EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 1144-1157
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fractional Order Techniques for Stiff Differential
Equations Arising from Chemistry Kinetics

Rania Wannan1, Muhammad Aslam2, Muhammad Farman3, Ali Akgül4,
Farhina Kouser5, Jihad Asad1,∗

1 Faculty of Applied Science, Palestine Technical University - Kadoorie, Tulkarem, Pales-
tine.
2 Key Laboratory of Synthetic and Natural Functional Molecule Chemistry of Ministry of
Education, Department of Chemistry and Materials Science, Northwest University, Xi’an
710127, P.R China
3 Department of Mathematics, Khawaja Fareed University of Engineering and Information
Technology, Rahim Yar Khan, Pakistan.
4 Department of Mathematics, Art and Science Faculty, Siirt University, 56100 Siirt, Turkey
5 Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan

Abstract. In this paper, we consider the stiff systems of ordinary differential equations arising
from chemistry kinetics. We develop the fractional order model for chemistry kinetics problems
by using the Caputo Fabrizio and Atangana-Baleanu derivatives in Caputo sense. We apply the
Sumudu transform to obtain the solutions of the models. Uniqueness and stability analysis of
the problem are also established by using the fixed point theory results. Numerical results are
obtained by using the proposed schemes which supports theoretical results. These concepts are
very important for using the real-life problems like Brine tank cascade, Recycled Brine tank cascade,
pond pollution, home heating and biomass transfer problem. These results are crucial for solving
the nonlinear model in chemistry kinetics.

2020 Mathematics Subject Classifications: 65P40, 68M07

Key Words and Phrases: Chemistry kinetics, fractional technique, stability, uniqueness, Sumudu
transform.

1. Introduction

Some problems with the fractional derivatives which include the trigonometric and
exponential functions [3–8, 12, 14] show some related approaches for models of epidemic.
Different fractional operator is used in literature to solve the real life problems [2, 9, 10,
13, 15] . The chemical reaction has been introduced by Robertson as [1]:

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4406

Email addresses: j.asad@ptuk.edu.ps (Jihad Asad)

https://www.ejpam.com 1144 © 2022 EJPAM All rights reserved.



J. Asad et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1144-1157 1145

A
k1→ B. (1)

B +B
k2→ C +B. (2)

B + C
k3→ A+ C. (3)

The problem has three equations, where k1,k2 and k3 describe the rate constants and
A,B and C are the chemical species contained. By using the mass action law, the get

y′1 = −M1y1 +M3y2y3,

y′2 = M1y1 −M2y
2
2 −M3y2y3,

y′3 = M2y
2
2.

(4)

In this system y1(t), y2(t) and y3(t) are the concentrations of the chemical species A,B and
C. The initial time t = 0 can be given by (y01, y02, y03)

T . Where M1 = 0.04,M2 = 3× 107

and M3 = 104, and the initial concentrations were y01 = 1/100000, y02 = 0 and y03 = 0.
Our new Caputo–Fabrizio fractional model for Robertson problem can therefore be written
as follows:

CF
0 Dρ

t y1 = −M1y1 +M3y2y3,
CF
0 Dρ

t y2 = M1y1 −M2y
2
2 −M3y2y3,

CF
0 Dρ

t y3 = M2y
2
2.

(5)

2. Basic Definitions

Some basic definitions are given in this section [3–5, 12].

Definition 1. Sumudu transform for any function ϕ(t) over a set is given as,
A = {ϕ(t) : there exist Λ, τ1, τ2 > 0, |ϕ(t)| < Λexp(|t|/τi), if t ∈ (−1)i × [0,∞)}
is described by

F (u) = ST [ϕ(t)] =

∫ ∞

0
exp(−t)ϕ(ut)dt, u ∈ (−τ1, τ2). (6)

Definition 2. Atangana-Baleanu derivative in Caputo sense is described as [13] :

ABC
a Dα

τ ϕ(t) =
AB(α)

n− α

∫ t

a

dn

dwn
ϕ(w)Eα

−α(t− w)α

(n− α)
dw, n− 1 < α < n. (7)

The Laplace transform of equation (7) is acquired as:

