EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1842-1854
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Comparative Analysis of Conformable,
Non-conformable, Riemann-Liouville, and Caputo

Fractional Derivatives

A. Ait Brahim1,∗, J. El Ghordaf1, A. El Hajaji2, K. Hilal1, J. E.Nápoles Valdes3

1 AMSC Laboratory, University of Sciences and Technology , Beni Mellal, Morocco
2 OEE Departement, ENCGJ, University of Chouaib Doukali, El jadida, Morocco
3 FACENA Laboratory Universidad Nacional del Nordeste , (3400) Corrientes, Argentina

Abstract. This study undertakes a comparative analysis of the non conformable and conformable
fractional derivatives alongside the Riemann-Liouville and Caputo fractional derivatives. It exam-
ines their efficacy in solving fractional ordinary differential equations and explores their applications
in physics through numerical simulations. The findings suggest that the conformable fractional
derivative emerges as a promising substitute for the non conformable, Riemann-Liouville and Ca-
puto fractional derivatives within the range of order α where 1/2 < α < 1.

2020 Mathematics Subject Classifications: 26A33, 34A08, 65R20, 35R11

Key Words and Phrases: Conformbale fractional derivative, non-conformable fractional deriva-
tive, Riemann-Liouville, Caputo fractional derivatives

1. Introduction

The principles of classical calculus, which form the bedrock of mathematical anal-
ysis, have been instrumental in understanding and modeling various phenomena across
science and engineering. However, there exist numerous real-world phenomena that defy
straightforward modeling using classical calculus due to their non-local behavior or long-
term memory effects. Fractional calculus emerges as a powerful framework to address
such complexities by extending the traditional notions of derivatives and integrals to
non-integer orders [1–3, 5]. In recent years, fractional calculus has garnered significant
attention across a multitude of scientific and engineering disciplines. This surge in inter-
est is driven by its remarkable capacity to capture and describe phenomena that exhibit
intricate temporal dependencies or spatial interactions. Whether it’s the diffusion of par-
ticles in porous media, the dynamics of complex networks, or the behavior of viscoelastic

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5237

Email addresses: abdessamad11117@gmail.com (A. Ait Brahim),
a elhajaji@yahou.fr (A. El Hajaji), hilalkhalid2005@yahou.fr (K. Hilal),
elg jalila@yahou.fr (J. El Ghordaf), jnapoles@exa.unne.edu.ar (J.N. Valdes)

https://www.ejpam.com 1842 © 2024 EJPAM All rights reserved.



A. Ait Brahim et all / Eur. J. Pure Appl. Math, 17 (3) (2024), 1842-1854 1843

materials, fractional calculus provides a versatile toolkit for modeling and analyzing these
systems with unprecedented accuracy and depth. The applications of fractional calcu-
lus span a wide spectrum, encompassing fields such as physics, biology, finance, signal
processing, and control theory, among others. From characterizing anomalous diffusion
processes in biological systems to optimizing the performance of control systems with
fractional-order controllers, the influence of fractional calculus permeates numerous facets
of modern science and technology. In this context, the significance of fractional calculus
extends far beyond its theoretical foundations; it serves as a practical bridge between the-
oretical insights and real-world applications. As researchers continue to explore and refine
the methodologies within this field, the potential for breakthroughs in understanding com-
plex phenomena and developing innovative solutions grows ever greater. In this dynamic
landscape, the exploration of fractional calculus promises to yield profound insights and
transformative advancements across diverse domains, shaping the future of science and
engineering in profound ways. During the late 1950s and early 1960s, local non-integer
differential operators were introduced, defined in terms of incremental quotients. However,
their utilization remained sporadic over the following five decades, hindering widespread
dissemination and understanding of their potential. It wasn’t until 2014 that the first
comprehensive formalization of these local operators emerged, marking a significant mile-
stone in the development of non-integer order calculus. This contribution addressed the
limitations of global operators. R. Khalil et al. in their article ”A new definition of the
fractional derivative” (refer to [9]), introduced the concept of the ”Conformable fractional
derivative” with the following definition: let α ∈]0, 1] and M :]0,+∞[−→ R . Therefore,
the conformable fractional derivative of order α at t0 ≻ 0 is defined by

M (α) (t0) = lim
ε−→0

M
(
t+ εt0

1−α
)
−M (t0)

ε
, (1)

if it exists.
In fact, this novel form of derivation upholds all properties of conventional derivation

except for semi-groups. R. Khalil et al., along with T. Abdeljawad, the ”pioneer” of
this innovative concept, laid the theoretical groundwork in their article ”On Conformable
fractional calculus” [3]. A. Abdelhakim [1], in a 2019 publication, demonstrated that the
existence of limits is tantamount to differentiability in the classical sense, highlighting
both the fractional nature of this approach and the broader significance of the theory
initiated by R. Khalil et al. and T. Abdeljawad. This has sparked ongoing debates among
proponents and critics of the methodology.

