lot
of engineering, physics, biology, viscoelasticity, and
fluid dynamics problems, [7], [8], [9], [10], [11]. Nu-
merous approaches have recently been established for
handling FDEs in their linear and nonlinear cases in-
cluding finite difference approach, differential trans-
form approach, variational iteration approach, homo-
topy perturbation approach, Adomain decomposition
approach, predictor corrector approach, and many
others, [12], [13], [14], [15], [16], [17]. In the same
regard, there have been many nonlinear phenomena,
which have been formulated with the use of FDEs,
like fractional-order Stiff’s, Duffing’s, Rössler’s, and
Chuah’s systems.
This work is devoted to deal with a system of
FDEs using three recent numerical approaches; FEM,
MFEM, and IMFEM. Such a system can be repre-
sented by
Dδx1(s) = Θ1(s, x1(s), x2(s),··· , xn(s)),
Dδx2(s) = Θ2(s, x1(s), x2(s),··· , xn(s)),
.
.
.
Dδxn(s) = Θn(s, x1(s), x2(s),··· , xn(s)),
(1)
with initial conditions
x1(0) = v1, x2(0) = v2,··· , xn(0) = vn.
The following is how our paper is arrangement:
Part 2 remembers some necessary facts and prelim-
inaries connected to the fractional calculus. Part 3
is devoted to recollect the latest numerical versions
of the FEM, the MFEM and IMFEM, to handle the
systems of FDEs. In Part 4, the collected numerical
approaches are implemented to deal with fractional-
order systems. In Part 5, a certain numerical example
is provided for verifying the capability of the studied
approaches, succeeded by Part 6 that summarizes the
conclusion of this paper.
2 Fundamental facts
In this paper, we aim to implement certain numerical
approaches for approximating solutions to the follow-
ing Fractional Initial Value Problem (FIVP):
Dδ
∗z(s) = φ(s, z(s)),(2)
with initial condition
z(0) = z0,(3)
where δ∈(0,1]. To this end, we remember some fun-
damental concepts and facts connected with the frac-
tional calculus.
AMJED ZRAIQAT1, IQBAL M. BATIHA1,2, SHAMESEDDIN ALSHORM1
1Department of Mathematics, Al Zaytoonah University, Amman 11733, JORDAN
2Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, UAE
Abstract: - For dealing with systems of Fractional Differential Equations (FDEs), the primary goal of this work
is to offer several graphical comparisons performed with the use of specific recent adjustments of the so-called
Fractional Euler Method (FEM). These numerical adjustments are the Modified FEM (or MFEM) and Improved
Modified FEM (or IMFEM) coupled with the FEM itself. This would reveal the best approximate solution to
the fractional-order system consisting of FDEs in comparison with its exact solution.
Key-Words: -Caputooperator; FDEs; FEM; Caputo fractional differentiator; Riemann-Liouville fractional
integrator, Fractional differential equations; Fractional Euler method; Modified Fractional Euler method;
Improved fractional Euler method.
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Definition 1 [18] For 0< δ ≤1and x > 0, the
Riemann-Liouville integrator can be expressed by
Jδϑ(s) = 1
Γ(δ)s
0
(s−u)δ−1ϑ(u).du. (4)
The subsequent properties are satisfied by the
aforesaid operator, [18]:
•JδJβϑ(s) = JβJδϑ(s), δ, β > 0.
•JδJβϑ(s) = Jδ+βϑ(s), δ, β > 0.
•Jδxφ=Γ(δ+φ)
Γ(δ+φ+ 1)xφ+δ, φ > −1.
Definition 2 [18] The Caputo differentiator can be
expressed by
Dδ
∗ϑ(s) = Jm−δDmϑ(s)(5)
=1
Γ(m−δ)x
0
(s−u)(m−δ−1)ϑ(m)(u)du,
(6)
where m−1< δ ≤mfor which ϑ∈Cm[0, b]and
m∈N.
