EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1552-1567
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

The Homotopy Perturbation Method for Solving
Nonlocal Initial-Boundary Value Problems for Parabolic

and Hyperbolic Partial Differential Equations

Waleed Al-Hayani1,∗, Mahasin Thabet Younis1

1 Department of Mathematics, College of Computer Science and Mathematics,
University of Mosul, Iraq

Abstract. To obtain approximate-exact solutions to nonlocal initial-boundary value problems
(IBVPs) of linear and nonlinear parabolic and hyperbolic partial differential equations (PDEs)
subject to initial and nonlocal boundary conditions of integral type, the homotopy perturbation
method (HPM) is utilized in this study. The HPM is used to solve the specified nonlocal IBVPs,
which are then transformed into local Dirichlet IBVPs. Some examples demonstrate how accurate
and efficient the HPM.

2020 Mathematics Subject Classifications: 35-XX, 35K20, 35L04, 35L20

Key Words and Phrases: Nonlocal IBVPs, Parabolic PDEs, Hyperbolic PDEs, HPM, He’s
polynomials

1. Introduction

The transport equation, often known as the one-way wave equation, is an illus-
tration of a first-order linear partial differential equation with constant coefficient:

vτ − kvξ = 0, c ≤ ξ ≤ d, τ ≥ 0,

in which k is a fixed number that specifies constant-speed motion. We establish v (τ, ξ) at
time τ , which we set to 0, i.e. v (0, ξ) is equal to a specific function v0 (ξ) on c ≤ ξ ≤ d, and
the boundary conditions (BCs) known as the nonlocal BCs of integral type which connect

the solution of the differential equation to data of the integral type
∫ d
c v (τ, ξ) dξ = γ (τ) ,

in which v (τ, ξ) indicates the pollutants concentration in gr/cm (ratio of mass to length)

at time τ0 and
∫ d
c v (τ, ξ) dξ indicates the pollutants amount in the interval [c, d] at time

τ.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4794

Email addresses: waleedalhayani@uomosul.edu.iq, waleedalhayani@yahoo.es (W. Al-Hayani),
mahasin thabet@uomosul.edu.iq (M. Th. Younis)

https://www.ejpam.com 1552 © 2023 EJPAM All rights reserved.



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1553

In the case of beginning data and nonlocal BCs, a nonlocal IBVP is the problem
of finding a solution to a PDE. The nonlocal IBVPs with integral BCs can be used to
describe a wide range of problems in conduction of heat [14], engineering with chemicals
[20], thermo-elasticity [17], and physics of plasma [3]. The parabolic PDE with nonlocal
BCs has been studied in [4, 16, 21, 22, 26] and for hyperbolic PDEs [4, 23]. These topics
were looked into, and proper existence and uniqueness theorems were established.

In the last three decades, semi-analytical approximation methods have emerged, such
as HPM, homotopy analysis method (HAM), Adomian decomposition method (ADM),
and variational iteration method (VIM), etc. to solve linear and nonlinear (algebraic,
differential, partial differential, integral, etc.) equations. It has been shown that these
methods yield a rapid convergence of the solutions series.

Ji-Huan He proposed the HPM in 1998. Many authors have relied on him to solve
linear/non-linear ordinary differential equations (ODEs) and PDEs of integer and frac-
tional order [7, 8, 11–13, 18, 19]. If the exact answer exists, the approach converges to it
through repeated approximations.

Recently, Al-Hayani and Younis [2] have applied the HPM with green’s function to solve
the fuzzy system of boundary value problems. Ahmed et al. [1] have solved the nonlinear
system of Volterra integral equations and applied the genetic algorithm to enhance the
solutions by the HAM. Hamoud and Ghadle have used HAM for solving the first order
fuzzy Volterra-Fredholm integro-differential equations [9] and fractional Volterra-Fredholm
integro-differential equation of the second kind [10]. Fiza et al. [6] have applied the
multistep optimal homotopy asymptotic method to some nonlinear KdV-equations. Younis
and Al-Hayani [25] utilized the ADM to solve a fuzzy system of volterra integro-differential
equations. Turkyilmazoglu [24] has proven the accelerating the convergence of ADM.
Finally, Dawood et al. [5] have exercised VIM and MHPM to solve higher-order integro
differential equations.

The principal goal of this study is to use the HPM to find approximate-exact solutions
to solve nonlocal IBVPs for linear/non-linear parabolic and hyperbolic PDEs with initial
and nonlocal BCs of integral type.

