Journal of Applied Analysis and Computation
Volume 14, Number 2, April 2024, 964–975

Website:http://www.jaac-online.com
DOI:10.11948/20230221

GENERALIZED CAPUTO–FABRIZIOFRACTIONAL DIFFERENTIAL EQUATION∗
Masakazu Onitsuka1,† and Iz-iddine EL-Fassi2
Abstract In this paper, a generalization of the Caputo–Fabrizio fractional
derivative is proposed. The purpose of this study is to derive a solution formula for ordinary differential equations with the generalized Caputo–Fabrizio
fractional derivative. The main result can be applied to solve the Caputo–
Fabrizio fractional differential equation Dα y = f (y). That is, a new result
even for common Caputo–Fabrizio fractional differential equations is obtained.
The strength of the results obtained in this study is that the solution to thedifferential equation can be given using only the kernel included in the derivative and the right-hand side f of the equation. In other words, rather thanproviding a method to solve the solution, this study provides a formula forthe solution. This study is proposed as a tool for solving many nonlinear
equations, including the logistic type fractional differential equations.
Keywords Caputo–Fabrizio fractional derivative, fractional differential equation, fractional calculus, nonsingular kernel, logistic equation.
MSC(2010) 34A08, 26A33, 34A05.

1. Introduction
In this section, we first introduce the definitions of known fractional derivatives(Caputo–Fabrizio derivative and Caputo derivative), and then define a new fractional derivative that will be dealt with in this study. We call it a generalized
Caputo–Fabrizio derivative. Next, we consider a fractional order differential equation involving the generalized Caputo–Fabrizio derivative and introduce logistic
fractional differential equations. In addition, the solution formula, which is themain theorem of this study, is provided. Finally, we give remarks about the properties of solutions to nonlinear equations, which are important for understanding themain theorem.
† The corresponding author.
1 Department

of Applied Mathematics, Okayama University of Science,
Okayama, 700-000, Japan
2 Department of Mathematics, Faculty of Sciences and Techniques, S. M. Ben
Abdellah University, B.P. 2202, Fez, Morocco∗ The first author was supported by the Japan Society for the Promotion of
Science (JSPS) KAKENHI (grant number JP20K03668).
Email: onitsuka@ous.ac.jp, onitsuka@xmath.ous.ac.jp(M. Onitsuka), iziddmath@hotmail.fr(Iz. EL-Fassi)

Generalized CF fractional differential equation

965

1.1. Fractional derivatives
In 2015, the fractional derivative Dα φ(x) with 0 < α < 1 was introduced by Caputo
and Fabrizio [8]. The definition of Dα φ(x) is as follows.
Definition 1.1 (Caputo–Fabrizio derivative). Define
Z xα1
e− 1−α (x−ξ) φ0 (ξ)dξ,Dα φ(x) :=1−α 0

x ≥ 0,

with 0 < α < 1.
Needless to say, the Caputo–Fabrizio derivative Dα φ is a completely different
derivative than the Caputo derivative Dα φ(x) which is defined by the following.
Definition 1.2 (Caputo derivative). Define
Z x1
α(x − ξ)−α φ0 (ξ)dξ,D φ(x) :=Γ(1 − α) 0x ≥ 0,

with 0 < α < 1.
Of course, there are many other fractional derivatives, but the kernelαKα (x, ξ) := e− 1−α (x−ξ)
of the Caputo–Fabrizio derivative Dα φ is characterized by the absence of singularity. Although the Caputo–Fabrizio fractional derivative was recently proposed,
it has been applied in various fields. For example, it is useful for elucidating various phenomena including infectious diseases [1, 4, 5, 11, 20, 26, 31]. In particular,
Alinei-Poiana, Dulf, and Kovacs [1] have recently demonstrated the superiority offractional order models in tumor growth modeling in mathematical oncology. In
these applications, derivatives with singular kernels and derivatives with nonsingular kernels are compared in various situations and compete for superiority with eachother. In many cases, the Caputo–Fabrizio fractional derivative is compared withthe derivative with a singular kernel, as a representative of the derivative with anonsingular kernel. However, it is possible that there are nonsingular kernels thatcan reproduce the phenomenon better than the Caputo–Fabrizio fractional derivative. Therefore, in this study, we propose the following new derivative to expand
the class of derivatives with nonsingular kernels.
Definition 1.3 (Generalized Caputo–Fabrizio derivative). DefineZ x1κα (ξ) 0G αD φ(x) :=φ (ξ)dξ, x ≥ 0,
1 − α 0 κα (x)with 0 < α < 1 for continuously differentiable function φ, where κα is a positive
continuously differentiable function on [0, ∞).
Note here that G Dα φ has the nonsingular kernelGαKα (x, ξ) :=κα (ξ).
κα (x)

