EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3539-3556
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Bernstein Polynomials for Solving Fractional
Differential Equations with Two Parameters

A.K. Alomari1, Abdul Rhman Al-Shatnawi1, Adel Almalki2, Nidal Anakira3,4,∗

1 Department of Mathematics, Yarmouk University, Irbid, Jordan
2 Al-Gunfudah University College, Umm Al- Qura University, Mecca 21955, Saudi Arabia
3 Faculty of Education and Arts, Sohar University, Sohar 3111, Oman
4 Jadara University Research Center, Jadara University, Jordan

Abstract. This work presents a general framework for solving generalized fractional differential
equations based on operational matrices of the generalized Bernstein polynomials. This method
effectively obtains approximate analytical solutions of many fractional differential equations. The
generalized fractional derivative of the Caputo type with two parameters and its properties are
studied. Using orthonormal Bernstein polynomials has led to the development of fractional poly-
nomials, which offer an approximate solution for ordinary fractional differential equations. The
approach employs the generalized orthogonal Bernstein polynomials (FOBPs) and constructs their
operational matrices for fractional integration and derivative in the generalized Caputo sense to
achieve this objective. Operational matrices convert ordinary differential equations into a system
of algebraic equations, which can be solved using Newton’s method. The convergence analysis and
error estimate associated with the proposed problem have been investigated using the approxima-
tion of generalized orthogonal Bernstein polynomials (FOBPs). The effect of the new parameters
of the fractional derivative is presented in several examples. Finally, several examples are included
to clarify the proposed technique’s validity, efficiency, and applicability via generalized orthogonal
Bernstein polynomials (FOBPs) approximation.

2020 Mathematics Subject Classifications: 26A33, 11Cxx, 34A08

Key Words and Phrases: Generalized Fractional Derivative, Bernstein Polynomials,
Operational Matrices, Riccati Equation

1. Introduction

Differential equations are an essential tool for modeling several phenomena in sciences,
engineers, and other fields [16]. Studying this topic will help the researchers to understand
those models. Generalizing this topic from natural to fractional derivatives will help the

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5476

Email addresses: abdalkareem@yu.edu.jo (A.K. Aloimari),
abd1997shatnawi@gmail.com (A.R. Al-Shatnawi),
aaamalki@uqu.edu.sa (A. Almalki), nanakira@su.edu.om (N. Anakira)

https://www.ejpam.com 3539 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3540

researcher fit their actual data within the expected results using a nonstandard derivative
[4].

In 1695, a new concept of differential equations appeared, fractional differential equa-
tions, and it became the subject of interest to many scientists. Several fractional derivative
models have been introduced, such as Leibnitz, Euler, Fourier, Abel, Liouville, Riemann,
and Hadamard [12, 13, 18, 19, 42]. The difference between them is the kernel that used
[1, 2, 6, 11, 14]. The researchers have studied these new fractional operators by introducing
new definitions and discussing their important properties. These definitions have applica-
tions in various fields, such as physics [3], dynamical systems, engineering [29], mechanics,
signal processing, images, and control theory [34].

Definition 1.1. The Riemann-Liouville fractional integral of order α > 0 is defined by

Iα+a =
1

Γ(α)

∫ t

a
(t− s)α−1f(s)ds, t > a (1)

Using this definition of fractional integration, the Riemann-Liouville derivative and
the Caputo fractional derivative of order α > 0 are defined by:

RDα
a f(t) =

1

Γ(m− α)

dm

dtm
(

∫ t

a
(t− s)m−α−1f(s)ds) (2)

CDα
a+f(t) =

1

Γ(m− α)

∫ t

a
(t− s)m−α−1f (m)(s)ds (3)

respectively, where m− 1 < α ≤ m, and m ∈ N. The Caputo definition is one of the most
important definitions used to treat many physical problems related to fractional calculus,
due to its properties similar to those of ordinary derivatives. The fractional integral of the
fractional derivative is given by

IαaD
α
a+f(t) = f(t)−

m−1∑
n=0

f (k)(a)

k!
(t− a)k, t > 0. (4)

A generalized form of the fractional derivative is with two parameters recently pre-
sented by Udita N. Katugampola [23], and Odibat and Baleanu [41]. The new version
of the derivative contains two parameters; thus, the solution of the fractional differential
equation depends on those parameters. In this paper, we will use Odibat and Baleanu
definitions.