2. Applications of the HPM

The HPM will be used to solve nonlocal IBVPs for linear/non-linear variable-
coefficient parabolic and hyperbolic PDEs in this section. There will be five instances
shown.

2.1. Nonlocal IBVP for the linear/non-linear parabolic PDE

Let us consider the inhomogeneous linear/non-linear parabolic PDE

vτ −m (τ, ξ) vξξ + n (τ, ξ) v = h (τ, ξ) + F (v) , c ≤ ξ ≤ d, τ ≥ 0, (1)

subjecting to the IC
v (0, ξ) = α (ξ) , (2)



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1554

and the inhomogeneous nonlocal integral type BCs∫ d
c ψ1 (ξ) v (τ, ξ) dξ = γ1 (τ) and

∫ d
c ψ2 (ξ) v (τ, ξ) dξ = γ2 (τ) , (3)

in which ψi (ξ) , γi (τ) , i = 1, 2 and α (ξ) are specified as continuous functions.
Converting Equations (1)-(3) into local IBVP by using the method of introducing a

function w (τ, ξ) so that

w (τ, ξ) =

∫ ξ

c
ψ (ξ) v (τ, ξ) dξ, (4)

in which ψ (ξ) = ψ1 (ξ) + ψ2 (ξ) . Thus, we have

v (τ, ξ) =
1

ψ (ξ)
wξ (τ, ξ) , ψ (ξ) ̸= 0, (5)

vτ (τ, ξ) =
1

ψ (ξ)
wτξ (τ, ξ) , vττ (τ, ξ) =

1

ψ (ξ)
wττξ (τ, ξ) , (6)

vξ (τ, ξ) =
1

ψ (ξ)
wξξ (τ, ξ) +

(
1

ψ (ξ)

)′
wξ (τ, ξ) , (7)

vξξ (τ, ξ) =
1

ψ (ξ)
wξξξ (τ, ξ) + 2

(
1

ψ (ξ)

)′
wξξ (τ, ξ)

+

(
1

ψ (ξ)

)′′
wξ (τ, ξ) . (8)

Replacing Equations (5)-(8) into Equation (1) we conclude

Lemma 1. The nonlocal IBVP (1)-(3) may be reduced to a local IBVP of the form{
wτξ + r (τ, ξ)wξ + s (τ, ξ)wξξ −m (τ, ξ)wξξξ = g (τ, ξ) +N (w) ,

wξ (0, ξ) = h1 (ξ) , w (τ, c) = 0, w (τ, d) = γ (τ) ,
(9)

in which

r (τ, ξ) = −m (τ, ξ)ψ (ξ)

(
1

ψ (ξ)

)′′
+ n (τ, ξ) , (10)

s (τ, ξ) = −2m (τ, ξ)ψ (ξ)

(
1

ψ (ξ)

)′
, (11)

g (τ, ξ) = ψ (ξ)h (τ, ξ) , (12)

h1 (ξ) = ψ (ξ)α (ξ) , (13)

γ (τ) = γ1 (τ) + γ2 (τ) . (14)



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1555

and the non-linear term N (w) = ψ (ξ)F

(
wξ

ψ (ξ)

)
is assumed to be an analytic function.

This problem’s solution will lead to the original problem’s solution, in which v (τ, ξ) is
provided by Eq (5). By the HPM [7, 8, 11–13, 18, 19], we write

wτξ − (v0)τξ + p
[
(v0)τξ + r (τ, ξ)wξ + s (τ, ξ)wξξ

−m (τ, ξ)wξξξ − g (τ, ξ)−N (w)] = 0,
(15)

Define the solution w (τ, ξ) by an infinite series in the form

w (τ, ξ) =
∞∑
j=0

pjwj , (16)

and the non-linear term N (w) can be decomposed as

N (w (τ, ξ)) =
∞∑
j=0

pjHj (w) , (17)

in which the Hj are He’s polynomials of w0, w1, . . . , wj and are calculated by the defini-
tional formula [8, 15]

Hj (w0, w1, . . . , wj) =
1

j!