L[ABC
a Dα

τ ϕ(t)](s) =
AB(α)

1− α

(sαL[ϕ(t)](s)− sα−1ϕ(0))

sα + α
1−α

. (8)

By using Sumudu transform (ST) for equation (7), we obtain

ST [ABC
0 Dα

τ ϕ(t)](s) =
B(α)

1− α
αΓ(α+ 1)Eα(

−1

1− α
wα)[ST (ϕ(t))− ϕ(0)]. (9)



J. Asad et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1144-1157 1146

3. Caputo Fabrizio Derivative

By using Sumudu Transform on system (5), we have

M(ρ)
ST (y1(t))− y1(0)

1− ρ+ ρu
= ST [−M1y1 +M3y2y3],

M(ρ)
ST (y2(t))− y2(0)

1− ρ+ ρu
= ST [M1y1 −M2y

2
2 −M3y2y3],

M(ρ)
ST (y3(t))− y3(0)

1− ρ+ ρu
= ST [M2y

2
2].

(10)

Rearranging the above equations yields:

ST (y1(t)) = y1(0) +
1− ρ+ ρu

M(ρ)
ST [−M1y1 +M3y2y3],

ST (y2(t)) = y2(0) +
1− ρ+ ρu

M(ρ)
ST [M1y1 −M2y

2
2 −M3y2y3],

ST (y3(t)) = y3(0) +
1− ρ+ ρu

M(ρ)
ST [M2y

2
2].

(11)

Taking inverse transform for system (11) gives:

y1(t) = y1(0) + ST−1[
1− ρ+ ρu

M(ρ)
ST (−M1y1 +M3y2y3)],

y2(t) = y2(0) + ST−1[
1− ρ+ ρu

M(ρ)
ST (M1y1 −M2y

2
2 −M3y2y3)],

y3(t) = y3(0) + ST−1[
1− ρ+ ρu

M(ρ)
ST (M2y

2
2)].

(12)

We get this recursive form as:

y1(n+1)(t) = y1(n)(0) + ST−1[
1− ρ+ ρu

M(ρ)
ST (−M1y1(n) +M3y2(n)y3(n))],

y2(n+1)(t) = y2(n)(0) + ST−1[
1− ρ+ ρu

M(ρ)
ST (M1y1(n) −M2y

2
2(n) −M3y2(n)y3(n))],

y3(n+1)(t) = y3(n)(0) + ST−1[
1− ρ+ ρu

M(ρ)
ST (M2y

2
2(n))].

(13)

And the solution of system (13) is obtained as:

y1(t) = lim
n→∞

y1(n)(t), y2(t) = lim
n→∞

y2(n)(t), y3(t) = lim
n→∞

y3(n)(t). (14)

3.1. Stability Analysis of the Problem

Theorem 1. Let (X1, .) be a Banach space and P be a self-map of X1 satisfying
∥Px − Py∥ ≤ C∥x− Px∥+ c∥x− y∥



J. Asad et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1144-1157 1147

for all x, y ∈ X1, where 0 ≤ C, 0 ≤ c < 1. Consider that P is a P-Stable. We have

y1(n+1)(t) = y1(n)(0) + ST−1[
1− ρ+ ρu

M(ρ)
ST (−M1y1(n) +M3y2(n)y3(n))],

y2(n+1)(t) = y2(n)(0) + ST−1[
1− ρ+ ρu

M(ρ)
ST (M1y1(n) −M2y

2
2(n) −M3y2(n)y3(n))],

y3(n+1)(t) = y3(n)(0) + ST−1[
1− ρ+ ρu

M(ρ)
ST (M2y

2
2(n))].

(15)

Where 1−ρ+ρu
M(ρ) is the fractional Lagrange multiplier.

Theorem 2. Let us describe a self-map P as

P (y1(n)(t)) = y1(n+1)(t) = y1(n)(0) + ST−1[
1− ρ+ ρu

M(ρ)
ST (−M1y1(n) +M3y2(n)y3(n))],

P (y2(n)(t)) = y2(n+1)(t) = y2(n)(0) + ST−1[
1− ρ+ ρu

M(ρ)

ST (M1y1(n) −M2y
2
2(n) −M3y2(n)y3(n))],

P (y3(n)(t)) = y3(n+1)(t) = y3(n)(0) + ST−1[
1− ρ+ ρu

M(ρ)
ST (M2y

2
2(n))].