Lemma 1 [18] Consider, for m∈N, we have m−
1< δ ≤mand ϑ∈Cm[0, b]. Then
Dδ
∗Jδϑ(s) = ϑ(s),(7)
for s > 0. and
JδDδ
∗ϑ(s) = ϑ(s)−
m−1

k=1
Ï‘k(0+)sk
k!.(8)
Theorem 1 [19] For δ∈(0,1] and k=
0,1,2,··· , n + 1, consider Dkδ
∗ϑ(s)∈C(0, b], the
function Ï‘might be expanded around the node s0in
the following manner:
Ï‘(s) =
n

i=0
(s−s0)iδ
Γ(iδ + 1) Diδ
∗ϑ(s0)
+(s−s0)(n+1)δ
Γ((n+ 1)δ+ 1)D(n+1)δ
∗ϑ(ξ),
(9)
for which 0< ξ < s,∀x∈(0, b].
For further clarification, one might rewrite the pre-
vious equality in the following manner:
ϑ(s) = ϑ(s0) + (s−s0)δDδ
∗ϑ(s0)
Γ(δ+ 1)
+(s−s0)2δD2δ
∗ϑ(s0)
Γ(2δ+ 1) +··· +(s−s0)nδDnδ
∗ϑ(s0)
Γ(nδ + 1)
+(s−s0)(n+1)δD(n+1)δ
∗ϑ(ξ)
Γ((n+ 1)δ+ 1) .
(10)
3 Novel modifications of FEM
The main aim of this part is to recollect some current
numerical approaches (FEM, MFEM and IMFEM)
that can be employed to handle FIVPs. For this pur-
pose, we discretize the interval [0, b]as 0 = s0<
s1=s0+h < s2=s0+2h < ··· < sn=s0+nh =
bfor which si=s0+ih and h=b−a
nare respectively
called mesh points and step size, for i= 1,2, . . . , n.
In this regard, a useful generalized version of the tra-
ditional Euler method, called later on the FEM, was
proposed in [19], with the help of the first three terms
of Theorem 1 to deal with the FIVP (2-3). Such a
method consists of the following formula:
χ0=z0
χi+1 =χi+hδ
Γ(δ+ 1)φ(si, χi),(11)
for i= 0,1,··· , n −1. It should be noted that χi
indicates an approximate solution of problem (2-3).
In more recent time, the researchers in [12], have
effectively evolved a novel adjustment of the FEM,
named by the MFEM, for handling FIVP (2-3). Such
a formula has the following expression:
χ0=z0
χi+1 =χi+hδ
Γ(δ+ 1)
×φsi+hδ
2Γ(δ+ 1), χi+hδ
2Γ(δ+ 1)φ(si, χi),
(12)
for i= 1,2,··· , n −1.
In a similar manner, a further novel numerical ad-
justment, called IMFEM, has recently been estab-
lished in [20], for handling FIVP (2-3). Such a for-
mula has the following expression:
χ0=z0
χi+1 =χi+hδ
Γ(δ+ 1)
×φsi+hδ
2Γ(δ+ 1), χi+hδ
2Γ(δ+ 1)
×φsi, χi+hδ
Γ(δ+ 1)
×φsi, χi+hδ
Γ(δ+ 1)φ(si, χi)
(13)
for i= 0,1,2,··· , n −1.
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4 Dealing with fractional-order
systems
In order to provide a numerical solution to system (1)
with the use of the MFEM, one might consider the
first equation and consequently implement (12) in the
following manner:
x1(si+1) = x1(si) + hδ
Γ(δ+ 1)Θ1
×si+hδ
2Γ(δ+ 1), x1(si) + hδ
2Γ(δ+ 1)
×Θ1(si,x(s)) , x2(si) + hδ
2Γ(δ+ 1)
Θ1(si,x(s)) , xn(si) + hδ
2Γ(δ+ 1)Θ1(si,x(s)) .