Moreover, if κα (x) ≡ e 1−α x or G Kα = Kα , then the derivative G Dα φ is just the
Caputo–Fabrizio derivative Dα φ. Then we say that G Dα φ is an α-th order generalized Caputo–Fabrizio fractional derivative with the kernel G Kα (x, ξ).

966

M. Onitsuka & Iz. EL-Fassi

1.2. Generalized Caputo–Fabrizio fractional differential equation
This study deals with the generalized Caputo–Fabrizio fractional differential equation
G α
D y = f (y)(1.1)
with the kernel G Kα (x, ξ). Here f : Y → R is continuously differentiable, where Y
is an interval of R; that is, Y is the domain of definition of f . If G Kα = Kα , then
(1.1) becomes the Caputo–Fabrizio fractional differential equation
Dα y = f (y).

(1.2)

Studies on fractional ordinary differential equations with the Caputo–Fabrizio
derivative are in progress [6, 19, 21, 28, 33]. In addition, there are applications for
the Caputo–Fabrizio derivative in the field of partial differential equations [9, 12].
Against this background, it is an important question to know whether it is possible
to find an exact solution to the Caputo–Fabrizio fractional differential equations.
When α → 1− in (1.2), equation (1.2) is the well-known autonomous ordinary
differential equationy 0 = f (y).
Needless to say, the method of solving this equation when the integral of 1/f (y) is
explicitly given is widely known in many textbooks on ordinary differential equations. For example, using the method of separation of variables, the solution of thelogistic equationy 0 = y(1 − y)with the initial condition y(0) = y0 is displayed in the formy(x) =y0.
y0 + (1 − y0 )e−x

In 2022, Nieto [23] derived a formula for the implicit solution of the fractionalversion of the logistic differential equation
Dα y = y(1 − y)(1.3)

with y(0) = y0 . The form of the solution given by Nieto is as follows:
y(x) − y 2 (x)(1 − y(x))

2α=y0 − y022(1 − y0 ) α

· ex

(1.4)

with 0 < α < 1 and y(x) 6= 0, 1 for all x ∈ [0, ∞). We can find that othertypes of Caputo–Fabrizio fractional differential equations have also been solved
recently [3, 10, 14, 32]. In particular, Cui [10] uses the same technique as Nieto’s
paper [23] to obtain implicit solutions to the equations
Dα y = −y 2 + 1,

Dα y = 2y − y 2 + 1,

Dα y = −y + y 2 ,andDα y = y(1 − y 2 ).
See also [7, 15–17, 22, 27] for solutions of fractional differential equations with another fractional derivatives. In particular, for the fractional logistic equation, see [2,Generalized CF fractional differential equation967