(16)

is P -Stable in L1(a, b) if

C = ([1−M1f(γ) +M3(K + L)h(γ)], [1 +M1f(γ)−M2g(γ)−M3(K + L)h(γ)],

[1 +M2g(γ)]),

c = (0, 0, 0).

(17)

Proof. We prove that P has a fixed point. For this, we evaluate the following for all
(m,n) ∈ N× N.

P (y1(n)(t))− P (y1(m)(t)) = y1(n)(t)− y1(m)(t) + ST−1[
1− ρ+ ρu

M(ρ)
ST (−M1y1(n) +M3y2(n)y3(n))]

− ST−1[
1− ρ+ ρu

M(ρ)
ST (−M1y1(m) +M3y2(m)y3(m))],

P (y2(n)(t))− P (y2(m)(t)) = y2(n)(t)− y2(m)(t) + ST−1[
1− ρ+ ρu

M(ρ)
ST (M1y1(n) −M2y

2
2(n)

+M3y2(n)y3(n))]− ST−1[
1− ρ+ ρu

M(ρ)
ST (M1y1(m) −M2y

2
2(m) −M3y2(m)y3(m))],

P (y3(n)(t))− P (y3(m)(t)) = y3(n)(t)− y3(m)(t) + ST−1[
1− ρ+ ρu

M(ρ)
ST (M2y

2
2(n))]

− ST−1[
1− ρ+ ρu

M(ρ)
ST (M2y

2
2(m))]

.

(18)



J. Asad et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1144-1157 1148

Considering equation (18) , we get

∥P (y1(n)(t))− P (y1(m)(t))∥ = ∥y1(n)(t)− y1(m)(t)|

+ST−1[
1− ρ+ ρu

M(ρ)
ST (−M1y1(n) +M3y2(n)y3(n))]

−ST−1[
1− ρ+ ρu

M(ρ)
ST (−M1y1(m) +M3y2(m)y3(m))]∥.

(19)

Using triangular inequality for equation (19), we get

∥P (y1(n)(t))− P (y1(m)(t))∥ = ∥y1(n)(t)− y1(m)(t)∥

+ ∥ST−1[
1− ρ+ ρu

M(ρ)
ST (−M1y1(n) +M3y2(n)y3(n))]

− ST−1[
1− ρ+ ρu

M(ρ)
ST (−M1y1(m) +M3y2(m)y3(m))]∥

(20)

Upon further simplification gives:

∥P (y1(n)(t))− P (y1(m)(t))∥ = ∥y1(n)(t)− y1(m)(t)∥

+ ∥ST−1[
1− ρ+ ρu

M(ρ)
ST (−M1y1(n) +M3y2(n)y3(n) +M1y1(m)

−M3y2(m)y3(m))]∥

(21)

∥P (y1(n)(t))− P (y1(m)(t))∥ = ∥y1(n)(t)− y1(m)(t)∥

+ ST−1[
1− ρ+ ρu

M(ρ)
ST (∥ −M1(y1(n) − y1(m))∥+ ∥M3y2(n)(y3(n) − y3(m))∥

+ ∥M3y3(m)(y2(n) − y2(m))∥)].

(22)

∥y1(n)(t)− y1(m)(t)∥ ≈ ∥y2(n)(t)− y2(m)(t)∥
∥y1(n)(t)− y1(m)(t)∥ ≈ ∥y3(n)(t)− y3(m)(t)∥
Replacing this in equation (22) gives:

∥P (y1(n)(t))− P (y1(m)(t))∥ = ∥y1(n)(t)− y1(m)(t)∥

+ST−1[
1− ρ+ ρu

M(ρ)
ST (∥ −M1(y1(n) − y1(m))∥+ ∥M3y2(n)(y1(n) − y1(m))∥

+∥M3y3(m)(y1(n) − y1(m))∥)]