∂j

∂pj

[
N

( ∞∑
i=0

piwi

)]
p=0

, j = 0, 1, . . . . (18)

in which p ∈ [0, 1] is an embedding parameter. Replacing (16) and (17) into Equation
(15), we get

∞∑
j=0

pj (wj)τξ − (v0)τξ + p

[
(v0)τξ + r (τ, ξ)

∞∑
j=0

pj (wj)ξ + s (τ, ξ)
∞∑
j=0

pj (wj)ξξ

−m (τ, ξ)
∞∑
j=0

pj (wj)ξξξ − g (τ, ξ)−
∞∑
j=0

pjHj (w)

]
= 0,

(19)

and when we combine the terms in the same power of p, we get

p0 :

{
(w0)τξ − (v0)τξ = 0,

(w0)ξ (0, ξ) = h1 (ξ) , w0 (τ, c) = 0, w0 (τ, d) = γ (τ) ,

p1 :


(w1)τξ + (v0)τξ + r (τ, ξ) (w0)ξ + s (τ, ξ) (w0)ξξ −m (τ, ξ) (w0)ξξξ

−g (τ, ξ)−H0 (w) = 0,

(w1)ξ (0, ξ) = 0, w1 (τ, c) = 0, w1 (τ, d) = 0,

(20)

pj :

{
(wj)τξ + r (τ, ξ) (wj−1)ξ + s (τ, ξ) (wj−1)ξξ −m (τ, ξ) (wj−1)ξξξ −Hj−1 (w) = 0,

(wj)ξ (0, ξ) = 0, wj (τ, c) = 0, wj (τ, d) = 0, j ≥ 2



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1556

Solving the Equations (20) with choosing the initial approximation v0 = α (ξ). Applying

the inverse linear operators L−1
c,τξ (·) =

∫ ξ
c

∫ τ
0 (·) dτdξ to both sides of Equations (20), we

obtain

w0 (τ, ξ) =

∫ ξ

c
h1 (ξ) dξ + L−1

c,τξ (v0)τξ ,

w1 (τ, ξ) = L−1
c,τξ

[
m (τ, ξ) (w0)ξξξ − s (τ, ξ) (w0)ξξ − r (τ, ξ) (w0)ξ

+g (τ, ξ) +H0 (w)] ,

wj (τ, ξ) = L−1
c,τξ

[
m (τ, ξ) (wj−1)ξξξ − s (τ, ξ) (wj−1)ξξ

−r (τ, ξ) (wj−1)ξ +Hj−1 (w)
]
,

j ≥ 2 (21)

Applying the inverse linear operator L−1
d,τξ (·) =

∫ d
ξ

∫ τ
0 (·) dτdξ to both sides of Equations

(20), as previously, we get

w0 (τ, ξ) = γ (τ)−
∫ d

ξ
h1 (ξ) dξ + L−1

d,τξ (v0)τξ ,

w1 (τ, ξ) = −L−1
d,τξ

[
m (τ, ξ) (w0)ξξξ − s (τ, ξ) (w0)ξξ − r (τ, ξ) (w0)ξ

+g (τ, ξ) +H0 (w)] ,

wj (τ, ξ) = −L−1
d,τξ

[
m (τ, ξ) (wj−1)ξξξ − s (τ, ξ) (wj−1)ξξ

−r (τ, ξ) (wj−1)ξ +Hj−1 (w)
]
,

j ≥ 2 (22)

By combining the relationships in (21) and (22) and dividing by 2, we arrive at the equal-
weight average as the solution

w0 (τ, ξ) =
1

2

[∫ ξ

c
h1 (ξ) dξ + γ (τ)−

∫ d

ξ
h1 (ξ) dξ

]
+

1

2

[
L−1
c,τξ (v0)τξ + L−1

d,τξ (v0)τξ

]
,

w1 (τ, ξ) =
1

2
L−1
c,τξ

[
m (τ, ξ) (w0)ξξξ − s (τ, ξ) (w0)ξξ − r (τ, ξ) (w0)ξ + g (τ, ξ) +H0 (w)

]
−1

2
L−1
d,τξ

[
m (τ, ξ) (w0)ξξξ − s (τ, ξ) (w0)ξξ − r (τ, ξ) (w0)ξ + g (τ, ξ) +H0 (w)

]
,

wj (τ, ξ) =
1

2
L−1
c,τξ

[
m (τ, ξ) (wj−1)ξξξ − s (τ, ξ) (wj−1)ξξ − r (τ, ξ) (wj−1)ξ +Hj−1 (w)

]



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1557

−1

2
L−1
d,τξ

[
m (τ, ξ) (wj−1)ξξξ − s (τ, ξ) (wj−1)ξξ − r (τ, ξ) (wj−1)ξ +Hj−1 (w)

]
,

j ≥ 2 (23)

The best approximation for the solution is

w (τ, ξ) = lim
p→1

∞∑
j=0

pjwj = w0 + w1 + w2 + w3 + · · · .

we may use Equation (5) to return to the original dependent variable v (τ, ξ) once the
function w (τ, ξ) has been determined.

