EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 2, 2024, 1155-1167
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On a novel fractional calculus and its applications to
well-known problems

A. Ait Brahim1,∗, A. El Hajaji2, K. Hilal 1, J. El Ghordaf1

1 Laboratory of Applied Mathematics and Scientific Calculus, University of Sciences and
Technology, Béni Mellal , Morocco.
2 OEE Departement, ENCGJ, University of Chouaib Doukali, El jadida, Morocco

Abstract. The objective of this study is to introduce three novel and comprehensive models
utilizing the concept of conformable fractional calculus. These models encompass the fractional
population growth model (FPGM), the fractional body cooling model (FBCM), and the fractional
heat differential equation (FHDE). Firstly, using new results, a type of new conformable fractional
derivative introduced in [5]:

(DαH) (y) = limz−→0
H(y+ze(α−1)y)−H(y)

z ,

where α ∈ (0, 1] and H is a function. We obtain exact solutions of the considered models. The
results feature excellent agreement of exact solutions. Finally, it is shown that the proposed method
provides a more powerful mathematical tool for solving models in mathematical physics.

2020 Mathematics Subject Classifications: 26A33, 34A08, 65M70, 35R11, 92D25

Key Words and Phrases: New conformable fractional derivative, population grow model, body
coolig model, fractional heat equation.

1. Introduction

Historically, the origins of fractional calculus began with the correspondence between
L’Hopital and Leibniz [1, 2], where the former answered the question of whether the order
of derivation in an expression could be of a fractional nature. Obviously, this question
is self-contradictory, because the order of derivation indicates the frequency with which
the differential operator is applied, that is, it counts the number of repetitions of the
entire differential process. This origin was followed by further contributions by eminent
mathematicians, including the European Euler, Lacroix, Laplace, etc. One of the most

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5159

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)

https://www.ejpam.com 1155 © 2024 EJPAM All rights reserved.



A. Ait brahim et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1155-1167 1156

interesting, comprehensive, and well-founded the work is Riemman-Liouville [2], the cor-
responding fracture model starts from Fractional differential operators with two forms

Dα
c+H(z) =

1

Γ(m− α)

dm

dtm

∫ z

c

H(t)

(z − t)α+1−m
dt,m− 1 < α < m ∈ N, (1)

Dα
d−H(z) =

1

Γ(m− α)

dm

dtm

∫ d

z

H(t)

(z − t)α+1−m
dt,m− 1 < α < m ∈ N. (2)

Recently, the authors of [3] and [4] defined the classical fractional derivatives, suppos-
edly conformable derivatives fractional derivatives, depending only on the fundamental
limit definition derivation of Khalil et al. [3] presented a new derivation of H of order α,
known as :

TαH(t) = lim
ε−→0

H
(
t+ ϵt1−α

)
−H(t)

ε
, (3)

at 0 the fractional derivative is defined as H(α)(0) = limt−→0+ Tα(H)(t).
Also, Katugampola [4] proposed a new container defined as

Dα(H)(t) = lim
ε−→0

H
(
teεt

−α
)
−H(t)

ε
, (4)

with t > 0 and α ∈ [0, 1].
If H is differentiable in order α in some ([0, a], a > 0, and limt→0+ Dα(H)(t) exists, then
we obtain :

Dα(H)(0) = lim
t−→0+

Dα(H)(t). (5)

The studies cited in references [3, 4, 20] have significantly contributed to our understanding
of the properties of derivatives with respect to the parameter α. These investigations have
revealed that the derivatives of α adhere to fundamental rules such as the quotient rule
and product rule, akin to their counterparts in traditional calculus. Notably, these findings
evoke parallels with well-known theorems like Rolle’s theorem and the mean value theorem,
underscoring the robustness and applicability of the principles of calculus in the context of
fractional calculus. By elucidating these parallels, these studies provide valuable insights
into the behavior and characteristics of fractional derivatives, enriching our toolkit for
analyzing and solving complex problems across various scientific disciplines.