(14)
for i= 0,1,2,··· , n −1, where x(s) =
(x1(s), x2(s),··· , xn(s)). To solve the remaining
states numerically, the above technique might be re-
peated similarly to get their corresponding approxi-
mate solutions. Eventually, the subsequent recursive
formula that provides an approximate solution to sys-
tem (1) can be deduced:
x1(si+1) = x1(si) + hδ
Γ(δ+ 1)
×ΘiΘi+hδ
2Γ(δ+ 1), x1(si) + hδ
2Γ(δ+ 1)
×Θ1(si,x(s)) , x2(si) + hδ
2Γ(δ+ 1)
×Θi(si,x(s)) ,··· , xn(si)
+hδ
2Γ(δ+ 1)Θ1(si,x(s)) 
.
.
.
xn(si+1) = xn(si) + hδ
Γ(δ+ 1)
×Θnsi+hδ
2Γ(δ+ 1), x1(si) + hδ
2Γ(δ+ 1)
×Θn(si,x(s)) , x2(si) + hδ
2Γ(δ+ 1)
×Θn(si,x(s)) ,··· , xn(si)
+hδ
2Γ(δ+ 1)Θn(si,x(s)) .
(15)
On the other side, with the aim of solving system
(1) with the help of using IMFEM (13), one might
track the same identical way mentioned earlier to get
the desired IMFEM’s solution.
x1(si+1) = x1(si) + hδ
Γ(δ+ 1)
×Θisi+hδ
2Γ(δ+ 1), x1(si) + hδ
2Γ(δ+ 1)
×Θ1(si,x(s)) , x2(si) + hδ
2Γ(δ+ 1)
×Θi(si,x(s)) ,··· , xn(si)
+hδ
2Γ(δ+ 1)Θi(si,x(s)) , x2(si)
+hδ
2Γ(δ+ 1)Θi(si,x(s)) ,··· , xn(si)
+hδ
2Γ(δ+ 1)Θi(si,x(s)) ,··· , xn(si)
+hδ
2Γ(δ+ 1)Θi(si,x(s)) 
.
.
.
xn(si+1) = xn(si) + hδ
Γ(δ+ 1)
×Θnsi+hδ
2Γ(δ+ 1), x1(si) + hδ
2Γ(δ+ 1)
×Θn(si,x(s)) , x2(si) + hδ
2Γ(δ+ 1)
×Θn(si,x(s)) ,··· , xn(si)
+hδ
2Γ(δ+ 1)Θn(si,x(s)) ,··· , xn(si)
+hδ
2Γ(δ+ 1)Θn(si,x(s)) ,··· , xn(si)
+hδ
2Γ(δ+ 1)Θn(si,x(s)) .
(16)
In fact, the previous formulas can be valid for each
of x2(si+1), s3(si+1),··· , xn(si+1), that is when
the first term x1(si)of such formula is replacing
by x2(si), x3(si),··· , xn(si)coupled with replac-
ing Θ1by Θ2,Θ3,··· ,Θn, respectively, for i=
0,1,2,··· , n.
In conclusion, formula (15) implies a numerical
solution to fractional-order system (1) achieved with
the use of MFEM, whereas formula (16) implies an-
other numerical solution to the same system achieved
with the use of IMFEM.
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5 Numerical findings
This section is devoted to provide an illustrative ex-
ample to verify the validity of the both numerical pro-
cedures discussed previously in Part 4.To this end, we
take into consideration the following fractional-order
system:
Dδx(t) = x(t)−2y(t)+2t,
Dδy(t) = 2x(t)−0.9y(t)−3,(17)
with initial conditions:
x(0) = 1, y(0) = 0.(18)
The following form represents the exact solutions of
problem (17-18) for δ= 1:
x(t) = 1
56699−90683et
20 cos √1239t
20 
+ 147382 cos √1239t
20 2
+1
5669932922tcos √1239t
20 2
+ 457√1239et
20 sin √1239t
20 
+1
56699147382 sin √1239t
20 2
+ 32922tsin √1239t
20 2,
(19)
and
y(t) = −10
566995723et
20 cos √1239t
20 
−5723 cos √1239t
20 2
+−10
56699−7316tcos √1239t
20 2
+ 205√1239et
20 sin √1239t
20 
+−10
56699−5723 sin √1239t
20 2
−7316tsin √1239t
20 2.