2.2. Nonlocal IBVP for the linear/non-linear hyperbolic PDE

We consider the inhomogeneous linear/non-linear hyperbolic PDE

vττ −m (τ, ξ) vξξ + n (τ, ξ) v = h (τ, ξ) + F (v) , c ≤ ξ ≤ d, τ ≥ 0, (24)

subjecting to the ICs

v (0, ξ) = α1 (ξ) , vτ (0, ξ) = α2 (ξ) (25)

with the BCs (3). Replacing Equations (5)-(8) into Equation (24) we conclude

Lemma 2. The nonlocal IBVP (24) subjecting to (25) and (3) may be reduced to a
local IBVP of the form{

wττξ + r (τ, ξ)wξ + s (τ, ξ)wξξ −m (τ, ξ)wξξξ = g (τ, ξ) +N (w) ,

wξ (0, ξ) = h2 (ξ) , wτξ (0, ξ) = h3 (ξ) , w (τ, c) = 0, w (τ, d) = γ (τ) ,
(26)

in which hi (ξ) = ψ (ξ)αi (ξ) , i = 2, 3.
This problem’s solution will lead to the original problem’s solution, in which v (τ, ξ) is

provided by Eq (5). By the HPM, we write

wττξ − (v0)ττξ + p
[
(v0)ττξ + r (τ, ξ)wξ + s (τ, ξ)wξξ

−m (τ, ξ)wξξξ − g (τ, ξ)−N (w)] = 0,
(27)

Replacing (16) and (17) into Equation (27), we get

∞∑
j=0

pj (wj)ττξ − (v0)ττξ + p

[
(v0)ττξ + r (τ, ξ)

∞∑
j=0

pj (wj)ξ + s (τ, ξ)
∞∑
j=0

pj (wj)ξξ

−m (τ, ξ)
∞∑
j=0

pj (wj)ξξξ − g (τ, ξ)−
∞∑
j=0

pjHj (w)

]
= 0,

(28)



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1558

and when we combine the terms in the same power of p, we get

p0 :

{
(w0)ττξ − (v0)ττξ = 0,

(w0)ξ (0, ξ) = h2 (ξ) , (w0)τξ (0, ξ) = h3 (ξ) , w0 (τ, c) = 0, w0 (τ, d) = γ (τ) ,

p1 :


(w1)ττξ + (v0)ξτ + r (τ, ξ) (w0)ξ + s (τ, ξ) (w0)ξξ −m (τ, ξ) (w0)ξξξ

−g (τ, ξ)−H0 (w) = 0,

(w1)ξ (0, ξ) = 0, (w1)τξ (0, ξ) = 0, w1 (τ, c) = 0, w1 (τ, d) = 0,

(29)

pj :

{
(wj)ττξ + r (τ, ξ) (wj−1)ξ + s (τ, ξ) (wj−1)ξξ −m (τ, ξ) (wj−1)ξξξ −Hj−1 (w) = 0,

(wj)ξ (0, ξ) = 0, (wj)τξ (0, ξ) = 0, wj (τ, c) = 0, wj (τ, d) = 0, j ≥ 2

Solving the Equations (29) with choosing the initial approximation v0 = α1 (ξ) + τα2 (ξ).

Applying the inverse linear operators L−1
c,ττξ (·) =

∫ ξ
c

∫ τ
0

∫ τ
0 (·) dτdξ to both sides of Equa-

tions (29), we obtain

w0 (τ, ξ) =

∫ ξ

c
h2 (ξ) dξ + τ

∫ ξ

c
h3 (ξ) dξ + L−1

c,ττξ (v0)ττξ ,

w1 (τ, ξ) = L−1
c,ττξ

[
m (τ, ξ) (w0)ξξξ − s (τ, ξ) (w0)ξξ − r (τ, ξ) (w0)ξ

]
+g (τ, ξ) +H0 (w)] ,

wj (τ, ξ) = L−1
c,ττξ

[
m (τ, ξ) (wj−1)ξξξ − s (τ, ξ) (wj−1)ξξ

]
−r (τ, ξ) (wj−1)ξ +Hj−1 (w)

]
,

j ≥ 2 (30)