The aim of this study is to leverage the outcomes derived in [5] and extend their appli-
cation to physics, specifically in heat differential formulas, body cooling, and population
growth models. These applications have previously been explored by Khalil et al. in [6],
J. E. Napoles Valdes et al. in [19], and also in various references [20–23]. Our work builds
upon these previous studies by applying simpler models using the novel results of the
conformable fractional derivative outlined in [5].



A. Ait brahim et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1155-1167 1157

2. New results of fractional calculus

From calculus we see that the derivative of the function H at a given point z = d in
its domain is defined by the following restriction:

H ′(d) = lim
h→0

H(d+ h)−H(d)

h
. (6)

We define H as a differentiable function at z = d if its limit exists, denoted as H ′(d),
furthermore, if H is differentiable at this point, it is also continuous at the same point.

Definition 1. The new fractional derivative of H in order α is represented by

(DαH) (t) = lim
v→0

H
(
t+ ve(α−1)t

)
−H(t)

v
, (7)

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

DαH(0) = lim
t−→0+

(DαH)(t). (8)

Theorem 1. (see[5])
If H is differentiable of order α at t0 > 0, and H : [0,+∞) −→ R, then H is continuous

at t0.

Theorem 2. (see[5])
If H be α differentiable at a point t > 0, we have :

1) Dα(aH + bH) = a (DαH) + b (DαH), for all a, b ∈ R
2) Dα (tn) = ne(α−1)ttn−1 for all n ∈ R
3) Dα(β) = 0, for all constant H(t) = β.
4) Dα(HG) = H (DαG) +G (DαH).
5) Dα(H/G) = H (DαG) +G (DαH) /G2.
6) If H is differentiable, then (DαH) (t) = e(α−1)tH ′(t).

Theorem 3. Let α ∈ (0, 1], then:

1) Dα
(

1
1−αe

(1−α)t
)
= 1.

2) Dα
(
sin 1

1−αe
(1−α)t

)
= cos 1

1−αe
(1−α)t.

3) Dα
(
cos 1

1−αe
(1−α)t

)
= − sin 1

1−αe
(1−α)t.

4) Dα
(
e

1
1−α

e(1−α)t
)
= e

1
1−α

e(1−α)t

.

3. Applications to physics

3.1. Model of population growth

Population growth models are essential tools in understanding the dynamics of popu-
lations over time. They are used in various fields such as biology, ecology, epidemiology,



A. Ait brahim et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1155-1167 1158

and sociology to predict and analyze changes in population size. We define the problem
of population growth model by

dZ

dt
= kZ(t), Z(0) = N. (9)

Here L is constant, N is the initial value of population and Z is the population function,
and we can write the solution of this problem

Z(t) = NeLt. (10)

The following figure shows this solution :

Now, we present this problem in the form of new conformable fractional :

Z(α)(t) = LZ(t), Z(0) = N. (11)

From property (6) theorem(2) , we have :

e(α−1)tZ ′(t) = LZ(t). (12)

Therefore, we can continue to use the following expression

Z ′(t) = ke(α−1)tZ(t) or
dZ(t)

dt
= Le(α−1)tZ(t). (13)

By applying the separation of variables, we obtain:

dZ(t)

y(t)
= Le(α−1)tdt. (14)



A. Ait brahim et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1155-1167 1159

Now by integration we have :

lnZ(t) =
k

α− 1
e(α−1)t +B, (15)

we obtain:
Z(t) = eBe

L
α−1

e(α−1)t

, (16)

and we use the initial condition:

Z(t) = Ne
L

α−1
e(α−1)t

. (17)

In this example, we present the solution for different values of α: α = 0, α = 0.25, α = 0.5,
and α = 0.75 using the new conformable fractional calculus (see [5]), and we compare it
with other methods like Khalil’s conformable fractional derivative (see [3]) and ordinary
derivatives.