∥P (y1(n)(t))− P (y1(m)(t))∥ = ∥y1(n) − y1(m)∥[1 + ST−1ST (
1− ρ+ ρu

M(ρ)
)∥ −M1∥+

ST−1(ST
1− ρ+ ρu

M(ρ)
)∥M3(y2(n) + y3(m))∥]

(23)



J. Asad et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1144-1157 1149

Since y2(n), y3(m) are bounded as they are convergent sequence, therefore, we can obtain
two different constants K,L for all “t” such that

∥y2(n)∥ < K, ∥y3(m)∥ < L, (m,n) ∈ N× N. (24)

Now considering equation (22) with equation (23), we get

∥P (y1(n)(t))− P (y1(m)(t))∥ = (1−M1f(γ) +M3(K + L)h(γ))∥y1(n) − y1(m)∥ (25)

Where f,g and h are functions from ST−1ST (1−ρ+ρu
M(ρ) ). Similarly we get

∥P (y2(n)(t))− P (y2(m)(t))∥ = (1 +M1f(γ)−M2g(γ)−M3(K + L)h(γ))∥y2(n) − y2(m)∥
(26)

∥P (y3(n)(t))− P (y3(m)(t))∥ = (1 +M2g(γ))∥y3((n))− y3(m)∥ (27)

Where

1−M1f(γ) +M3(K + L)h(γ) < 1,
1+M1f(γ)−M2g(γ)−M3(K + L)h(γ) < 1,
1+M2g(γ) < 1,

Then, we get c = (0, 0, 0),
C = (1−M1f(γ) +M3(K + L)h(γ),
1 +M1f(γ)−M2g(γ)−M3(K + L)h(γ),
1 +M2g(γ)).

4. Atangana-Baleanu Derivative in Caputo Sense

We consider

ABC
0 Dα

t y1 = −M1y1 +M3y2y3,
ABC
0 Dα

t y2 = M1y1 −M2y
2
2 −M3y2y3,

ABC
0 Dα

t y3 = M2y
2
2.

(28)

By using Sumudu transform, we have

B(α)αΓ(α+ 1)

(1− α)
Eα(−

1

(1− α)
wα)ST (y1(t)− y1(0)) = ST [−M1y1 +M3y2y3],

B(α)αΓ(α+ 1)

(1− α)
Eα(−

1

(1− α)
wα)ST (y2(t)− y2(0)) = ST [M1y1 −M2y

2
2 −M3y2y3],

B(α)αΓ(α+ 1)

(1− α)
Eα(−

1

(1− α)
wα)ST (y3(t)− y3(0)) = ST [M2y

2
2].

(29)



J. Asad et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1144-1157 1150

Rearranging the above equations gives:

ST (y1(t)) = y1(0) +
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)
× ST [−M1y1 +M3y2y3],

ST (y2(t)) = y2(0) +
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)
× ST [M1y1 −M2y

2
2 −M3y2y3],

ST (y3(t)) = y3(0) +
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)
× ST [M2y

2
2].

(30)

y1(t) = y1(0) + ST−1[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)
× ST (−M1y1 +M3y2y3)],

y2(t) = y2(0) + ST−1[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)
× ST (M1y1 −M2y

2
2 −M3y2y3)],

y3(t) = y3(0) + ST−1[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)
× ST (M2y

2
2)].

(31)

Then, we get

y1(n+1)(t) = y1(n)(0) + ST−1[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)
×

ST (−M1y1(n) +M3y2(n)y3(n))],

y2(n+1)(t) = y2(n)(0) + ST−1[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)
×

ST (M1y1(n) −M2y
2
2(n) −M3y2(n)y3(n))],

y3(n+1)(t) = y3((n))(0) + ST ( − 1)[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)
× ST (M2y

2
2(n))].

(32)

And the solution of system (32) is obtained as:

y1(t) = lim
n→∞

y1(n)(t), y2(t) = lim
n→∞

y2(n)(t), y3(t) = lim
n→∞

y3(n)(t). (33)

4.1. Stability and Uniqueness of the Proposed Scheme

Assume that (X, |.|) is a Banach space and H a self-map of X. Let rn+1 = f(Hrn) be
specific recursive procedure. The following condition must be fulfilled for rn+1 = Hrn

• The fixed point set of H possesses at least one element.