This paper examines the research by D. R. Anderson and D. J. Ulness published in 2015
under the title ”Newly defined conformable derivatives” [4]. Building upon the provided
definition, we presume ordinary differentiability. Furthermore, it reflects the viewpoints
of R. Khalil et al. as referenced in [9]. Since differentiability aligns with the principles
outlined in [4], we present the primary computations without proofs, establish the rules
specified in [4], and introduce additional properties.
In response to these limitations, Khalil et al. introduced a new definition of fractional
derivative that satisfies these properties, which are absent in Riemann-Liouville and Ca-



A. Ait Brahim et all / Eur. J. Pure Appl. Math, 17 (3) (2024), 1842-1854 1844

puto fractional derivatives. This new derivative is known as the conformable fractional
derivative. Abdeljawad (2015) further refined the concept of the conformable fractional
derivative, providing a more comprehensive understanding of its properties. Notably, the
conformable fractional derivative is defined in a simpler manner compared to Riemann-
Liouville and Caputo fractional derivatives. Among these definitions, two of the most
prominent are:
If n is a positive integer and α ∈ [n− 1, n] derivative of N is given by:

• Riemann-Liouville definition:

Dα
a (M)(t) =

1

Γ(n− α)

dn

dtn

∫ t

a

M(u)

(t− u)α−n+1
du. (2)

• Caputo definition:

If n is a positive integer and α ∈ [n− 1, n), α derivative of N is given by

Dα
a (M)(t) =

1

Γ(n− α)

∫ t

a

M (n)(u)

(t− u)α−n+1
du. (3)

In their work referenced as [9], R. Khalil et al. introduced a novel concept termed the
”conformable fractional derivative.” Unlike the traditional Riemann-Liouville and Caputo
derivatives, which operate as nonlocal operators, the conformable fractional derivative
functions locally, closely resembling the conventional derivative.

Furthermore, a new definition of ”non-conformable” derivatives has been introduced in
a separate work [11]. Although these definitions effectively apply in the case of 0 < α < 1,
there arises a necessity for a general definition catering to conformable derivatives of any
order, integer or otherwise, thereby extending the scope of conformable derivatives to
higher orders [11]. Interested readers can also refer to additional sources such as [7], [16],
and [17] for further insights. Moreover, the realm of fractional and generalized calculus has
witnessed a surge in applications across various scientific and technological domains [6, 10,
13], showcasing advancements beyond the traditional integer orders. For a comprehensive
understanding of historical and technical aspects, readers are encouraged to explore [14].

This study delves into the analysis and comparison of conformable, Riemann-Liouville,
and Caputo fractional derivatives with order α, where 0 < α < 1. To enable a comprehen-
sive comparison, we conduct numerical simulations to explore the solutions of fractional
ordinary differential equations incorporating these three types of fractional derivatives.

2. Main results of conformable derivative

First, let’s recall the definition of Nα
1 M(t), which represents a non-conformable frac-

tional derivative of a function at a point t, as defined in [13]. This definition forms the
basis of our results, which closely resemble those found in classical calculus.



A. Ait Brahim et all / Eur. J. Pure Appl. Math, 17 (3) (2024), 1842-1854 1845

Definition 1. (See[13])
Given a function M : [0,+∞) → R. Then the N -derivative of M of order α is defined

by

Nα
1 M(t) = lim

ε→0

M
(
t+ εet

−α
)
−M(t)

ε
, (4)

for all t > 0, α ∈ (0, 1). If M is α-differentiable in some (0, a), and limt→0+ N
(α)
1 M(t)

exists, then define

N
(α)
1 M(0) = lim

t→0+
N

(α)
1 M(t). (5)

By following an analogous procedure to that of ordinary calculus, we can establish the
following result.

Definition 2. (See[11]) Be the function M : [0,+∞) → R. The N -derivative of function
M of order α is defined by

Nα
FM(t) = lim

ε→0

M(t+ εF (t, α))−M(t)

ε
, (6)

for all t > 0, α ∈ (0, 1) being F (α, t) is some function. Here we will use some cases of
F defined in function of Ea,b() .

