EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1940-1955
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Solving nth-order integro-differential equations by novel
generalized hybrid transform

Sana Ullah Khan1, Asif Khan1, Aman Ullah1, Shabir Ahmad1, Fuad A. Awwad2,
Emad A. A. Ismail2, Shehu Maitama3, Huzaifa Umar4, Hijaz Ahmad4,5,6,∗

1 Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa,
Pakistan
2 Department of Quantitative Analysis, College of Business Administration,
King Saud University, P.O. Box 71115, Riyadh 11587, Saudi Arabia
3 School of Mathematics, Shandong University, Jinan, Shandong, China
4 Operational Research Centre in Healthcare, Near East University, TRNC Mersin 10,
Nicosia, 99138, Turkey
5 Department of Computer Science and Mathematics, Lebanese American University, Beirut,
Lebanon
6 Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio
Emanuele II, 39,00186 Roma, Italy

Abstract. Recently, Shehu has introduced an integral transform called Shehu transform, which
generalizes the two well-known integrals transforms, i.e. Laplace and Sumudu transform. In
the literature, many integral transforms were used to compute the solution of integro-differential
equations (IDEs). In this article, for the first time, we use Shehu transform for the computation of
solution of nth-order IDEs. We present a general scheme of solution for nth-order IDEs. We give
some examples with detailed solutions to show the appropriateness of the method. We present the
accuracy, simplicity, and convergence of the proposed method through tables and graphs.

2020 Mathematics Subject Classifications: 44A10, 44A35, 44A40, 45E10, 45G10, 47G10

Key Words and Phrases: Integro-differential equations, Shehu transform, Integral transforms

1. Introduction and motivation

Volterra identified the genetic factors when studying a population growth model. He
introduced a new topic in which both differential and integral operators appeared in the
same equation, known as Volterra IDEs [27, 29]. IDEs have recently piqued the interest
of researchers due to their wide range of applications in fields such as fluid dynamics,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4840

Email addresses: sanamath898@gmail.com (S. U. Khan),
lifekhan507@gmail.com (Asif Khan), amanswt@gmail.com (A. Ullah),
shabirahmad2232@gmail.com (S. Ahmad), ahmad.hijaz@uninettuno.it (H. Ahmad)

https://www.ejpam.com 1940 © 2023 EJPAM All rights reserved.



H. Ahmad et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1940-1955 1941

circuit analysis, epidemiology, infectious diseases, and heat flow [30]. The solution to such
problems is linked to the solution of the Volterra form of IDEs. As a result, various methods
for solving IDEs have been used by researchers. Laplace and Fourier introduced integral
transforms, which are the most widely used in the literature and recently applied to many
other integral transforms that can solve differential and integral equations [6, 14, 18]. The
main difference between the Laplace from the FT, FT (Fourier transform) can only be
defined on a stable system, while the Laplace transform can be defined for the system which
is stable or unstable. Another integral transform is the Mellin transform, which is used in
applied sciences due to its invariant property [21]. Many integral transforms introduced
in the last few decades, including the Hankel’s integral transform [17], Sumudu integral
transform [41], Elzaki transforms [20], natural transform [26], Abdon-Kilicman integral
transform [15], the Yang transform [42], and others. Some existing integral transforms,
however, are incapable of solving models containing nonlinear terms. As a result, several
researchers are interested in a different approach to solving real-world problems. The
double LADM is used to solve linear and nonlinear PDEs in [19]. Belgacem et al. in
2017 [16] applied Natural transform (NT) and Sumudu transform (ST) to solve Stokes
equation and diffusion equation of fractional order. However, since physical phenomena
are almost nonlinear, we’re interested in nonlinear integro-differential equations. The
nth-order nonlinear IDE is given by:

G(n)(x) + u(x)G(x) +

∫ d

c
K(x, t)G(p)(t)dt = h(x), c < x < d, (1)

with initial conditions Gj(0) = βj , where βj ∈ R for j = 0, 1, ..., n− 1,and p ≥ 1.
Maitama and Zhao successfully derived an integral transform known as the Shehu

transform from the classical FT in 2019 and demonstrated its accuracy, validity, and
simplicity by applying it to both ODEs and PDEs [31]. Further, Adomian [22, 23] in-
troduced a novel and efficient approach (named the Adomian decomposition method) for
solving linear as well as nonlinear equations at the beginning of the 1980s. This ap-
proach quickly converges the solutions sequence to linear and nonlinear deterministic and
stochastic equations. The purpose of the current study is to solve nth-order IDEs by using
a novel generalized transform called Hybrid Shehu transform (HST). The HST consists of
approximating the solution as

G(x) =

∞∑
q=0

Gq(x), (2)

and for p ≥ 2, the nonlinear term N(G(p)) (if any) will decomposed by

N(G(p)) =

∞∑
q=0

Aq, (3)

where Aq(Adomian polynomials) is defined as

Aq =
1

Γ(q + 1)

dq

dηq

N
 ∞∑

q=0

ηqGq

p
η=0

, q = 0, 1, 2, · · · .