13, 24, 30]. Logistic-type differential equations are known to be extremely effective
not only as models for representing population growth, but also for understanding
the mechanisms of infectious diseases. For example, Nishiura, Tsuzuki, Yuan, Yamaguchi, and Asai [25] analyzes the dynamics of cholera in Yemen. Wang, Wu,
and Yang [29] proved that the exponential term in a generalized logistic equation
has a one-to-one nonlinear correspondence to the basic reproduction number of the
SIR model. Therefore, it is very important to study logistic equations and theirgeneralized equations in order to understand phenomena. In particular, the papers [1–4, 14, 23, 24] deal with fractional-order differential equations of the logistic
type, mention the importance of fractional-order differential equations of the logistic
type, and compare solutions of fractional-order differential equations with solutionsof ordinary differential equations.
Inspired by the above results, in order to respond to the demands of society,the present study attempts to apply the technique used in [23] to more generalequations. That is, we aim to derive a formula for the implicit solution of (1.1).
The following theorem and corollary realize the expression of the non-trivial solution determined only by κα , which constitutes the kernel given to the fractional
differential equation, and the form f on the right-hand side of the equation. This
fact extracts the essence that cannot be found just by looking
at the solution to aR
concrete equation. Our results have the strength that if fdη(η) is given explicitly,
the differential equation can be solved. The obtained result is as follows.
Theorem 1.1. LetR S := {η ∈ Y : f (η) 6= 0}, and let F : S → R be a function
defined by F (η) := fdη(η) . Then an implicit solution of (1.1) with y(0) = y0 ∈ Y isgiven byκα (x)f (y(x))
F (y(x)) − F (y0 )exp=1κα (0)f (y0 )α−1for all x ∈ [0, T ), with 0 < α < 1, whenever y(x) ∈ S for all x ∈ [0, T ), where
0 < T ≤ ∞ is the smaller of the maximal existence time M of the solution y(x)and inf{x ∈ [0, ∞) : f (y(x)) = 0} if {x ∈ [0, ∞) : f (y(x)) = 0} 6= ∅; M if
{x ∈ [0, ∞) : f (y(x)) = 0} = ∅.
αWhen κα (x) ≡ e 1−α x , we can obtain the following result.
Corollary 1.1. Let
R S := {η ∈ Y : f (η) 6= 0}, and let F : S → R be a function
defined by F (η) := fdη(η) . Then an implicit solution of (1.2) with y(0) = y0 ∈ Y isgiven by
f (y(x))F (y(x)) − F (y0 ) − αxexp=1
f (y0 )
α−1
for all x ∈ [0, T ), with 0 < α < 1, whenever y(x) ∈ S for all x ∈ [0, T ), where
0 < T ≤ ∞ is the smaller of the maximal existence time M of the solution y(x)and inf{x ∈ [0, ∞) : f (y(x)) = 0} if {x ∈ [0, ∞) : f (y(x)) = 0} 6= ∅; M if
{x ∈ [0, ∞) : f (y(x)) = 0} = ∅.

1.3. Remarks
Remark 1.1. Throughout this paper, let [0, M ) be the maximum existence intervalof the solution y(x) of (1.1), and 0 < M ≤ ∞ be called the maximal existence timeof y(x). If there exists a constant y1 ∈ Y such that f (y1 ) = 0, then we see that theconstant function y(x) = y1 for x ∈ [0, M ) is a trivial solution. Especially, M of

968

M. Onitsuka & Iz. EL-Fassi

this solution can be choose M = ∞. That is, the trivial solutions of (1.1) exists for
all x ∈ [0, ∞).
On the other hand, an implicit solution given in Theorem 1.1 can be said to bea formula for nontrivial solutions. There are some facts to note about 0 < T ≤ ∞that appears in Theorem 1.1 because (1.1) is nonlinear. First, we note that there
may be nontrivial solutions that blow up in finite time. For example, the equationy 0 = −y(1 − y)with y(0) = y0 has the solution
y(x) =y0.
y0 + (1 − y0 )ex−y0This implies that if y0 > 1 holds, then the solution y(x) blows up in x = ln 1−y
.
0


−y0
That is, M = ln 1−y
if y0 > 1, but M = ∞ if y0 ≤ 1. In addition, notice0that in the first case, M is depend on the initial value y0 . Next, we note that
nontrivial solutions may match a trivial solution after a finite time. In other words,
the uniqueness of the solution may be lost at some finite time. For example, theequation√y0 = − ywith y(0) = 41 has the solution(
y(x) =(1−x)2
40

if 0 ≤ x < 1,
if x ≥ 1.