Theorem 1. (See[11]) Let M and G be N -differentiable at a point t > 0 and α ∈ (0, 1] :
a) Nα

F (aM + bG)(t) = aNα
F (M)(t) + bNα

F (G)(t).
b) Na

F

(
tP
)
= et−αptt−1, p ∈ R.

c) If, in addition, M is differentiable then NF (M) = F (t, α)M ′(t).

Definition 3. The conformable fractional derivative of M in order α is represented by

(DαM) (t) = lim
v→0

M
(
t+ ve(α−1)t

)
−M(t)

v
, (7)

with , α ∈ [0, 1] and t > 0.
On the other hand if M is differentiable of order α in [0, a], a > 0, and limt→0+ (DαM) (t)
exists, we gets

DαM(0) = lim
t−→0+

(DαM)(t). (8)

Theorem 2. (See [8])
If M differentiable of order α at t0 > 0 and M : [0,+∞) −→ R then M is continuous

at t0.

Theorem 3. (See[8]) If M be α differentiable at a point t > 0. We have

(i) Dα(aM + bM) = a (DαM) + b (DαM), for all a, b ∈ R

(ii) (Dα(M/H)) = (M (DαH) +H (DαM)) /H2.

(iii) If M is differentiable, then (DαM) (t) = e(α−1)tM ′(t).



A. Ait Brahim et all / Eur. J. Pure Appl. Math, 17 (3) (2024), 1842-1854 1846

3. Fractional ordinary differential equations :

3.1. Fractional ordinary differential equation with non-conformable frac-
tional derivative

The initial value problem of fractional ordinary differential equations with generalized
derivative, for 0 < α < 1

Nα
F y(t) = λy(t), (9)

with the initial condition y(t0) = y0 has the solution

y(t) = y0e
λ(t−t0). (10)

In general form, if instead of (9) we have :

Nα
F y(t) = f(t), (11)

with the initial condition y(t0) = y0, the solution is

y(t) = y0e
∫ t
t0

f(s)dαF s
= y0e

NF
Jα
t0
f(t). (12)

3.2. Fractional ordinary differential equation with Riemann Liouville, Ca-
puto, and conformable fractional Derivatives

This section discusses solutions to fractional Ordinary Differential Equations using Rie-
mann Liouville, Caputo, and conformable fractional Derivatives and then their numerical
simulations. For more detail concerning these, one can refer to Abdeljawad (2015), Khalil
et al. (2014), Podlubny (1999), and Kilbas (2006).

Solutions The initial value problem of fractional ordinary differential equations with
Riemann-Liouville fractional derivative, for 0 < α < 1,

RDα
t w(t) = λw(t), t > 0, (13)

with the initial condition
I1−α
t w(t)

∣∣
t=0

= w0, (14)

has the solution
w(t) = w0t

α−1Eα,α (λt
α) , (15)

where Eα,β is the Mittag-Leffler Function defined by

Eα,β(t) =

∞∑
k=0

tk

Γ(αk + β)
, α, β > 0. (16)



A. Ait Brahim et all / Eur. J. Pure Appl. Math, 17 (3) (2024), 1842-1854 1847

Figure 1 : Comparative solutions for serval values of α, with Riemann-Liouville
fractional derivative.

The initial value problem of fractional ordinary differential equations with Caputo
fractional derivative, for 0 < α < 1,

CDα
t w(t) = λw(t), t > 0, (17)

with the initial condition
w(t)|t=0 = w0, (18)

has the solution
w(t) = w0Eα,1 (λt

α) . (19)

Figure 2 : Comparative solutions for serval values of α, with Caputo fractional derivative.



A. Ait Brahim et all / Eur. J. Pure Appl. Math, 17 (3) (2024), 1842-1854 1848

The initial value of fractional ordinary differential equations employing the conformable
fractional derivative, with 0 < α < 1,

Tα
t w(t) = λw(t), t > 0, (20)

with the initial condition
w(t)|t=0 = w0, (21)

has the solution
w(t) = w0e

λ tα

α . (22)

Figure 3 :Comparative solutions for serval values of α, with the conformable fractional
derivative.

This study Figure 1, Figure 2 and Figure 3 delves into the analysis and comparison of
conformable, Riemann-Liouville, and Caputo fractional derivatives of order α, where

0 < α < 1. To facilitate a comprehensive comparison, we conduct numerical simulations
to examine the solutions of fractional ordinary differential equations that incorporate

these three types of fractional derivatives.