Definition 1.2. The generalized fractional derivative of a continues function f is Dα,ρ
a+ f

of order α > 0, and ρ > 0 can be written as

Dα,ρ
a+ f(t) =

ρα−m+1

Γ(m− α)

∫ t

a
sρ−1(tρ − sρ)m−α−1(s1−ρ d

ds
)mf(s)ds, (5)

where a > 0, ρ > 0, m− 1 < α ≤ m, and m ∈ N.



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3541

This generalization has investigated several applications. For example, Kumar et al.
[28] introduced the solution of the coronavirus disease model in Brazil via the new defini-
tion. The solution of influences infection dynamics for a butterfly pathogen and COVID-19
model restively are presented by Kumar and Erturk [20, 27] respectively. There is now no
research paper handling the approximations of the solution using analytical methods.

One of the essential main branches of numerical analysis is approximation. Polynomi-
als are useful mathematical tools that are easily defined, characterized, combined, derived,
and grouped to form curves that can approximate any function to any required precision.
Also, different bases can characterize the polynomials. Each basis type possesses unique
strengths, including the monomial power, Jacobi, Bernstein, and Hermite forms. A wide
range of problems can be effectively addressed by selecting the appropriate basis, and
various complexities can be mitigated or eliminated. One of those polynomials is Bern-
stein polynomials, which are widely used to approximate the solutions of ordinary and
fractional differential equations [32, 35, 39, 40]. The method assumes the solution can
be approximated via a linear combination of Bernstein polynomials. Those polynomials
have been used to solve several linear and nonlinear fractional differential equations such
as Khan et al. [25] solved Brusselator system, Mirzaee, and Alipour [30] solved Volterra
integro-differential equations and others [7–10, 17, 21, 22, 24, 33, 36–38, 43]. Up to now,
Bernstein polynomials do not use to approximate the generalized Caputo-type fractional
derivative.This work will build this algorithm. Unlike other methods, the algorithm does
not need to select an initial guess like other methods or use help from other techniques.
It can be used directly to solve the problem.

Finding the approximate analytic solution based on the orthogonal polynomials for
FDE of generalized form will help the scientist fit their real data of the model with several
parameters that make it easy to differentiate, integrate, and analyze the models. The main
objective of this work is to build a convergent numerical algorithm for solving the fractional
differential equation (FDE) with two parameters in terms of an analytic approach. In
this paper, we utilized the operational matrix of FDEs to solve the fractional model of
differential equations. By employing an operational matrix, we transform the fractional
differential equation into a set of linear or nonlinear algebraic equations. Solving this
system yields an approximate solution for the original equation.The algorithm can give the
best approximation solution to linear and nonlinear FDE in two parameters, as increases
the degree of the polynomial the solution converges to the exact one.

2. Bernstein Polynomials

Bernstein polynomials are one of the most important groups of polynomials because
they have many properties such as continuity and base group unit of B polynomials over the
period [0, R]. At x = 0 the bases of the Bernstein polynomial disappear except for the first
polynomial, which is equal to 1, and at x = R these polynomials disappear except for the
last polynomial, which also equals 1 during the interval [0, R]. This property is important
for increasing the flexibility to enforce boundary conditions at interval endpoints. It also
ensures that the sum at any point x of all Bernstein polynomials is unity.



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3542

Definition 2.1. The B-polynomial of n-th degree are defined on the interval [0, 1] as [15]

Bi,n(t) =

(
n

i

)
(1− t)n−iti, (6)

Bernstein polynomials are considered positive, forming the unit for every real x that
belongs to the period [0, 1], which can be proved easily. These polynomials can be written
in terms of the linear combination of the basic functions by using the binomial expansion
of (1− t)n−i, as:

Bi,n(t) =
(
n
i

)
(1− t)n−iti,

=
(
n
i

)
ti

(
n−i∑
k=0

(−1)k
(
n− i

k

)
tk

)
,

=

n−i∑
k=0

(−1)k
(
n

i

)(
n− i

k

)
tk+i, i = 1, 2, .., n.