This solution will match the trivial (constant) solution y(x) ≡ 0 after the finite
time x = 1. This says that we can choose M = ∞, but we must choose T = 1 in
Theorem 1.1. Of course, if all solutions exist for all x ∈ [0, ∞) and the uniquenessof all solutions is guaranteed, then we can choose T = ∞.
Remark 1.2. Define U := {η ∈ R : η 6= 0, 1}. Let y(x) ∈ U for all x ∈ [0, T ) be asolution of (1.2) with f (y) = y(1 − y) and y(0) = y0 , where T is a suitable value or∞. Since!

Z y(x)F (y(x)) − F (y0 )
−1
1
exp
= exp
dη
α−1
1 − α y0f (η)
 !
Z y(x) 
−1
1
−1
= exp
−
dη
1 − α y0η 1−η−1

(1 − y0 )y(x) 1−α=y0 (1 − y(x))−α(1 − y0 )y(x) 1−α=y0 (1 − y(x))
and
0<

−1

f (y(x))y(x)(1 − y(x))y(x)(1 − y(x))==
f (y0 )y0 (1 − y0 )y0 (1 − y0 )

(1.5)

(1.6)

Generalized CF fractional differential equation969

hold for x ∈ [0, T ), by Corollary 1.1, we obtain
f (y(x))F (y(x)) − F (y0 ) − αx1=expf (y0 )
α−1
α

2
x 1−α
y0 (1 − y(x))e1 − y(x)=(1 − y0 )y(x)1 − y0α" 2 # 1−αy0 (1 − y0 )ex
1 − y(x) α=y(x)(1 − y(x))
1 − y0
for x ∈ [0, T ). This equation implies (1.4). That is, we obtain an implicit solutionof (1.3) with y(x) ∈ U for all x ∈ [0, T ). In addition, we note that using Corollary
1.1, Cui’s results [10] are also derivable.
This paper is organized as follows. In the next section, we will introduce somebasic facts about a fractional integral. In Section 3, we prove the main theorem.
In Section 4, we present important examples. In the last section, we will state ourconclusions and future outlook.

2. Fractional integral
It is known that the fractional integral [18] corresponding to Caputo–Fabrizio derivative is given byZ x
I α φ(x) = (1 − α)(φ(x) − φ(0)) + α
φ(ξ)dξ.
(2.1)
0Then we see thatI α Dα φ(x) = φ(x) + C,

(2.2)

where C is an arbitrary constant. But, note thatαDα I α φ(x) = φ(x) − φ(0)e− 1−α xholds (see, [19]). Two properties of fractional integrals, (2.1) and (2.2), play animportant role in this paper.

3. Proof of main theorem
We will prove the main theorem.
Proof. [Proof of Theorem 1.1] Let S = {η ∈ Y : f (η) 6= 0}, and let y(x) forx ∈ [0, M ) be a solution of (1.1) with y(0) = y0 ∈ Y , where Y is the domain of
definition of f and M is the maximal existence time of y(x). Defineαg(x) := κα (x)e− 1−α x ,with 0 < α < 1. Then we see thatGDα y(x) =

1
1−α

Z x
0

α
g(ξ) − 1−α(x−ξ) 0ey (ξ)dξg(x)970

M. Onitsuka & Iz. EL-Fassi

=

1
Dα
g(x)

Z


g(x)y 0 (x)dx .

From this with (1.1), we obtain the equation
Z

α
0
D
g(x)y (x)dx = g(x)f (y(x)).
Thus, by (2.1) and (2.2), the equality
Z

Z
0
α α
0
g(x)y (x)dx = I α g(x)f (y(x))g(x)y (x)dx + C1 =I D=(1 − α)(g(x)f (y(x)) − g(0)f (y(0)))
Z x+αg(ξ)f (y(ξ))dξ0holds, where C1 is an arbitrary constant. Since κα is continuously differentiable, gis also continuously differentiable. Differentiating both sides of the above equation
with respect to x, we haveg(x)y 0 (x) =(1 − α)g 0 (x)f (y(x)) + (1 − α)g(x)f 0 (y(x))y 0 (x)+ αg(x)f (y(x))(3.1)