4. Application to physics

4.1. Newton’s Cooling Problem:

Newton’s Cooling Problem is a fundamental concept in physics that studies how an
object loses heat when exposed to a colder environment. This phenomenon, named after
the famous physicist Sir Isaac Newton, is crucial in many fields, from engineering to
meteorology to biology. We define the problem of Newton’s cooling law states that the
cooling by equation

dT

dt
= −κ (T (t)− Tc) , T (0) = T0. (23)



A. Ait Brahim et all / Eur. J. Pure Appl. Math, 17 (3) (2024), 1842-1854 1849

In t = 0 we have the initial temperature T0, and we gets the solution:

T (t) = Tc + (T0 − Tc) e
−κt. (24)

First of all, we must ensure that in the pass of equation (23) to its corresponding fractional
equation, the physical parameters in it retain their physical units. Following [12], it is
necessary to take into account “the distortion” suffered by the derivative when going from
the ordinary case to the non-integer. We can use the physical parameter [k] = s−1 to pass
from the ordinary differential operator to the fractional one [11], as follows

d

dt
→ F

(
k−1, α

)
Nα

F . (25)

In this way, the differential operator retains its dimensionality [d/dt] = s−1, and α is the
order of derivative. Susbstituting (25) in (23), we obtain the fractional model as

Nα
FT (t) + F−1

(
k−1, α

)
κT (t) = F−1

(
k−1, α

)
κϵ(t)

F (t, α)T ′(t) + F−1
(
k−1, α

)
κT (t) = F−1

(
k−1, α

)
κϵ(t)

T ′(t) +
F−1

(
k−1, α

)
F (t, α)

κT (t) =
F−1

(
k−1, α

)
F (t, α)

κϵ(t).

Since

P (t) =
κF−1

(
k−1, α

)
F (t, α)

,

then

exp

(∫
κF−1

(
k−1, α

)
F (t, α)

dt

)
= exp

(
κF−1

(
k−1, α

) ∫ dt

F (t, α)

)
.

d

dt

[
exp

(
κF−1

(
k−1, α

) ∫ dt

F (t, α)

)
T (t)

]
= exp

(
κF−1

(
k−1, α

) ∫ dt

F (t, α)

)
F−1

(
k−1, α

)
F (t, α)

κϵ(t)exp

(
κF−1

(
k−1, α

) ∫ dt

F (t, α)

)
T (t)

= κF−1
(
k−1, α

) ∫
exp

(
κF−1

(
k−1, α

) ∫ dt

F (t, α)

)
ϵ (t)

F (t, α)
dt.

Therefore,

T (t, α) = exp

(
−κF−1

(
k−1, α

) ∫ dt

F (t, α)

)
κF−1

(
k−1, α

) ∫
exp

(
κF−1

(
k−1, α

) ∫ dt

F (t, α)

)
ϵ (t)

F (t, α)
dt.

(26)
The representation of this solution of the new fractional derivative for the values of

α = 0, α = 0.25, α = 0.5, and α = 0.75 is compared with results obtained using the factor
T (t, α) = e(α−1)t directly, as well as non-conformable fractional derivatives using the factor
T (t, α) = t−α (see [5, 15]). These comparisons illustrate the differences and similarities



A. Ait Brahim et all / Eur. J. Pure Appl. Math, 17 (3) (2024), 1842-1854 1850

in behavior between the new fractional derivative approach and the established meth-
ods. Specifically, the comparison with T (t, α) = e(α−1)t showcases how the exponential
factor influences the solutions for various values of α. Meanwhile, the comparison with
non-conformable fractional derivatives, which utilize the algebraic factor T (t, α) = t−α,
highlights the impact of a different functional form on the solution’s characteristics. These
graphical comparisons are essential for understanding the nuances and potential advan-
tages of the new fractional derivative method over traditional approaches, providing deeper
insights into its applicability and performance in various scenarios.

4.2. Falling body problem:

The problem of falling bodies, also known as the free fall problem, is a classic physics
scenario that examines the motion of objects under the influence of gravity alone, disre-
garding other forces such as air resistance. It serves as a fundamental concept in physics,
providing insights into motion, acceleration, and gravitational effects. When an object
is in free fall, it experiences a constant acceleration due to gravity, which on the surface
of the Earth is approximately 9.8 m.s−1 directed towards the center of the Earth. This
acceleration is represented by the symbol g.