H. Ahmad et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1940-1955 1942

The convergence of the series (2) and (3) in [1, 5].

2. Preliminaries

Definition 1. [31] The Shehu transform of the function G(x) by the following integral

S [G(x)] = H(s, u) =

∫ ∞

0
exp

(
−sx

u

)
G(x)dx, (4)

provided that the integral converges.

Definition 2. [31] The Shehu transform is linear, i.e, for any constants k1,k2 ̸= 0,we
have

S [k1G(x) + k2J(x)] = k1S [G(x)] + k2S [J(x)] .

Definition 3. [31] The formula for the Shehu transform of nth−order derivative of G(x)
is represented as:

S
[
G(n)(x)

]
=

sn

un
G(s, u)−

( s
u

)(n−1)
G(0)−

( s
u

)(n−2)
G′(0)− · · · −G(n−1)(0). (5)

3. Solution procedure using HST

The nth order nonlinear IDE (1) can also written as

G(n)(x) = h(x)− u(x)G(x)−
∫ d

c
K(x, t)G(p)(t)dt, c < x < d. (6)

Applying Shehu transform to both side of the (6) and keep in mind the fact that the
Convolution theorem holds for Shehu transform

sn

un
G(s, u)−

( s
u

)(n−1)
G(0)−

( s
u

)(n−2)
G′(0)− · · · −G(n−1)(0)

= S [h(x)]− S [u(x) ∗G(x)] (s, u)− S
[∫ d

c
K(x, t)G(p)(t)dt

]
= S [h(x)]− S [u(x)]S [G(x)]−

∫ d

c
S [K(x, t)]G(p)(t)dt,

this can be reduce to

G(s, u) =


un

[
( s
u)

(n−1)
G(0)−( s

u)
(n−2)

G′(0)−···−G(n−1)(0)
]

sn+unS[u(x)] + unS[h(x)]
sn+unS[u(x)]

− un

sn+unS[u(x)]
∫ d
c S [K(x, t)]G(p)(t)dt,

(7)

substituting Equ. (2) and (3) into Eq. (7 ), we get

S

 ∞∑
q=0

Gq(x)

 =
un
[(

s
u

)(n−1)
G(0)−

(
s
u

)(n−2)
G′(0)− · · · −G(n−1)(0)

]
sn + unS [u(x)]



H. Ahmad et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1940-1955 1943

+
unS [h(x)]

sn + unS [u(x)]
− un

sn + unS [u(x)]

∫ d

c
S [K(x, t)]

∞∑
q=0

Aq(t)dt,

the HST method and comparing terms givesS [G0(x)] =
un

[
( s
u)

(n−1)
G(0)−( s

u)
(n−2)

G′(0)−···−G(n−1)(0)
]

sn+unS[u(x)]

+ unS[h(x)]
sn+unS[u(x)]

(8)

The general can be obtained as

S [Gq+1(x)] = − un

sn + unS [u(x)]

∫ d

c
S [K(x, t)]

∞∑
q=0

Aq(t)dt, (9)

for q = 0, 1, 2, · · · . A sufficient condition for (9) to comply is that

lim
s→∞

un

sn + unS [u(x)]
= 0.

Application of the inverse Shehu Transform (8) gives G0(x), and using the recursive rela-
tion (9) gives the other terms Gq(x), q ≥ 0 as

ϕq [G(x)] =

q−1∑
r=0

Gr(x),

with
lim
q→∞

ϕq [G(x)] = G(x).

The following theorem and examples show the convergence of the proposed method.

3.0.1. Convergence theorem and error estimate

Theorem 1 (Convergence of the proposed method). Let H be a Hilbert space and “G” be
the exact salution of the problem (6) and

∑∞
q=0Gqbe approximate solution of the problem

(6) which is obtained by (HST), will converges to “G” when ∃0 ≤ α ≤ 1,∥Gk+1∥ ≤ α∥G∥,
∀ k ϵ Z+.