for x ∈ [0, M ).
Let T be the smaller of M and inf{x ∈ [0, ∞) : f (y(x)) = 0} if {x ∈ [0, ∞) :f (y(x)) = 0} =6 ∅; M if {x ∈ [0, ∞) : f (y(x)) = 0} = ∅. If y0 ∈ Y satisfies f (y0 ) = 0,then y(x) = y0 for all x ∈ [0, ∞) is a trivial solution, and T = 0. Now we assume
that y(x) ∈ S for all x ∈ [0, T ), where T is positive by the definition of S and T .
From this with the fact that g(x) 6= 0 holds, we obtain
1 − (1 − α)f 0 (y(x)) 0
g 0 (x)y (x) = (1 − α)+ α,f (y(x))g(x)for x ∈ [0, T ). Integrating both sides by x and by the definition of F , we get
F (y(x)) − (1 − α) ln |f (y(x))| = (1 − α) ln |g(x)| + αx + C2 ,where C2 is an arbitrary constant. This implies that|g(x)f (y(x))|−(1−α) = exp (−F (y(x)) + αx + C2 ) ,and so thatg(x)f (y(x)) = ± exp−F (y(x)) + αx + C2−(1 − α).

Substituting x = 0, and using y(0) = y0 , we have
−F (y0 ) + C2
g(0)f (y0 ) = ± exp.
−(1 − α)Henceg(x)f (y(x))exp
g(0)f (y0 )

F (y(x)) − F (y0 ) − αx−(1 − α)α=1holds for x ∈ [0, T ). Recalling that g(x) = κα (x)e− 1−α x , we obtain the solution
formula for (1.1). This completes the proof.

Generalized CF fractional differential equation

971

Remark 3.1. If f (y) = 0 for all y ∈ Y in (1.1), then we see that y(x) = y0 for
all x ∈ [0, ∞) is the solution of (1.1) with y(0) = y0 ∈ Y . In fact, in this case, we
obtainy 0 (x) = 0
from (3.1). Thus, y(x) is a constant, and so that we have y(x) = y0 for all x ∈ [0, ∞)by y(0) = y0 .

4. Examples
Example 4.1. Consider the linear differential equation
GDα y = λy(4.1)

with the kernel G Kα (x, ξ), where λ 6= 0. By Theorem 1.1, we obtain the following
implicit solution of (4.1) with y(0) = y0 :κα (x) y(x)κα (0) y0

1+λ(α−1)λ(α−1)=1with 0 < α < 1, whenever y(x) ∈ V1 := {η ∈ R : η 6= 0} for all x ∈ [0, T ), where T
is a suitable value or ∞. Since this equation can be solved for y(x), we can get the
explicit solution λ(α−1)κα (0) 1+λ(α−1)(4.2)y(x) = y0
κα (x)
with 0 < α < 1. Note here that we can choose T = ∞ because κα is positive andy0 ∈ V1 . Moreover, when y0 = 0 this function satisfies y(x) = 0 for all x ∈ [0, ∞).
Combining the above facts, we see that for any y0 ∈ R, the explicit solution of (4.1)
with y(0) = y0 is given by (4.2)α for all x ∈ [0, ∞).
In particular, if κα (x) ≡ e 1−α x or G Kα = Kα , then (4.2) is
λαy(x) = y0 e 1+λ(α−1) x
for all x ∈ [0, ∞). When α → 1− it becomesy(x) = y0 eλx
for all x ∈ [0, ∞). Clearly this is the solution of the first-order linear differential
equation y 0 = λy with y(0) = y0 ∈ R.
Example 4.2. Consider the logistic differential equation
GDα y = y(1 − y)with the kernelGαKα (x, ξ) =α
ξ + 1 − 1−α
(x−ξ)
e
.
x+1

(4.3)

(4.4)

Since κα (x) = (x + 1)e 1−α x holds, using Theorem 1.1 and (1.5) and (1.6), we obtainthe following implicit solution of (4.3) with (4.4) and y(0) = y0 :α2
y0 (1 − y(x))ex 1−α 1 − y(x)(x + 1)=1(1 − y0 )y(x)1 − y0972