The motion of falling bodies can be described using equations derived from Newton’s
laws of motion and kinematic equations. These equations allow us to predict various
aspects of the motion, such as the time taken to fall, the velocity at any given time, and
the distance traveled during the fall.

This problem considers the fall of a body of massm, starting from rest, under the action
of gravity. Suppose that the chosen reference system has as its origin the starting point
(rest of the body) at a height A from the floor at the moment the fall begins, that is, at t =
0. The downward movement will be chosen as positive. At any point P on its trajectory,
the distance traveled will be the function y dependent on time t, consequently, using
ordinary derivatives, the instantaneous velocity and acceleration snapshot, respectively,
are given by

v(t) =
dy(t)

dt
, a =

dv(t)

dt
=

d2y(t)

dt2
. (27)

According to Newton’s Law, we have F = mg, dv(t)dt = g. Then this problem is modeled
by the following differential equation with initial condition as follows

dv(t)

dt
= g, v(0) = 0, y(0) = A. (28)

Now consider the problem with the new conformable fractional derivative defined in Def-
inition 1.

Dαv(t) = g, v(0) = 0, y(0) = A. (29)

Using Theorem 2.4, part 6) we get:

e(α−1)tv′(t) = g. (30)



A. Ait Brahim et all / Eur. J. Pure Appl. Math, 17 (3) (2024), 1842-1854 1851

Integrating, we obtain :

v(t) = g(
1

1− α
e(1−α)t) + c. (31)

Using the initial condition, namely v(0) = 0, we find that the constant is zero. Similary,
we obtain :

y′(t) = ge(1−α)t(
1

1− α
e(1−α)t). (32)

Then, we have :

y(t) =
g

2(1− α)2
e2(1−α)t +M. (33)

Using the condition initial, we find that :

y(t) =
g

2(1− α)2
e2(1−α)t +A. (34)

Figure 4: Comparative solutions for serval values of α, with the new conformable
fractional derivative.

In this example, we present the solution for various values of α: α = 0, α = 0.25,
α = 0.5, and α = 0.75, utilizing the new conformable fractional calculus (see [1, 12]). We
then compare these solutions with those obtained using other methods, such as Khalil’s
conformable fractional derivative (see [13]), as well as ordinary derivatives. This com-
parison highlights the differences in the behavior of the solutions across these methods,
illustrating the potential advantages and limitations of the new conformable fractional
calculus in contrast to more traditional approaches.



A. Ait Brahim et all / Eur. J. Pure Appl. Math, 17 (3) (2024), 1842-1854 1852

5. Conclusion

Numerical simulations conducted on the solutions to initial value problems of fractional
ordinary differential equations reveal that the graphs of solutions using Riemann-Liouville
and conformable fractional derivatives exhibit significant proximity for 0 < α < 1. Fur-
thermore, the solutions’ graphs employing all three fractional derivatives show substantial
similarity for 1/2 < α < 1. Hence, conformable fractional derivatives offer a viable alter-
native to Riemann-Liouville and Caputo derivatives within the range of 1/2 < α < 1.

The simplicity of the conformable fractional derivative’s definition stands out as a no-
table advantage over Riemann-Liouville and Caputo derivatives. Unlike these traditional
fractional derivatives, the conformable fractional derivative is more intuitive and straight-
forward to implement, reducing the complexity often associated with fractional calculus.
Additionally, the conformable fractional derivative aligns with several fundamental prop-
erties of the usual derivative, such as the product rule, quotient rule, and chain rule,
which are not entirely satisfied by Riemann-Liouville and Caputo derivatives. This com-
patibility with classical derivative properties further enhances its appeal and broadens its
applicability in various fields.

Furthermore, the conformable fractional derivative offers a unique flexibility by allow-
ing for the consideration of both conformable and non-conformable kernels, as well as the
ability to vary the order of α. This dual capability provides a significant advantage in
modeling and analysis, enabling researchers and practitioners to tailor their approach to
the specific characteristics of the problem at hand. The ability to switch between con-
formable and non-conformable kernels and to adjust the fractional order introduces a level
of versatility that is not available in traditional ordinary or even other conformable models.

This adaptability makes the conformable fractional derivative particularly useful in
applications where the behavior of the system or process can be better captured through
a variable-order derivative, or where the inclusion of different kernel types can offer a more
accurate representation. Consequently, the conformable fractional derivative stands out as
a powerful tool in the realm of fractional calculus, providing both simplicity in definition
and extensive flexibility in application.

Authors’ Contributions

All authors have read and approved the final version of the manuscript. The authors
contributed equally to this work.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of
this paper.



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