3.2. Model of body cooling

As he shifted his sweater over time, Newton’s law of cold was a body law designed
to Body warming is proportional to the temperature difference between the body and its



A. Ait brahim et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1155-1167 1160

surroundings. The procedure is only a slight difference, since one must take into account
the impossibility of temperature differences and the way the sweater transfers in the same
way. This is equivalent to confirming that the heat transfer coefficient between the heat
sink and the temperature difference is a constant.

We define the problem of Newton’s cooling law states that the cooling by equation

dT

dt
= −k (T (t)− Tc) , T (0) = T0. (18)

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

T (t) = Tc + (T0 − Tc) e
−kt. (19)

The following figure shows this solution :

Utilizing property (6) of theorem (2), we derive the novel conformable fractional dif-
ferential formula for this particular problem:

dαT (t)

dtα
= −k (T (t)− Tc) , (20)

so,
e(α−1)tT ′(t) = −k (T (t)− Tc) . (21)

Then, we have:
dT (t)

dt
= −e(1−α)tk (T (t)− Tc) , (22)



A. Ait brahim et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1155-1167 1161

and
dT (t)

T (t)− Tc
= −e(1−α)tk. (23)

Therefore,

Ln (T (t)− Tc) =
−ke(1−α)t

1− α
+ C, (24)

then,

T (t) = Tc + eCe
−ke(1−α)t

1−α , (25)

with the initial condition, we have the final solution:

T (t) = Tc + (T0 − Tc) e
−ke(1−α)t

1−α . (26)

The graphical 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 [19]).



A. Ait brahim et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1155-1167 1162

3.3. Fractional Heat equation

The heat equation is a fundamental concept in physics and thermal engineering. It
describes how the temperature in a material varies over time depending on the heat ap-
plied to it and its thermal properties. This equation is widely used in many fields such as
thermodynamics, geophysics, meteorology, materials engineering, and many others.

Solving the heat equation makes it possible to predict the temporal evolution of the
temperature in a given material. Solutions can be obtained analytically in some simple
cases, but in most cases numerical methods are required to obtain accurate solutions.
Techniques such as the finite difference method, finite element method or finite volume
method are commonly used for this purpose.
The heat equation has many practical applications. For example :

• Modeling heat transfers in buildings to improve energy efficiency. Prediction of
temperature distribution in materials during manufacturing processes.

• Study of heat propagation in soils for geothermal energy. Understanding of weather
phenomena such as cloud formation and ocean currents.

The heat equation is an essential tool for understanding and predicting thermal phenomena
in a wide variety of scientific fields and technological applications. Its simple but powerful
mathematical formulation makes it a pillar of thermal modeling and contributes to many
advances in science and engineering. Then form of fractional heat equation represented
by :

∂α

∂tα
∂αz(x, t)

∂tα
=

∂2z(x, t)

∂x2
, (27)

z(0, t) = 0, t > 0, (28)

z(1, t) = 0, t > 0, (29)

∂z(x, 0)

∂t
= 0, (30)

z(x, 0) = F (x), 0 < x < 1. (31)

Consider the conformable fractional linear differential equations with constant coefficients
using the findings presented in [20, 23]:

dα

dyα
dαz

dyα
± µ2z = 0. (32)

We use the equation R2 ± µ2 = 0, and associate to this equation, then we have R = ±ν,
or R = ±µi. Furthermore, according to theorem (3), we establish that

z = e∓
µ

1−α
e(1−α)t

, (33)

constitute two distinct solutions of the equation. Also, in the second case we obtain by
properties (2) and (3) above (see theorem(3)), that

z1 = sin(
µ

1− α
e(1−α)t) and z2 = cos(

µ

1− α
e(1−α)t). (34)



A. Ait brahim et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1155-1167 1163



A. Ait brahim et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1155-1167 1164

Let z1 = sin( µ
1−αe

(1−α)t) and z2 = cos( µ
1−αe

(1−α)t) are two independent solutions, we use
the new fractional [5], for different value of α : α = 0, α = 0.25, α = 0.5, and α = 0.75,
and we compare with ordinary derivative and Khalil conformable fractional [3].