Definition 2.2. The Orthonormal Bernstein polynomial (OBPs) on the interval [0, 1] of
degree n in terms of original orthonormal Bernstein basis functions can be defined as [31]

ϕi,n(s) =
√

2(n− i) + 1
i∑

k=0

(−1)k
(
2n+1−k

i−k

)(
i
k

)(
n−k
i−k

) Bi−k,n−k(s), (7)

where

Bi,n(s) =

n−i∑
k=0

(−1)k
(
n

i

)(
n− i

k

)
si+k. (8)

Definition 2.3. Fractional Orthonormal Bernstein polynomials (FOBPs) are derived by
transforming s to tv, where (v > 0), using the foundation of OBPs. These FOBPs are
represented as ϕv

i,n(t) for i = 0, . . . , n. The analytical expression of FOBPs can be obtained
through the following formula [31]:

ϕv
i,n(t) =

√
2(n− i) + 1(1− tv)n−i

i∑
k=0

(−1)k
(
2n+ 1− k

i− k

)(
i

k

)
tv(i−k), (9)

for i = 0, 1, · · · , n.

Now we define ϕv(t) be a set of FOBPs of degree n as:

ϕv(t) = [ϕv
0,n(t), ϕ

v
1,n(t), · · · , ϕv

n,n(t)]
T , (10)

where ϕv
i,n(t) for i = 0, . . . , n are FOBPs defined in Equation (9), and its matrix form as:

ϕv(t) = AT v
n (t), (11)

where
T v
n (t) = [1, tv, t2v, · · · , tnv], (12)



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3543

And A = [ai,j ] is (n+ 1)× (n+ 1) matrix where its elements are

ai.j =
√

2(n− i) + 1

min{i,j}∑
k=max{0,j−n+i}

mi,j−kni,k, i, j = 0, 1, 2, · · · , n (13)

Where mi,j and ni,j are defined as follows:

mi,j = (−1)j
(
n−i
j

)
, j = 0, 1, · · · , n− i,

ni,j = (−1)i−j
(
2n+1−i+j

j

)(
i

i−j

)
, j = 0, 1, · · · , i.

.

2.1. Approximation of Function for FOBPs

The set of Bernstein polynomials {B0,m, B1,m, · · · , Bm,m} in Hilbert space L2[0, 1] is
a complete basis [26]. Therefore, The set of FOBPs {ϕ0,m, ϕ1,m, · · · , ϕm,m} is a complete
basis in Hilbert space L2[0, 1]. So any function f(t) in Hilbert space L2[0, 1] can be
represented by FOBPs as

f(t) ≃
m∑
0

ciϕi,m(t) = CTϕ(t) (14)

where ϕT (t) = [ϕ0,m, ϕ1,m, · · · , ϕm,m] and CT = [c1, c2, · · · , cm]. The vector C can be
obtained by

CT ⟨ϕv(t), ϕv(t)⟩w(t) = ⟨f(t), ϕv(t)⟩w(t), (15)

but ϕ(t) are orthogonal with respect to the weight function w(t), so Equation (15) becomes

CT = v⟨f(t), ϕv(t)⟩w(t). (16)

3. generalized Caputo-type fractional derivatives

Definition 3.1. The generalized fractional integral of a continues function f is Iα,ρa+ f of
order α > 0, and ρ > 0 can be written as [41].

Iα,ρa+ f(t) =
ρ1−α

Γ(α)

∫ t

a
sρ−1(tρ − sρ)α−1f(s)ds. (17)

Odibat and Baleanu were able to determine the relationship between the generalized
fractional integral given in (17) and the given generalized fractional derivative given in (5)
as [41]:

Iα,ρ
a+

Dα,ρ
a+ f(t) = f(t)−

m−1∑
n=0

1

ρnn!
(tρ − aρ)n[(x1−ρ d

dx
)nf(x)]|x=a, (18)



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3544

where 0 < α ≤ m, ρ > 0 and a ≥ 0.