(20)
With assuming that δ= 1, h = 0.1and using
formulae (15) and (16), we depict approximate solu-
tions of system (17-17) in Figure 1 and Figure 2 got-
ten by MFEM and IMFEM. These figures have also
a comparison between such solutions and the exact
solutions together with the FEM’s solutions.
0 1 2 3 4 5 6 7 8 9 10
−2
0
2
4
6
8
10
12
14
16
18
t
x(t)
x(t) according to α=1 and and h=0.1
x(t) with FEM with α=1
x(t) with MFEM with α=1
x(t) with IMFEM with α=1
Exact x(t) with α=1
Fig.1: The solution x(t)obtained with the use of
FEM, MFEM and IMFEM when δ= 1 and h= 0.1
0 1 2 3 4 5 6 7 8 9 10
−5
0
5
10
15
20
25
t
y(t)
y(t) according to α=1 and h=0.1
y(t) with FEM with α=1
y(t) with MFEM with α=1
y(t) with IMFEM with α=1
Exact y(t) with α=1
Fig.2: The solution y(t)obtained with the use of
FEM, MFEM and IMFEM when δ= 1 and h= 0.1
It should be noted, based on the previous figures,
that it is difficult to distinguish which method is the
best. To this aim, we re-plot in Figure 3 and Figure
4 the MFEM and IMFEM’s solutions and compared
them with the FEM’s solution coupled with the exact
solutions twice again, but this time we take h= 0.01.
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0 1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
11
t
x(t)
x(t) according to α=1 and h=0.01
x(t) with FEM with α=1
x(t) with MFEM with α=1
x(t) with IMFEM with α=1
Exact x(t) with α=1
Fig.3: The solution x(t)obtained with the use of
FEM, MFEM and IMFEM when δ= 1 and h= 0.01
0 1 2 3 4 5 6 7 8 9 10
−2
0
2
4
6
8
10
12
14
16
18
t
y(t)
y(t) according to α=1 and h=0.01
y(t) with FEM with α=1
y(t) with MFEM with α=1
y(t) with IMFEM with α=1
Exact y(t) with α=1
Fig.4: The solution y(t)obtained with the use of
FEM, MFEM and IMFEM when δ= 1 and h= 0.01
On the basis of the above simulations, it might be
observed that the MFEM’s and IMFEM’s solutions
are closer to the exact solutions of system (17-17) than
the other solution generated by the FEM. However, to
see how numerical solutions of the considered system
look like for different fractional-order values, we plot
the MFEM’s solution in Figure 5 and the IMFEM’s
solution in Figure 6 with δ= 0.8,0.9,1and h= 0.01.
0 1 2 3 4 5 6 7 8 9 10
−5
0
5
10
15
20
t
x(t)
The solution x(t) by using MFEM for diferent values of α with h=0.01
α=0.8
α=0.9
α=1
Fig.5: The solution x(t)obtained with the use of
MFEM when δ= 0.8,0.9,1and h= 0.01
0 1 2 3 4 5 6 7 8 9 10
−5
0
5
10
15
20
25
t
y(t)
The solution y(t) by using MFEM for diferent values of α with h=0.01
α=0.8
α=0.9
α=1
Fig.6: The solution y(t)obtained with the use of
MFEM when δ= 0.8,0.9,1and h= 0.01
In fact, the usefulness of the previous numerical
simulations lies in illustrating the system’s dynamic
behavior and how it changes over time.
6 Conclusion
For the purpose of handling the fractional-order sys-
tems, this work has performed several graphical com-
parisons between certain recent adjustments of the
FEM, called the MFEM and IMFEM. It has appeared
that the IMFEM can generate high accuracy solutions
to the fractional-order systems in comparison with the
other studied approaches, succeeded by the MFEM
and finally FEM.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Iqbal M. Batiha has implemented the
Algorithm 1.1 and 1.2 in MATLAB.
Amjed Zraiqat was responsible for the
theoretical framework of this part.
Shameseddin Alshorm was responsi-
ble for writing the original draft.
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Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflicts of Interest
The authors have no conflicts of interest to
declare that are relevant to the content of this
article.
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