Proof. Let we have

U0 = G0,

U1 = G0 +G1,

U2 = G0 +G1 +G2,
...



H. Ahmad et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1940-1955 1944

Uq = G1 +G2 + . . .+Gq,

and we have to show that {Uq}∞q=0 is Cauchy Sequance in the Hilbert space “H” .Therfore
consider

∥Uq+1 −Uq∥ = ∥Gq+1∥ ≤ α∥Gq∥ ≤ α2∥Gq−1∥ ≤ . . . ≤ αq+1∥G0∥

But for every q,mϵN , such that q ≥ m, So we have

∥Uq −Um∥ = ∥(Uq −Uq−1) + (Uq−1 −Uq−2) + . . .+ (Um+1 −Um)∥
≤ ∥(Uq −Uq−1)∥+ ∥(Uq−1 −Uq−2)∥+ . . .+ ∥(Um+1 −Um)∥
≤ αq∥G0∥+ αq−1∥G0∥+ . . .+ αq+1∥G0∥

≤ (αq+1 + αq+2 . . .)∥G0∥ =
αq+1

1− α
∥G0∥

Hance
lim

q,m→∞
∥Un −Um∥ = 0

i-e {Un}∞q=0 is Cauchy Sequance in the Hilbert space “H” and it implise that ∃UϵH, such
that limq→∞Uq = U, i-e U =

∑∞
q=0Gq. This ends the proof.

Theorem 2 (Error estimate). Let
∑j

i=0Gi < ∞ and “G” be its approximate solution .
Let ζ > 0 such that ∥Gi+1∥ ≤ ζ∥Gi∥, then the maximum absolute error is

∥G−
j∑

i=0

Gi∥ <
ζj+1

1− ζ
∥G0∥.

Proof. Since
∑j

i=0Gi < ∞ this indicates that
∑j

i=0Gi is finite. Consider

∥G−
j∑

i=0

Gi∥ = ∥
∞∑

i=j+1

Gi∥

≤
∞∑
i=0

∥Gi∥

≤
j∑

i=0

ζj∥G0∥

≤ ζj+1(1 + ζ + ζ2 + · · · )∥G0∥

≤ ζj+1

1− ζ
∥G0∥.

This ends the proof.



H. Ahmad et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1940-1955 1945

4. Applications

The proposed method for solving nth-order IDEs is demonstrated in this section with
three examples. To demonstrate validity and efficiency of the results obtained using the
current method, we provide comparison between exact and approximate solution. For
numerical values of absolute error, we define absolute error as:

Eq = |Gexact − ϕq(x)| ,

where q = 0, 1, 2, 3 · · · represent the number of the iterations.

Example 1. Consider the second-order IDE as

G′′(x) = exp(x)− x+

∫ 1

0
xtG(t)dt, (10)

under the initial conditions (ICs) G(0) = 1, G′(0) = 1.

Solution. Applying Shehu transform to (10), we have

S
[
G′′(x)

]
= S [exp(x)]− S [x] + S

[∫ 1

0
xtG(t)dt

]
s2

u2
Y (s, u) =

s

u
G(0) +G′(0)

u

s− u
− u2

s2
+

∫ 1

0
S [x] tG(t)dt,

where S [G(x)] (s, u) = Y (s, u), using ICs, we have

Y (s, u) =
u2

s2
+

u

s
+

u3

s2(s− u)
− u4

s4
+

u4

s4

∫ 1

0
tG(t)dt,

putting the series solution (2) for Y (s, u) in the above equation, one can get

Y0(s, u) =
u2

s2
+

u

s
+

u3

s2(s− u)
− u4

s4
, (11)

and by recursive relation, we obtain

S [Gq+1(x)] (s, u) =
u4

s4

∫ 1

0
tGq(t)dt.

Now applying Shehu inverse of both side of (11) giveG0(x) and using the recursive relation
for n = 1, 2, 3, . . ., we get

G0(x) = exp(x)− x3

3!
,

G1(x) =
29

3! · 30
x3,

G2(x) =
29

3! · 302
x3,



H. Ahmad et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1940-1955 1946

...

Gq(x)
29

3!30q
x3.

Thus, the desired approximate solution for q = 1, 2, 3, · · · is given by

ϕq(x) =

q−1∑
r=0

Gr(x) = exp(x)− x3

6 · 30q−1
.