In the following, we will show some theorems and essential properties of generalized frac-
tional derivatives and integrals with two parameters that we will need in the next section.

(i) Iα,ρ0 C = C ρ−α

Γ(α+1) t
αρ.

(ii) Iα,ρ0 tβ =
ρ−αΓ(1+β

ρ
)

Γ(1+β
ρ
+α)

tβ+αρ.

(iii) Iα,ρa [Cf(t) + g(t)] = CIα,ρa f(t) + Iα,ρa g(t).

(iv) Dα,ρ
a+

C = 0.

(v) Dα,ρ
0 tβ = {

0 : β < m, m− 1 ≤ α < m
ρα−mΓ(β+1)Γ(β+ρ−mρ

ρ
)

Γ(β−m+1)Γ(m−α+β+ρ−mρ
ρ

)
tβ−αρ : otherwise

.

(vi) Dα,ρ
a [Cf(t) + g(t)] = CDα,ρ

a+
f(t) +Dα,ρ

a+
g(t).

4. generalized operational matrix of orthogonal Bernstein polynomials

The main objective of this section is to derive the FOBPs operational matrices for
generalized Caputo fractional derivatives. We derive the FOBPs operational matrices of
fractional integration and derivative, the identities, and zero matrices of order (n+1) are
I and O, respectively.

4.1. Operational Matrix of Fractional Integration Based on FOBPs

The generalized fractional integral of ϕv(t) is defined as:

Iα,ρϕv(t) = Iα,ρ,vϕv(t), (19)

where Iα,ρ,v is called the operational matrix of generalized fractional integration of order
α > 0, ρ > 0. From Equation (11) and the properties of the operator Iα,ρ we get.

Iα,ρϕv(t) = Iα,ρAT v
n (t) = AIα,ρ(T v

n (t)) = A[Iα,ρ1, Iα,ρtv, · · · , Iα,ρtnv]T ,

= A[ρ
−αΓ[1]
Γ[1+α] t

αρ, ρ−α Γ[1+ v
ρ
]

Γ[1+α+ v
ρ
] t
v+αρ, · · · , ρ−α Γ[1+nv

ρ
]

Γ[1+α+nv
ρ
] t
nv+αρ]T ,

= ABT̄ v
n (t).

(20)

where B = [bi,j ] is an (n+ 1)× (n+ 1) matrix, and for i, j = 0, . . . , n, the bi,j is given by:

bij =

 ρ−α Γ[1+ iv
ρ
]

Γ[1+α+ iv
ρ
]

: i = j

0 : otherwise
(21)



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3545

And
T̄ v
n (t) = [tαρ, · · · , tnv+αρ]T . (22)

Now, we approximate tiv+αρ for i = 0, 1, . . . , n, in terms of FOBPs as

tiv+αρ ≃ ET
i ϕ

v(t), (23)

where

Ei = v

∫ 1

0
tiv+αρϕv(t)w(t)dt,

= v

[∫ 1

0
tiv+αρϕv

0,n(t)t
v−1dt, · · · ,

∫ 1

0
tiv+αρϕv

n,n(t)t
v−1dt

]T
.

We will define a new matrix of order (n + 1) × (n + 1) denoted E = [Eij ] where Ei,j is
given by:

Eij =
√
2(n− j) + 1

j∑
k=0

(−1)k
(
2n+ 1− k

j − k

)(
j

k

)
Γ[i+ j + αρ

v − k + 1]Γ[n− j + 1]

Γ[i+ n+ αρ
v − k + 2]

. (24)

We have
Iα,ρϕv(t) = ABETϕv(t) (25)

So, Iα,ρ,v = ABET .

4.2. Operational Matrix of Fractional Differentiation Based on FOBPs

The generalized fractional differential of ϕv(t) is defined as:

Dα,ρ
t ϕv(t) = Dα,ρ,vϕv(t). (26)

Where Dα,ρ,v is called the operational matrix of generalized fractional differential of order
α > 0, ρ > 0. From Equation (11) and the properties of the operator Dα,ρ

t we get.