Applying the inverse linear operator L−1
d,ττξ (·) =

∫ d
ξ

∫ τ
0

∫ τ
0 (·) dτdξ to both sides of Equa-

tions (29), as previously, we get

w0 (τ, ξ) = γ (τ)−
∫ d

ξ
h2 (ξ) dξ − τ

∫ d

ξ
h3 (ξ) dξ + L−1

d,ττξ (v0)ττξ ,

w1 (τ, ξ) = −L−1
d,ττξ

[
m (τ, ξ) (w0)ξξξ − s (τ, ξ) (w0)ξξ − r (τ, ξ) (w0)ξ

+g (τ, ξ) +H0 (w)] ,

wj (τ, ξ) = −L−1
d,ττξ

[
m (τ, ξ) (wj−1)ξξξ − s (τ, ξ) (wj−1)ξξ

−r (τ, ξ) (wj−1)ξ +Hj−1 (w)
]
,



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1559

j ≥ 2 (31)

By combining the relationships in (30) and (31) and dividing by 2, we arrive at the equal-
weight average as the solution

w0 (τ, ξ) =
1

2

[∫ ξ

c
h2 (ξ) dξ + τ

∫ ξ

c
h3 (ξ) dξ + γ (τ)−

∫ d

ξ
h2 (ξ) dξ − τ

∫ d

ξ
h3 (ξ) dξ

]

+
1

2

[
L−1
c,ττξ (v0)ττξ + L−1

d,ττξ (v0)ττξ

]
,

w1 (τ, ξ) =
1

2
L−1
c,ττξ

[
m (τ, ξ) (w0)ξξξ − s (τ, ξ) (w0)ξξ − r (τ, ξ) (w0)ξ + g (τ, ξ) +H0 (w)

]
−1

2
L−1
d,ττξ

[
m (τ, ξ) (w0)ξξξ − s (τ, ξ) (w0)ξξ − r (τ, ξ) (w0)ξ + g (τ, ξ) +H0 (w)

]
,

wj (τ, ξ) =
1

2
L−1
c,ττξ

[
m (τ, ξ) (wj−1)ξξξ − s (τ, ξ) (wj−1)ξξ − r (τ, ξ) (wj−1)ξ +Hj−1 (w)

]
−1

2
L−1
d,ττξ

[
m (τ, ξ) (wj−1)ξξξ − s (τ, ξ) (wj−1)ξξ − r (τ, ξ) (wj−1)ξ +Hj−1 (w)

]
,

j ≥ 2 (32)

Similarly, once the function w (τ, ξ) has been established, we may utilize Equation (5) to
go back to the initial dependant variable v (τ, ξ).

3. Problems

Problem 1. We first consider the linear nonlocal inhomogeneous IBVP [4]

vτ − vξξ + v = 0, 0 ≤ ξ ≤ π, τ ≥ 0,

v (0, ξ) = sin (ξ) ,∫ π
0 ξv (τ, ξ) dξ = πe−2τ ,∫ π
0 (1− ξ) v (τ, ξ) dξ = (2− π) e−2τ ,

(33)

in which c = 0, d = π, m (τ, ξ) = 1, n (τ, ξ) = 1, h (τ, ξ) = 0, α (ξ) = sin (ξ) , γ (τ) =
2e−2τ and ψ (ξ) = 1. Replacing Equations (5)-(8) into Equation (33), we get a local
inhomogeneous IBVP of the form

wτξ + wξ − wξξξ = 0, wξ (0, ξ) = sin (ξ) , w (τ, 0) = 0, w (τ, π) = 2e−2τ

in which r (τ, ξ) = 1, s (τ, ξ) = 0, m (τ, ξ) = 1, g (τ, ξ) = 0 and h1 (ξ) = sin (ξ) . Following
the algorithm (23), the iterations are

w0 (τ, ξ) =
1

2

[∫ ξ

0
h1 (ξ) dξ + γ (τ)−

∫ π

ξ
h1 (ξ) dξ

]
= − cos (ξ) + e−2τ ,



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1560

w1 (τ, ξ) =
1

2
L−1
0,τξ

[
(w0)ξξξ − (w0)ξ + g (τ, ξ)

]
− 1

2
L−1
π,τξ

[
(w0)ξξξ − (w0)ξ + g (τ, ξ)

]
= 2τ cos (ξ) ,

wj (τ, ξ) =
1

2
L−1
0,τξ

[
(wj−1)ξξξ − (wj−1)ξ

]
− 1

2
L−1
π,τξ

[
(wj−1)ξξξ − (wj−1)ξ

]
=

(−1)j−1

j!
(2τ)j cos (ξ) , j ≥ 2.

Thus, the series form’s approximate solution is

w (τ, ξ) = −
(
1− 2τ + 2τ2 − 4

3
τ3 +

2

3
τ4 − 4

15
τ5 + · · ·

)
cos (ξ) + e−2τ .