Hence,

G(x) = lim
q→∞

ϕq(x) = lim
q→∞

exp(x)− lim
q→∞

x3

6 · 30q−1
= exp(x),

which the exact solution of (10).

0 0.5 1 1.5 2 2.5 3 3.5 4

x

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 1: Solution curves of Example 1.

0 0.5 1 1.5 2 2.5 3 3.5 4

x

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Approximate

Exact

Figure 2: Comparison between approximate and exact solution of Example 1.

Example 2. Consider the third-order IDE as{
G′′′(x) = sin(x)− x−

∫ Π
2

0 xtG′(t)dt, G(0) = 1, G′(0) = 0,G′′(0) = −1 . (12)



H. Ahmad et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1940-1955 1947

x E3 E6 E8

0.1 1.8518E-06 6.8587E-12 7.6207E-15

0.2 1.3714E-06 5.3971E-11 6.1877E-14

0.3 5.0000E-05 1.8518E-10 2.0576E-13

0.4 1.1852E-05 4.2998E-10 4.9321E-13

0.5 2.3148E-05 9.0000E-11 9.5259E-13

0.6 4.1011E-05 1.3998E-09 1.5982E-12

0.7 6.3518E-05 2.3525E-09 2.6139E-12

0.8 9.5001E-05 3.4998E-09 3.8979E-12

0.9 1.3500E-04 5.0000E-09 5.5555E-12

1.0 1.7809E-04 6.9001E-09 9.5998E-12

Table 1: Absolute error for Example 1.

Solution. Taking the Shehu transform of (12), we obtain

S
[
G′′′(x)

]
= S [sin(x)− x]− S

[∫ Π
2

0
xtG′(t)dt

]
,

so that

s3

u3
Y (s, u)− s2

u2
G(0)− s

u
G′(0)−G′′(0) =

u2

s2 + u2
− u2

s2
− u2

s2

∫ Π
2

0
tG′(t)dt,

by using the IC, we get

Y (s, u) =
u

s
− u3

s3
+

u5

s3(s2 + u2)
− u5

s5
− u5

s5

∫ Π
2

0
tG′(t)dt,

where S [G(x)] (s, u) = Y (s, u), substituting (2) for Y (s, u) and comparing terms, we have

Y0(s, u) =
u

s
− u3

s3
+

u5

s3(s2 + u2)
− u5

s5
− u5

s5
, (13)

using the recursive relation we get

S [Gq+1(x)] (s, u) = −u5

s5

∫ Π
2

0
tG′

q(t)dt. (14)

Taking the inverse Shehu transform of (13) and (14) gives :

G0(x) = cos(x)− x4

4!
,

G1(x) =
−(π5 + 960)

4! · (960)
x4,

G2(x) =
(π5 + 960) · π5

4! · (960)2
x4,



H. Ahmad et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1940-1955 1948

...

Gq(x) =
(−1)q · (π5 + 960) · π5(q−1)

4! · (960)q
x4, q = 1, 2, 3 · · ·

the desired solution is as follow

ϕq(x) =

q−1∑
r=0

Gr(x) = cos(x) +
(−1)q · π5(q−1)

4! · (960)q−1
x4, q = 1, 2, 3 · · ·

G(x) = lim
q→∞

ϕq(x) = lim
q→∞

(
cos(x) +

(−1)q · π5(q−1)

4! · (960)q−1
x4

)
= cos(x).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x

0

2

4

6

8

10

12

14

16

18

Figure 3: Solution curves of Example 2.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x

0

2

4

6

8

10

12

14

16

18

Exact

Approximate

Figure 4: Comparison between exact and approximate solution Example 2.

Example 3. Consider the 5th order integro-differential equation as

G(5)(x) = x+

∫ 1

0
(t− x)G2(t)dt,

with initial condition G(0) = G′(0) = G′′(0) = G′′′(0) = G(4)(0) = 0.