Dα,ρ
t ϕv(t) = Dα,ρ

t AT v
n (t) = ADα,ρ

t (T v
n (t)) = A[Dα,ρ

t 1, Dα,ρ
t tv, · · · , Dα,ρ

t tnv]T ,

= A[0, ρ−1+α Γ[1+v]Γ[ v+ρ−1ρ
ρ

]

Γ[v]Γ[1−α+ v
ρ
] tv−αρ, · · · , ρ−1+α Γ[1+nv]Γ[nv

ρ
]

Γ[n]Γ[1−α+n
ρ
] t
nv−αρ]T ,

= ABT̄ v
n (t).

(27)

where B = [bi,j ] is an (n + 1) × (n + 1) matrix, and for i, j = 0, 1, . . . , n, the bi,j is given
by:

bij =

 ρ−1+α Γ[1+iv]Γ[ iv
ρ
]

Γ[iv]Γ[1−α+ i
ρ
]

: i = j ̸= 0

0 : otherwise
(28)



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3546

And
T̄ v
n (t) = [0, tv−αρ, · · · , tnv−αρ]T . (29)

Now, we approximate tiv−αρ for i = 1, . . . , n, in terms of FOBPs as

tiv−αρ ≃ ĒT
i ϕ

v(t), (30)

where

Ēi = v

∫ 1

0
tiv−αρϕv(t)w(t)dt,

= v

[∫ 1

0
tiv−αρϕv

0,n(t)t
v−1dt, · · · ,

∫ 1

0
tiv−αρϕv

n,n(t)t
v−1dt

]T
.

Now, Ē can be written as (n+ 1)× (n+ 1) matrix in the form Ē = [Ēij ] where:

Ēij =

{
0 : i = j = 0√

2(n− j) + 1
∑j

k=0(−1)k
(
2n+1−k

j−k

)(
j
k

)Γ[i+j−αρ
v
−k+1]Γ[n−j+1]

Γ[i+n−αρ
v
−k+2]

: otherwise
(31)

Then,
Dα,ρ

t ϕv(t) = ABĒTϕv(t), (32)

and Dα,ρ,v = ABĒT . So, The generation of the above formula can be given by diagram 1

Figure 1: Operational Matrix of Fractional Differentiation.



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3547

5. Convergence Analysis and Error Estimate

In this section, we provide the convergence theorem of the method based on some
existing results in [5, 30].

Theorem 5.1. [5] Let f : [0, 1] → R such that f ∈ Cn+1[0, 1], and sn = span{ϕv
0,n, ϕ

v
1,n, · · · , ϕv

n,n},
now if KTϕv be the best approximation f out of Sn then

||f −KTϕv||2 ≤
C̄

(n+ 1)!2
√
2n+ 1

, (33)

where
C̄ = max

t∈[0,1]
|f (n+1)(t)|. (34)

6. Examples.

Example 1. Consider the following linear ordinary differential equation on [0, 1].

Dα,ρ
t u(t) = u(t) + t, u(0) = 1, 0 < α ≤ 1, ρ > 0 (35)

When α = ρ = 1, the equation has exact solution u(t) = 2e2t − t− 1.

First suppose that u(t) = KTϕv(t), and Dα,ρ
0 u(t) = KTDα,ρϕv(t), next write t in term

of 1D-FOBPs as t = cTϕv(t),where

cT = v

∫ 1

0
tϕv(t)Tdt. (36)

We can construct n linear equation using∫ 1

0
(KTDα,ρ −KT − cT )ϕv(t)ti+1dt = 0, i = 0, 1, · · ·n− 1. (37)

Apply the initial condition u(0) = 1 to get

KTϕv(0) = 1. (38)

Thus, we have (n+ 1) equations for (n+ 1) unknown variables of the vector K. After
solving the linear system, we can calculate the approximation solution u(t).