This series has been written in closed-form.

w (τ, ξ) = −e−2τ cos (ξ) + e−2τ .

Using Equation (5) to return to the original dependent variable, we get

v (τ, ξ) =
wξ (τ, ξ)

ψ (ξ)
= e−2τ sin (ξ) ,

is the exact solution of the nonlocal IBVP (33) compatible with ADM.

Problem 2. Let us consider the linear nonlocal inhomogeneous IBVP [4]

vτ − vξξ = sin (ξ) , 0 ≤ ξ ≤ π, τ ≥ 0,

v (0, ξ) = cos (ξ) ,∫ π
0 ξv (τ, ξ) dξ = − (2 + π) e−τ + π,∫ π
0 (k − ξ) v (τ, ξ) dξ = (2 + π − 2k) e−τ + 2k − π,

(34)

in which c = 0, d = π, m (τ, ξ) = 1, n (τ, ξ) = 0, h (τ, ξ) = sin (ξ) , α (ξ) = cos (ξ) ,
γ (τ) = 2k (1− e−τ ) and ψ (ξ) = k, k constant. Replacing Equations (5)-(8) into Equation
(34), we get a local inhomogeneous IBVP of the form

wτξ − wξξξ = k sin (ξ) , wξ (0, ξ) = k cos (ξ) , w (τ, 0) = 0, w (τ, π) = 2k (1− e−τ )

in which r (τ, ξ) = 0, s (τ, ξ) = 0, m (τ, ξ) = 1, g (τ, ξ) = k sin (ξ) and h1 (ξ) = k cos (ξ) .
Utilizing the algorithm (23), the iterations are

w0 (τ, ξ) =
1

2

[∫ ξ

0
h1 (ξ) dξ + γ (τ)−

∫ π

ξ
h1 (ξ) dξ

]
= k sin (ξ) + k

(
1− e−τ

)
,

w1 (τ, ξ) =
1

2
L−1
0,τξ

[
(w0)ξξξ + g (τ, ξ)

]
− 1

2
L−1
π,τξ

[
(w0)ξξξ + g (τ, ξ)

]
= −kτ (sin (ξ) + cos (ξ)) ,



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1561

wj (τ, ξ) =
1

2
L−1
0,τξ

[
(wj−1)ξξξ

]
− 1

2
L−1
π,τξ

[
(wj−1)ξξξ

]
=

(−1)j

j!
kτ j (sin (ξ) + cos (ξ)) , j ≥ 2.

Thus, the series form’s approximate solution is

w (τ, ξ) = k

(
1− τ +

τ2

2!
− τ3

3!
+ · · ·

)
sin (ξ)−k

(
τ − τ2

2!
+
τ3

3!
+ · · ·

)
cos (ξ)+k

(
1− e−τ

)
.

This series has been written in closed-form

w (τ, ξ) = ke−τ sin (ξ)− k
(
1− e−τ

)
cos (ξ) + k

(
1− e−τ

)
.

Using Equation (5) to return to the original dependent variable, we get

v (τ, ξ) =
wξ (τ, ξ)

ψ (ξ)
= e−τ cos (ξ) +

(
1− e−τ

)
sin (ξ) ,

is the exact solution of the nonlocal IBVP (34) compatible with ADM.

Problem 3. We consider the linear nonlocal inhomogeneous IBVP [4]

vττ − vξξ = 0, 0 ≤ ξ ≤ 1, τ ≥ 0,

v (0, ξ) = ξ2, vτ (0, ξ) = 0,∫ 1
0 v (τ, ξ) dξ =

1

3
+ τ2,∫ 1

0 ξv (τ, ξ) dξ =
1

4
+

1

2
τ2,

(35)

in which c = 0, d = 1, m (τ, ξ) = 1, n (τ, ξ) = 0, h (τ, ξ) = 0, α1 (ξ) = ξ2, α2 (ξ) = 0,

γ (τ) =
7

12
+

3

2
τ2 and ψ (ξ) = ξ + 1. Replacing Equations (5)-(8) into Equation (35), we

get a local inhomogeneous IBVP of the form
wττξ −

2

(ξ + 1)2
wξ +

2

ξ + 1
wξξ − wξξξ = 0,

wξ (0, ξ) = ξ3 + ξ2, wτξ (0, ξ) = 0,

w (τ, 0) = 0, w (τ, 1) =
7

12
+

3

2
τ2,

in which r (τ, ξ) =
−2

(ξ + 1)2
, s (τ, ξ) =

2

ξ + 1
, m (τ, ξ) = 1, g (τ, ξ) = 0, h2 (ξ) = ξ3 + ξ2

and h3 (ξ) = 0. Using the algorithm (32), the iterations are

w0 (τ, ξ) =
1

2

[∫ ξ

0
h2 (ξ) dξ + τ

∫ ξ

0
h3 (ξ) dξ + γ (τ)−

∫ 1

ξ
h2 (ξ) dξ − τ

∫ 1

ξ
h3 (ξ) dξ

]