H. Ahmad et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1940-1955 1949

x E3 E6 E8

0.1 4.23394E-07 1.37144E-08 1.39359E-09

0.2 6.77430E-06 2.19431E-07 2.229474E-08

0.3 3.42949E-05 1.11087E-06 1.12880E-07

0.4 1.08388E-04 3.51090E-06 3.56759E-07

0.5 2.64621E-04 8.57155E-06 8.70995E-07

0.6 5.48719E-04 1.77739E-05 1.80609E-06

0.7 1.01656E-03 3.29284E-05 3.34601E-06

0.8 1.73422E-03 5.61748E-05 5.70815E-06

0.9 2.77789E-03 8.99807E-05 9.14335E-06

1.0 4.23394E-03 1.37144E-04 1.39359E-06

Table 2: Maximum error for Example 2

Solution. Applying Shehu transform, we get

s5

u5
Y (s, u)− s4

u4
G(0)− s3

u3
G′(0)− s2

u2
G′′(0)− s

u
G′′′(0)−G(4)(0) =

u2

s2
− u2

s2

∫ 1

0
G2(t)dt,

by putting the initial condition, we get

Y (s, u) =
u7

s7
− u7

s7

∫ 1

0
G2(t)dt,

here S [G] (s, u) = Y (s, u). Putting (2) for Y (s, u) and comparing terms, we have

Y0(s, u) =
u7

s7
,

using the recursive relation we obtain

S [Gq+1(x)] (s, u) = −u7

s7

∫ 1

0
G2

q(t)dt.

Now applying Shehu inverse of both side give G0(x) and using the recursive relation gives

G0(x) =
x6

6!
,

G1(x) = − 1

(6!)313
x6,

G2(x) = − 1

(6!)7(13)3
x6,

...

Gq(x) = − 1

(6!)(2q+1−1)(13)(2q−1)
x6,



H. Ahmad et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1940-1955 1950

thus, the desired approximate solution is:

ϕq(x) =

q−1∑
r=0

Gr(x) =
x6

6!
− 1

(6!)(2q−1)(13)(2q−1−1)
x6,

G(x) = lim
q→∞

ϕq(x) = lim
q→∞

(
x6

6!
− 1

(6!)(2q−1)(13)(2q−1−1)
x6
)

=
x6

6!
.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x

-1

-0.5

0

0.5

1

Figure 5: Solution curve of Example 2.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x

-1

-0.5

0

0.5

1 Approximate

Exact

Figure 6: Comparison between approximate and exact solution of Example 3.



REFERENCES 1951

x E3 E6 E8

0.1 4.5377E-30 2.8563E-221 8.1172E-877

0.2 2.9041E-28 1.8280E-219 5.1950E-875

0.3 3.3056E-27 2.0822E-218 5.9174E-874

0.4 1.8586E-26 1.1699E-217 3.3248E-873

0.5 7.0902E-26 4.4630E-217 1.2683E-872

0.6 2.1171E-25 1.3326E-216 3.7871E-872

0.7 5.3386E-25 3.3604E-216 9.5498E-872

0.8 1.1895E-24 7.4877E-216 2.1278E-871

0.9 2.4115E-24 1.5179E-215 4.3138E-871

1.0 4.5377E-24 2.8563E-215 8.1172E-871

Table 3: Maximum error for Example 3.

5. Conclusion

In this article, we have used a more generalized novel transform called Shehu transform,
which is the generalization of Sumudu and Laplace transform, to solve higher-order IDEs.
We have presented a general scheme of solutions through the proposed transform. We have
given few examples with a detailed solution to show the accuracy and validity of the pro-
posed method. We have shown the convergence of the method through graphs and tables.
From graphs and tables, we can say that the approximate solution is very close to the exact
solution. Thus, the suggested method is more appropriate than other complex analytical
methods because the proposed method is highly accurate, less computational, and fast
convergent. Other numerical and analytical methods [3, 4, 9, 10, 13, 24, 25, 28, 34, 36–
40] can be applied to address such types of problems and other challenging problems
[2, 7, 8, 11, 12, 32, 33, 35]. In our next paper, we will use the Shehu transform to solve
other types of IDEs of integer and fractional orders.

Acknowledgements

Researchers Supporting Project number (RSPD2023R576), King Saud University, Riyadh,
Saudi Arabia.

References

[1] K Abbaoui and Y Cherruault. Convergence of adomian’s method applied to differen-
tial equations. Computers & Mathematics with Applications, 28(5):103–109, 1994.

[2] AE Abouelregals, H Ahmad, and S-W Yao. Functionally graded piezoelectric medium
exposed to a movable heat flow based on a heat equation with a memory-dependent
derivative. Materials, 13(18):3953, 2020.