Figure 2. Shows the exact solution and the approximate solutions of Example 1 for
α, ρ = 1, v = 1, and n = 6. The absolute error between the exact solution and approxima-
tion solutions of Example 1 when n = 6, with α, ρ =, and v = 1 is plotted in Figure 3. Of
course, by increasing the value of n of FOBPs, the approximate values of u(t) converge to
the exact solutions. Figure 4. Present the solution when n = 6, ρ = 1, v = 1, and different
values of α. The effect of ρ is presented in Figure ??. The solution does not depend only
on α but also on ρ. The solution for different values of α and ρ is presented in Figure



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3548

uexact

n=6

0.0 0.2 0.4 0.6 0.8 1.0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

t

u
(t
)

Figure 2: The exact solution and approximation solutions for Example 1 when n = 6, with α = 1, ρ = 1, and
v = 1 .

0.0 0.2 0.4 0.6 0.8 1.0
0

1.× 10-6

2.× 10-6

3.× 10-6

4.× 10-6

t

A
bs
ol
ut
e
E
ro
rr

Figure 3: The absolute error between the exact solution and approximation solutions, for example, 1 when
α = ρ = 1, and v = 1, for n = 6.

6. It is worth mentioning that the solution is changed for every single value of α and ρ.
Table 1 shown approximate solutions of example 1 when t = 1, v = 1, and different value
of α, ρ. Finally, we note that the CPU time for this example is 2.372s for n = 6 using
Mathematica software.

Example 2. Consider the following nonlinear Riccati Equation on [0,1], with a given
initial condition

Dα,ρ
0 u(t) = 2u(t)− u2(t) + 1, u(0) = 0, 0 < α ≤ 1, ρ > 0. (39)

The exact solution when α, ρ = 1 is

u(t) =
e2

√
2t − 1

−e2
√
2t +

√
2e2

√
2t + 1 +

√
2
. (40)



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3549

α=1

α=0.9

α=0.75

0.0 0.2 0.4 0.6 0.8 1.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5

t

u
(t
)

Figure 4: The approximate solution for Example 1 for fixed n = 6, v = 1, ρ = 1, and vary α.

ρ=1

ρ=0.9

ρ=0.75

0.0 0.2 0.4 0.6 0.8 1.0
1.0

1.5

2.0

2.5

3.0

3.5

t

u
(t
)

Figure 5: The approximate solutions for Example 1 for fixed n = 6, α = 1, v = 1, and vary ρ.

α=1,ρ=1

α=0.9,ρ=0.9

α=0.85,ρ=0.75

0.0 0.2 0.4 0.6 0.8 1.0
1.0

1.5

2.0

2.5

3.0

3.5

t

u
(t
)

Figure 6: The approximation solutions for Example 1 for fixed n = 6, v = 1, and vary α, ρ.



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3550

Table 1: Approximate solutions of Example 1 when t = 1, v = 1, and different value of α, ρ

n α = 1,ρ = 1 α = 1,ρ = 0.9 α = 0.95,ρ = 0.75 α = 0.9,ρ = 1.2

5 3.43666 3.35715 3.31114 4.02278

6 3.43656 3.36259 3.32984 4.01898

7 3.43656 3.36669 3.34406 4.01691

8 3.43656 3.36986 3.35519 4.01535

Now, we will Suppose that u(t) = KTϕv(t), and Dα,ρ
0 u(t) = KTDα,ρ,vϕv(t), next we

write 1 in term of FOBPs as 1 = cTϕv(t), where

cT = v

∫ 1

0
ϕv(t)dt. (41)

Substitute these assumptions into the Equation 39. Thus we get

KTDα,ρ,vϕv(t)− 2KTϕv(t) + (KTϕV (t))2 − CTϕv(t) = 0. (42)

Construct the n equation as

∫ 1

0
[KTDα,ρ,vϕv(t)− 2KTϕv(t) + (KTϕV (t))2 − CTϕv(t)]

i+1
√
t = 0, i = 1, 2, · · · , n,

(43)
the initial condition gives

KTϕv(0) = 0. (44)

Thus, we have (n+1) nonlinear equation for (n+1) unknown variable of vector K. After
solving the nonlinear system, we can calculate the approximation solution.

uexact

n=6

0.0 0.2 0.4 0.6 0.8 1.0
0.0

0.5

1.0

1.5

t

u
(t
)

Figure 7: The exact solution and approximation solutions for Example 2 when n = 6, with α = 1, ρ = 1, and
v = 1 .