Also, we have the form of new fractional differential formula of order 2α with constant
coefficients represented by

dα

dxα
dαz

dxα
+ a

dαz

dxα
+ bz = h(x). (35)

Can be write Dα for dα

dxα , then we have :

Dα (Dαz) + aDαz + bz = h(x). (36)

To obtain the solution, we consider the equation R2+aR+ b = 0. Utilizing the properties
of the new conformable fractional derivative as outlined in [5], along with properties (2),
(3), and (4) (refer to theorem (3)), we arrive at a theory akin to that of conventional linear
differential equations.

Now, let’s delve into the discussion of our heat equation (27). We will employ the
method of separation of variables. Assume z(x, t) = M(x)R(t). Substituting this into the
differential equation yields:

dα

dtα
dαR(t)

dtα
M(x) = R(t)

d2M(x)

dx2
, (37)

then, we obtain:
dα

dtα
dαR(t)

dtα
/R(t) =

d2M(x)

dx2
/P = β, (38)

for some constant. Then:

dα

dtα
dαR(t)

dtα
− βR = 0, and

d2M(x)

dx2
− βM = 0. (39)

We consider the formula :
d2M(x)

dx2
− βM = 0. (40)

As is well known, there are three cases for the values of β to be considered. β = 0, β = −ν2

and β = ν2. Conditions (28) and (29) forces

ν = nπ and Mn(x) = cn sin(nπx). (41)

Through equations (32) and (34), we obtain:

R(t) = b1 cos(
nπ

1− α
e(1−α)t) + b2 sin(

nπ

1− α
e(1−α)t). (42)

And the condition (30) now gives b2 = 0, and R(t) = b1 cos(
nπ
1−αe

(1−α)t), then, using
equation (41), we obtain :

z(x, t) =

∞∑
n=1

an sin(nπx) cos(
nπ

1− α
e(1−α)t), (43)



REFERENCES 1165

using condition (31) we find that an is the nth Fourier coefficient of the function h(x).
Finally, we can consider the following form of the new fractional heat differential

equation:
∂α

∂xα
∂αz(x, t)

∂xα
=

∂z

∂t
, (44)

with conditions:
z(x, 0) = h(x), 0 < x < 1,

z(0, t) = 0, t > 0,

z(1, t) = 0, t > 0.

In this context, equation (27) can be effectively applied, enabling the solution of prob-
lem (44) with the provided boundary conditions in a manner akin to solving the ordinary
heat equation. This approach considers properties (2), (3), and (4) (refer to theorem (3)).

Remark 1. One should note that if we set α = 1, problem (44) will become the original
problem treated by the fractional Black-Scholes equation (see [8]), a significant equation in
financial mathematics. Utilizing this solution of the heat equation, we deduce the solution
of the Black-Scholes model.

4. Conclusion

In conclusion, this informative article has shed light on the novel concept of con-
formable derivatives and their definitions, unveiling a rich landscape of possibilities in
mathematical analysis and applications across various scientific domains. By delving into
the fundamental properties of adaptive differential operators, we’ve unlocked new avenues
for understanding complex phenomena in physics and beyond.

The inclusion of graphical representations has not only enhanced our comprehension of
the proposed fractional differential equations but has also demonstrated their remarkable
resemblance to solutions obtained using traditional derivatives, underscoring the efficacy
of these novel mathematical tools.

Looking ahead, the insights gleaned from our exploration, particularly in the realm of
the fractional heat equation, hold promise for addressing significant challenges in financial
modeling. Leveraging these outcomes to tackle the fractional Black-Scholes model repre-
sents a natural progression, offering potential breakthroughs in pricing exotic options and
managing risk in modern financial markets.

As we continue to push the boundaries of mathematical innovation and its applications,
the journey of discovery sparked by conformable derivatives promises to be both exciting
and fruitful, paving the way for deeper insights into the intricacies of natural phenomena
and the complexities of financial systems.

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