REFERENCES 1952

[3] M Adels, ME Ramadans, H Ahmad, and T Botmart. Sobolev-type nonlinear Hilfer
fractional stochastic differential equations with noninstantaneous impulsive. AIMS
Mathematics, 7(11):20105–20125, 2022.

[4] M Adels, NH Sweilams, MM Khaders, SM Ahmeds, H Ahmad, and T Botmart. Nu-
merical simulation using the non-standard weighted average FDM for 2Dim variable-
order Cable equation. Results in Physics, 39:105682, 2022.

[5] G Adomian. Nonlinear stochastic systems theory and applications to physics, vol-
ume 46. Springer Science & Business Media, 1988.

[6] HA Agwas, FM Ali, and A Kılıçman. A new integral transform on time scales and
its applications. Advances in Difference Equations, 2012(1):1–14, 2012.

[7] H Ahmad and TA Khan. Variational iteration algorithm I with an auxiliary parameter
for the solution of differential equations of motion for simple and damped mass–spring
systems. Noise & Vibration Worldwide, 51(1-2):12–20, 2020.

[8] H Ahmads, TA Khan, and S-W Yao. An efficient approach for the numerical solution
of fifth-order KdV equations. Open Mathematics, 18(1):738–748, 2020.

[9] I Ahmads, AR Seadawys, H Ahmads, P Thounthongs, and F Wang. Numerical
study of multi-dimensional hyperbolic telegraph equations arising in nuclear material
science via an efficient local meshless method. International Journal of Nonlinear
Sciences and Numerical Simulation, 23(1):115–122, 2022.

[10] M Ahsans, AA Khans, S Dinibutuns, I Ahmads, H Ahmads, N Jarasthitikulchai, and
W Sudsutad. The Haar wavelets based numerical solution of Reccati equation with
integral boundary condition. Thermal Science, 27(Spec. issue 1):93–100, 2023.

[11] A Akgül and H Ahmad. Reproducing kernel method for Fangzhu’s oscillator for water
collection from air. Mathematical Methods in the Applied Sciences, 2020.

[12] R Alharbis, R Jans, S Alyobis, A Yousif, and Z Khan. Mathematical modeling and
stability analysis of the dynamics of monkeypox via fractional-calculus. Fractals,
30(10):2240266, 2022.

[13] B Almutairis, I Ahmads, B Almohsens, H Ahmads, and DU Ozsahin. Numerical
simulations of time-fractional PDEs arising in mathematics and physics using the local
Meshless differential quadrature method. Thermal Science, 27(Spec. issue 1):263–272,
2023.

[14] A Atangana. A note on the triple laplace transform and its applications to some kind
of third-order differential equation. In Abstract and Applied Analysis, volume 2013.
Hindawi, 2013.



REFERENCES 1953

[15] A Atanganas and A Kilicman. A novel integral operator transform and its application
to some fode and fpde with some kind of singularities. Mathematical Problems in
Engineering, 2013, 2013.

[16] FBM Belgacems, R Silambarasans, H Zakia, and T. Mekkaoui. New and extended
applications of the natural and sumudu transforms: Fractional diusion and stokes
fluid flow realms. Advances Real and Complex Analysis with Applications, 2017.

[17] B Davies. Integral transforms and their applications, volume 41. Springer Science &
Business Media, 2002.

[18] JM Davis, IA Gravagnes, BJ Jacksons, II Markss, J Robert J, and AA Ramos. The
laplace transform on time scales revisited. Journal of Mathematical Analysis and
Applications, 332(2):1291–1307, 2007.

[19] H Eltayeb. A note on double laplace decomposition method and nonlinear partial
differential equations. New Trends in Mathematical Sciences, 5(4):156–164, 2017.

[20] TM Elzaki. The new integral transform elzaki transform. Global Journal of pure and
applied mathematics, 7(1):57–64, 2011.

[21] P Flajolets, X Gourdon, and P Dumas. Mellin transforms and asymptotics: Harmonic
sums. Theoretical computer science, 144(1-2):3–58, 1995.

[22] JM Heris. solving the integro-differential equations using the modified laplace ado-
mian decomposition method. Journal of Mathematical Extension, 6, 2012.

[23] JM Heris. solving the integro-differential equations using the modified laplace ado-
mian decomposition method. Journal of Mathematical Extension, 6, 2012.

[24] H Irshads, M Shakeels, I Ahmads, H Ahmads, C Tearnbucha, and W Sudsutad.
Simulation of generalized time fractional Gardner equation utilizing in plasma physics
for non-linear propagation of ion-acoustic waves. Thermal Science, 27(Spec. issue
1):121–128, 2023.