• rn converges to a point p ∈ F (H).

• limn→∞ xn(t) = p.



J. Asad et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1144-1157 1151

.

Theorem 3. Suppose that (X, |.|) is a Banach space and H a self-map of X satisfying
∥Hx −Hr∥ ≤ Θ∥x−Hx∥+ θ∥x− r∥ ,

for all x, r ∈ X, where 0≤ Θ, 0≤ θ < 1. Then H is Picard H-Stable.

Theorem 4. Describe H as a self-map:

H[y1(n+1)(t)] = y1(n+1)(t) = y1(n)(t) + ST−1[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)

×ST (−M1y1(n) +M3y2(n)y3(n))],

H[y2(n+1)(t)] = y2(n+1)(t) = y2(n)(0) + ST−1[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)

×ST (M1y1(n) −M2y
2
2(n) −M3y2(n)y3(n))],

H[y3(n+1)(t)] = y3(n+1)(t) = y3(n)(0) + ST−1[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)

×ST (M2y
2
2(n))].

(34)

Proof. By using norm properties, then the iteration is H-Stable

∥H[y1(n)(t)]−H[y1(m)(t)]∥ ≤ ∥y1(n)(t)− y1(m)(t)∥+ ST−1[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)

×ST (−M1∥y1(n) − y1(m)∥+M3∥y2(n)y3(n) − y2(m)y3(m)∥)],

∥H[y2(n)(t)]−H[y2(m)(t)]∥ ≤ ∥y2(n)(t)− y2(m)(t)∥+ ST−1[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)

×ST (M1∥y1(n) − y1(m)∥ −M2∥y22(n) − y22(m)∥ −M3∥y2(n)y3(n) − y2(m)y3(m)∥)],

∥H[y3(n)(t)]−H[y3(m)(t)]∥ ≤ ∥y3(n)(t)− y3(n)(t)∥+ ST−1[
1− α

B(α)αΓ(α+ 1)Eα(− 1
(1−α)w

α)

×ST (M2∥y22(n) − y22(m)∥)].
(35)

Its satisfied in theorem 3, when

θ = (0, 0, 0), Θ = (∥y1(n)(t)− y1(m)(t)∥ × ∥ − (y1(n)(t) + y1(m)(t))∥ −M1∥y1(n) − y1(m)∥+
M3∥y2(n)y3(n) − y2(m)y3(m)∥ × ∥y2(n)(t)− y2(m)(t)∥ × ∥ − (y2(n)(t) + y2(m)(t))∥+

M1∥y1(n) − y1(m)∥ −M2∥y22(n) − y22(m)∥ −M3∥y2(n)y3(n) − y2(m)y3(m)∥×
∥y3(n)(t)− y3(m)(t)∥ × ∥ − (y3(n)(t) + y3(m)(t))∥+M2∥y22(n) − y22(m)∥)

(36)



J. Asad et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1144-1157 1152

Theorem 5. The special solution of system (28) using the iteration method is unique
singular solution.

Proof. By using Hilbert space H = L2((p, q)× (0, r)) which can be described as
h : (p, q)× (0, T ) → R

Considering θ = (0, 0, 0), Θ = (−M1y1 +M3y2y3, M1y1 −M2y
2
2 −M3y2y3, M2y

2
2). We

have

T ((y1(11)(t)− y1(12)(t), y2(21)(t)− y2(22)(t), y3(31)(t)− y3(32)(t)), (V1, V2, V3)). (37)

we get

(−M1(y1(11) − y1(12)) +M3(y2(21) − y2(22))(y3(31) − y3(32)), V1) ≤ M1∥y1(11) − y1(12)∥
∥V1∥+M3∥y2(21) − y2(22)∥∥y3(31) − y3(32)∥∥V1∥,
(M1(y1(11) − y1(12))−M2(y

2
2(21) − y22(22))−M3(y2(21) − y2(22))(y3(31) − y3(32)), V2)

≤ M1∥y1(11) − y1(12)∥∥V2∥+M2∥(y22(21) − y22(22))∥
∥V2∥+ ∥+M3∥y2(21) − y2(22)∥∥y3(31) − y3(32)∥∥V2∥,
(M2(y

2
2(21) − y22(22)), V3) ≤ M2∥(y22(21) − y22(22))∥∥V3∥.