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3551

n=3

n=4

n=5

n=6

0.0 0.2 0.4 0.6 0.8 1.0
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007

t

A
bs
ol
ut
e
E
rr
or

Figure 8: The absolute error between the exact solution and approximate solution, for example, 2 when α = 1,
ρ = 1, v = 1 and vary n.

α=1

α=0.9

α=0.75

0.0 0.2 0.4 0.6 0.8 1.0
0.0

0.5

1.0

1.5

t

u
(t
)

Figure 9: The approximation solutions for Example 2 for fixed n = 4, ρ = 1, and v = 1 and vary α.

Figures 7 show the exact solution together with the approximate solution, for example,
2 when n = 6, and the absolute error between them for α, ρ = 1, v = 1, and different
values of n are shown in Figures 8, We note, by increasing the value of n of FOBPs, the
approximate solution of u(t) converges to the exact solutions.

In Figure 9, we plot the approximate solution, for example, 2 when n = 4, ρ = 1,
v = 1, and different values of α. As α approaches 1, we note that the approximate
solution converges to the exact solution. The approximate solution for example 2 when
n = 4, α = 1, v = 1, and different value of ρ are shown in Figure 10. The solution
does not depend only on α but also on ρ. The effect of changing two values of α, ρ for
changing an approximate solution for example 2 is presented in figure 11, and we note
that as α, ρ approaches 1, the numerical solution converges to the exact solution. Table
2 shows Approximate solutions of fractional Riccati equation when t = 1, v = 1, and
different values of α, ρ. It is clear that the solution behaviors depend on the two fractional
parameters which give the scientists a benefit in choosing which one can fit the real data



N. Anakira et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3539-3556 3552

ρ=1

ρ=0.9

ρ=0.85

0.0 0.2 0.4 0.6 0.8 1.0
0.0

0.5

1.0

1.5

t

u
(t
)

Figure 10: The approximation solutions for Example 2 for fixed n = 4, α = 1, and v = 1, and vary ρ.

α=1,ρ=1

α=0.85,ρ=0.75

α=0.95,ρ=0.75

0.0 0.2 0.4 0.6 0.8 1.0
0.0

0.5

1.0

1.5

t

u
(t
)

Figure 11: The approximation solutions for Example 2 for fixed n = 4, and v = 1 and vary α, ρ.

more accurately.

7. Conclusion

In this paper, we apply the method of operational matrices for Bernstein polynomials
to solve the generalized fractional differential of the Caputo type with two parameters.
Operational matrices convert differential equations into algebraic equations to calculate
approximate solutions to linear and nonlinear fractional differential equations. The accu-
racy of the approximate solutions was verified by comparing the approximate solutions
when α, ρ = 1, and v = 1 with the exact solutions in the case of linear and nonlinear
equations. The method used to analyze and solve fractional differential equations is im-
plemented. The operational matrices method is effective and reveals the existence of the
approximate solution. From the solved linear and nonlinear problems, it was found that
the approximate solutions obtained using the presented algorithm are very close to the



REFERENCES 3553

Table 2: Approximate solutions of fractional Riccati equation when t = 1, and different values of α, ρ

n α = 1,ρ = 1 α = 1,ρ = 0.9 α = 0.95,ρ = 0.75 α = 0.9,ρ = 1.2

3 1.69116 1.68978 1.70788 1.79401

4 1.6892 1.68927 1.70997 1.78634

5 1.68945 1.68947 1.7106 1.78903

6 1.68937 1.68891 1.71064 1.79657

exact solutions. Therefore, this study will begin further implementing and investigating
generalized partial Caputo systems. Physical interpretation of the generalized fractional
differential equations can also be investigated in the future.

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