[25] R Jans, A Khans, S Boulaarass, and SA Zubair. Dynamical behaviour and chaotic
phenomena of HIV infection through fractional calculus. Discrete Dynamics in Nature
and Society, 2022, 2022.

[26] ZH Khan and WA Khan. N-transform properties and applications. NUST journal of
engineering sciences, 1(1):127–133, 2008.

[27] W Lederman. Handbook of applicable mathematics. vol. 5, a: Combinatorics and
geometry; vol. 5, b: Combinatorics and geometry. A Wiley-Interscience Publication,
1985.



REFERENCES 1954

[28] J-F Lis, I Ahmads, H Ahmads, D Shahs, Y-M Chus, P Thounthong, and M Ayaz.
Numerical solution of two-term time-fractional PDE models arising in mathematical
physics using local meshless method. Open Physics, 18(1):1063–1072, 2020.

[29] AC Macbride. LI. G. Chambers, Integral Equations: A Short Course (International
Textbook Company Limited, 1976), 198 pp.,£ 7· 00. Proceedings of the Edinburgh
Mathematical Society, 20(4):361–362, 1977.

[30] RC MacCamy. An integro-differential equation with application in heat flow. Quar-
terly of Applied Mathematics, 35(1):1–19, 1977.

[31] S Maitama and W Zhao. New integral transform: Shehu transform a generalization
of sumudu and laplace transform for solving differential equations. arXiv preprint
arXiv:1904.11370, 2019.

[32] NA Shahs, I Ahmads, O Bazighifans, AE Abouelregal, and H Ahmad. Multistage
optimal homotopy asymptotic method for the nonlinear Riccati ordinary differential
equation in nonlinear physics. Applied Mathematics, 14(6):1009–1016, 2020.

[33] M Shakeels, I Hussains, H Ahmads, I Ahmads, P Thounthong, and Ying-Fang Y-
F Zhang. Meshless technique for the solution of time-fractional partial differential
equations having real-world applications. Journal of Function Spaces, 2020:1–17,
2020.

[34] M Shakeels, MN Khans, I Ahmads, H Ahmads, N Jarasthitikulchai, and W Sudsutad.
Local meshless collocation scheme for numerical simulation of space fractional PDE.
Thermal Science, 27(Spec. issue 1):101–109, 2023.

[35] TA Sulaimans, A Yusufs, S Abdel-Khaleks, M Bayram, and H Ahmad. Nonau-
tonomous complex wave solutions to the (2+ 1)-dimensional variable-coefficients non-
linear Chiral Schrödinger equation. Results in Physics, 19:103604, 2020.

[36] T-Q Tang, Z Shahs, R Jan, and E Alzahrani. Modeling the dynamics of tumor–
immune cells interactions via fractional calculus. The European Physical Journal
Plus, 137(3):367, 2022.

[37] T-Q Tangs, Z Shahs, R Jans, W Deebani, and Meshal M Shutaywi. A robust study
to conceptualize the interactions of CD4+ T-cells and human immunodeficiency virus
via fractional-calculus. Physica Scripta, 96(12):125231, 2021.

[38] F Wangs, I Ahmads, H Ahmads, MD Alsulamis, KS Alimgeers, C Cesarano, and
TA Nofal. Meshless method based on RBFs for solving three-dimensional multi-term
time fractional PDEs arising in engineering phenomenons. Journal of King Saud
University-Science, 33(8):101604, 2021.

[39] F Wangs, NA Alis, I Ahmads, H Ahmads, KM Alams, , and P Thounthong. Solution
of Burgers’ equation appears in fluid mechanics by multistage optimal homotopy
asymptotic method. Thermal Science, 26(1 Part B):815–821, 2022.



REFERENCES 1955

[40] F Wangs, E Hous, I Ahmads, H Ahmad, and Y Gu. An efficient meshless method for
hyperbolic telegraph equations in (1+ 1) dimensions. CMES-Computer Modeling in
Engineering and Sciences, 128(2):687–98, 2021.

[41] GK Watugala. Sumudu transform: a new integral transform to solve differential
equations and control engineering problems. Integrated Education, 24(1):35–43, 1993.

[42] X-J Yang. A new integral transform method for solving steady heat-transfer problem.
Thermal Science, 20(suppl. 3):639–642, 2016.