(38)

By using conditions, we get

∥y1 − y1(11)∥, ∥y1 − y1(12)∥ ≤ χe1

ω
,

∥y2 − y2(21)∥, ∥y2 − y2(22)∥ ≤ χe2

ς
,

∥y3 − y3(31)∥, ∥y3 − y3(32)∥ ≤ χe3

υ
.

(39)

where

ω = 3(M1∥y1(11) − y1(12)∥+M3∥y2(21) − y2(22)∥∥y3(31) − y3(32)∥)∥V1∥,
ς = 3(M1∥y1(11) − y1(12)∥+M2∥(y22(21) − y22(22))∥+M3∥y2(21) − y2(22)∥
∥y3(31) − y3(32)∥)∥V2∥,

υ = 3(M2∥(y22(21) − y22(22))∥)∥V3∥.

(40)

But, it is obvious that

(M1∥y1(11) − y1(12)∥+M3∥y2(21) − y2(22)∥∥y3(31) − y3(32)∥) ̸= 0

(M1∥y1(11) − y1(12)∥+M2∥(y22(21) − y22(22))∥+M3∥y2(21) − y2(22)∥∥y3(31) − y3(32)∥) ̸= 0

(M2∥(y22(21) − y22(22))∥) ̸= 0

Where∥V1∥, ∥V2∥, ∥V3∥ ≠ 0

(41)



J. Asad et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1144-1157 1153

Therefore, we have

∥y1(11) − y1(12)∥ = 0, ∥y2(21) − y2(22)∥ = 0, ∥y3(31) − y3(32)∥ = 0, (42)

Which yields that

y1(11) = y1(12), y2(21) = y2(22), y3(31) = y3(32) (43)

This completes the proof of uniqueness.

5. Results and Discussion

The mathematical analysis of chemistry kinetics model with non-linear occurrence has
been offered. Some convergence of theoretical results with some numerical method is also
given in [11]. In Figures 1, 2, 3, 4, 5, 6 the memory effect of fractional order technique for
Robertson problem has been demonstrated. Figures 1, 2, 3 represent the results by using
Caputo-Fabrizio derivative and Figures 4, 5, 6 are obtained with ABC derivative. We can
get better concentration of the components by using the fractional derivative which are
very important for chemical problem to check the actual behavior of the concentration of
the chemical with smallest changes in derivative with respect to time.

Figure 1: Concentration results of y1(t) with CF operator at different fractional order .



J. Asad et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1144-1157 1154

Figure 2: Concentration results of y2(t) with CF operator at different fractional order .

Figure 3: Concentration results of y3(t) with CF operator at different fractional order .

6. Conclusion

In this paper, we considered the stiff systems of nonlinear ordinary equations which
are depend on time t with given initial conditions. The new fractional operator has been
implemented to several initial value problems arising from chemical reactions composed
of large systems of stiff ordinary differential equations. By using the fixed point theory
results, stability and uniqueness of the chemistry kinetic model have been researched.
The arbitrary derivative of fractional order has been taken in the Caputo-Fabrizio sense



J. Asad et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1144-1157 1155

Figure 4: Concentration results of y1(t) with ABC operator at different fractional order .

Figure 5: Concentration results of y2(t) with ABC operator at different fractional order .

with no singular kernel and Atangana-Baleanu in caputo sense with Mittag-Leffler kernel
respectively. Sumudu transform is used to obtain the results for proposed schemes. These
concepts are very important to use real life problems like Brine tank cascade, Recycled
Brine tank cascade, pond pollution, home heating and biomass transfer problem.



REFERENCES 1156

Figure 6: Concentration results of y3(t) with ABC operator at different fractional order.

Acknowledgements

Jihad Asad, and Rania Wannan would like to thank Palestine Technical University –
Kadoorie for supporting this project financially.

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