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1562

=
1

4
ξ4 +

1

3
ξ3 +

3

4
τ2,

w1 (τ, ξ) =
1

2
L−1
0,ττξ

[
(w0)ξξξ −

2

ξ + 1
(w0)ξξ +

2

(ξ + 1)2
(w0)ξ + g (τ, ξ)

]

−1

2
L−1
1,ττξ

[
(w0)ξξξ −

2

ξ + 1
(w0)ξξ +

2

(ξ + 1)2
(w0)ξ + g (τ, ξ)

]

=
1

2
τ2ξ2 + τ2ξ − 3

4
τ2,

wj (τ, ξ) =
1

2
L−1
0,ττξ

[
(wj−1)ξξξ −

2

ξ + 1
(wj−1)ξξ +

2

(ξ + 1)2
(wj−1)ξ

]

−1

2
L−1
1,ττξ

[
(wj−1)ξξξ −

2

ξ + 1
(wj−1)ξξ +

2

(ξ + 1)2
(wj−1)ξ

]
= 0, j ≥ 2.

Thus, the series form’s approximate solution is

w (τ, ξ) =
1

4
ξ4 +

1

3
ξ3 +

1

2
τ2ξ2 + τ2ξ,

Using Equation (5) to return to the original dependent variable, we get

v (τ, ξ) =
wξ (τ, ξ)

ψ (ξ)
= ξ2 + τ2,

is the exact solution of the nonlocal IBVP (35) compatible with ADM.

Problem 4. Consider the non-linear nonlocal inhomogeneous IBVP [4]
vττ − ξvξξ = 1− v2, 0 ≤ ξ ≤ 1, τ ≥ 0,

v (0, ξ) = 1, vτ (0, ξ) = 0,∫ 1
0 v (τ, ξ) dξ = 1,

∫ 1
0 (ξ − 1) v (τ, ξ) dξ = −1

2
,

(36)

in which c = 0, d = 1, m (τ, ξ) = ξ, n (τ, ξ) = 0, h (τ, ξ) = 1, F (v) = −v2, α1 (ξ) = 1,

α2 (ξ) = 0, γ (τ) =
1

2
and ψ (ξ) = ξ. Replacing Equations (5)-(8) into Equation (36), we

get a local inhomogeneous IBVP of the form
wττξ −

2

ξ
wξ + 2wξξ − ξwξξξ = ξ − 1

ξ
(wξ)

2 ,

wξ (0, ξ) = ξ, wτξ (0, ξ) = 0,

w (τ, 0) = 0, w (τ, 1) =
1

2
,



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1563

in which r (τ, ξ) =
−2

ξ
, s (τ, ξ) = 2, m (τ, ξ) = ξ, g (τ, ξ) = ξ, h2 (ξ) = ξ, h3 (ξ) = 0 and

the non-linear term N (w) = −1

ξ
(wξ)

2 is given by Equation (17). The corresponding He’s

polynomials by the formula Equation (18) are given by

Hj (w) = −1

ξ

j∑
i=0

(wξ)j−i (wξ)i , j ≥ i, j = 0, 1, . . . .

due to the fact that the nonlinear component N (w) exhibits quadratic nonlinearity in wξ.
It should be noted that only the dependent variable w and its derivatives are parametrized
in p, whereas τ and ξ are not. Following the algorithm (32), the iterations are

w0 (τ, ξ) =
1

2

[∫ ξ

0
h2 (ξ) dξ + τ

∫ ξ

0
h3 (ξ) dξ + γ (τ)−

∫ 1

ξ
h2 (ξ) dξ − τ

∫ 1

ξ
h3 (ξ) dξ

]
=

1

2
ξ2,

w1 (τ, ξ) =
1

2
L−1
0,ττξ

[
ξ (w0)ξξξ − 2 (w0)ξξ +

2

ξ
(w0)ξ + g (τ, ξ)−H0 (w)

]

−1

2
L−1
1,ττξ

[
ξ (w0)ξξξ − 2 (w0)ξξ +

2

ξ
(w0)ξ + g (τ, ξ)−H0 (w)

]
= 0,

wj (τ, ξ) =
1

2
L−1
0,ττξ

[
ξ (wj−1)ξξξ − 2 (wj−1)ξξ +

2

ξ
(wj−1)ξ −Hj−1 (w)

]

−1

2
L−1
1,ττξ

[
ξ (wj−1)ξξξ − 2 (wj−1)ξξ +

2

ξ
(wj−1)ξ −Hj−1 (w)

]
= 0, j ≥ 2.