M. Onitsuka & Iz. EL-Fassi

with 0 < α < 1, whenever y(x) ∈ U for all x ∈ [0, T ), where T is a suitable value or∞.
Figure 1 shows that solution curves of classical logistic equation (black), Caputo–
Fabrizio fractional logistic equation (1.3) (blue), and generalized Caputo–Fabrizio
fractional logistic equation (4.3) with (4.4) (red) with y(0) = 0.1 and α = 0.1. Afterintersecting the solution curve of classical logistic equation, the solution curve of
equation (4.3) with (4.4) lies between solution curves of classical logistic equation
and Caputo–Fabrizio fractional logistic equation (1.3). Figures 2 and 3 show thecase where α = 0.5 and 0.9. As α → 1− , the graphs of (1.3) and (4.3) with (4.4)
approach the graph of classical logistic equation.

y
1

0.5
0.1
5

10

15

20

x

Figure 1. Solution curves of classical logistic equation (black), Caputo–Fabrizio fractional logistic
equation (1.3) (blue), and generalized Caputo–Fabrizio fractional logistic equation (4.3) with (4.4) (red),with y(0) = 0.1 and α = 0.1.

y
1

0.5
0.1
5

10

15

20

x

Figure 2. Solution curves of classical logistic equation (black), Caputo–Fabrizio fractional logistic
equation (1.3) (blue), and generalized Caputo–Fabrizio fractional logistic equation (4.3) with (4.4) (red),with y(0) = 0.1 and α = 0.5.

More generally, we get the following result.
Example 4.3. Consider equation (4.3) with the kernelGKα (x, ξ) =αh(ξ) − 1−α
(x−ξ)
e
,
h(x)

(4.5)

where h(x) is a positive continuous function. Doing the same as in Example 4.1,

Generalized CF fractional differential equation

973

y
1

0.5
0.1
5

10

15

20

x

Figure 3. Solution curves of classical logistic equation (black), Caputo–Fabrizio fractional logistic
equation (1.3) (blue), and generalized Caputo–Fabrizio fractional logistic equation (4.3) with (4.4) (red),with y(0) = 0.1 and α = 0.9.

we obtain the following implicit solution of (4.3) with (4.5) and y(0) = y0 :α2h(x) y0 (1 − y(x))ex 1−α 1 − y(x)=1h(0) (1 − y0 )y(x)1 − y0with 0 < α < 1, whenever y(x) ∈ U for all x ∈ [0, T ), where T is a suitable value or∞.
Example 4.4. Consider the differential equation
GDα y =

y
y−1

(4.6)

with the kernel G Kα (x, ξ), where y 6= 1 i.e., y(x) ∈ Y1 := {η ∈ R : η 6= 1}. By
Theorem 1.1, we obtain the following implicit solution of (4.6) with y(0) = y0 :1κα (x)y(x)(y0 − 1) y(x) 1−αexpκα (0)y0 (y(x) − 1) y0

y(x) − y0
α−1


=1

with 0 < α < 1, whenever y(x) ∈ V2 := {η ∈ Y1 : η 6= 0} for all x ∈ [0, T ), where T
is a suitable value or ∞.

5. Conclusions and future outlook
This study introduces a novel generalized Caputo–Fabrizio fractional derivative andpresents a versatile method for solving generalized Caputo–Fabrizio fractional differential equations. Our findings extend to encompass Caputo–Fabrizio fractional
differential equations, offering a broad application range. Importantly, we provide
a theorem that offers a versatile solution framework, not restricted to specific equations like the logistic or Riccati equations.
When κα (x) ≡ 1, the expression
GDα φ(x) =1(φ(x) − φ(0))
1−αreveals that the new derivative, G Dα φ, can be viewed as an extension of the Caputo–
Fabrizio fractional derivative, albeit not necessarily as a traditional “derivative”.

974

M. Onitsuka & Iz. EL-Fassi

It’s worth noting that further research could potentially lead to characterizing thegeneralized Caputo–Fabrizio fractional derivative by imposing specific constraints
on the nonsingular kernel, G Kα (x, ξ).

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