Thus, the series form’s approximate solution is

w (τ, ξ) =
1

2
ξ2,

Using Equation (5) to return to the original dependent variable, we get

v (τ, ξ) =
wξ (τ, ξ)

ψ (ξ)
= 1,

is the exact solution of the nonlocal IBVP (36) compatible with ADM.

Problem 5. Finally, we consider the non-linear nonlocal inhomogeneous IBVP [4]

vτ − ξvξξ = −vvξ, 0 ≤ ξ ≤ 1, τ ≥ 0,

v (0, ξ) = ξ,∫ 1
0 v (τ, ξ) dξ =

1

2 (1 + τ)
,

∫ 1
0

(
eξ − 1

)
v (τ, ξ) dξ =

1

2 (1 + τ)
,

(37)



W. Al-Hayani, M. Th. Younis / Eur. J. Pure Appl. Math, 16 (3) (2023), 1552-1567 1564

in which c = 0, d = π, m (τ, ξ) = ξ, n (τ, ξ) = 0, h (τ, ξ) = 0, F (v) = −vvξ, α (ξ) = ξ,

γ (τ) =
1

1 + τ
and ψ (ξ) = eξ. Replacing Equations (5)-(8) into Equation (37), we get a

local inhomogeneous IBVP of the form
wτξ − ξwξ + 2ξwξξ − ξwξξξ = e−ξ

[
(wξ)

2 − wξwξξ

]
,

wξ (0, ξ) = ξeξ, w (τ, 0) = 0, w (τ, 1) =
1

1 + τ

in which r (τ, ξ) = −ξ, s (τ, ξ) = 2ξ, m (τ, ξ) = ξ, g (τ, ξ) = 0, h1 (ξ) = ξeξ and the non-

linear term N (w) = e−ξ
[
(wξ)

2 − wξwξξ

]
is given by Equation (17). The corresponding

He’s polynomials by the formula Equation (18) are given by

Hj (w) = e−ξ

[
j∑

i=0
(wξ)j−i (wξ)i −

j∑
i=0

(wξ)j−i (wξξ)i

]
, j ≥ i, j = 0, 1, . . . .

because the non-linear term N (w) is the difference between a quadratic nonlinearity in
wξ and a product nonlinearity in wξ and wξξ. Utilizing the algorithm (23), the iterations
are

w0 (τ, ξ) =
1

2

[∫ ξ

0
h1 (ξ) dξ + γ (τ)−

∫ 1

ξ
h1 (ξ) dξ

]
=

1

2
+ eξ (ξ − 1) +

1

2 (1 + τ)
,

w1 (τ, ξ) =
1

2
L−1
0,τξ

[
ξ (w0)ξξξ − 2ξ (w0)ξξ + ξ (w0)ξ + g (τ, ξ) +H0 (w)

]
−1

2
L−1
1,τξ

[
ξ (w0)ξξξ − 2ξ (w0)ξξ + ξ (w0)ξ + g (τ, ξ) +H0 (w)

]
= −1

2
τ − τeξ (ξ − 1) ,

wj (τ, ξ) =
1

2
L−1
0,τξ

[
ξ (wj−1)ξξξ − 2ξ (wj−1)ξξ + ξ (wj−1)ξ +Hj−1 (w)

]
−1

2
L−1
1,τξ

[
ξ (wj−1)ξξξ − 2ξ (wj−1)ξξ + ξ (wj−1)ξ +Hj−1 (w)

]
= (−1)j

[
1

2
τ j + τ jeξ (ξ − 1)

]
, j ≥ 2.

Thus, the series form’s approximate solution is

w (τ, ξ) =
1

2

(
1− τ + τ2 − τ3 + · · ·

)
+
(
1− τ + τ2 − τ3 + · · ·

)
eξ (ξ − 1) +

1

2 (1 + τ)
.

This series has been written in closed-form

w (τ, ξ) =
1

1 + τ
eξ (ξ − 1) +

1

1 + τ
, |τ | < 1.



REFERENCES 1565

Using Equation (5) to return to the original dependent variable, we get

v (τ, ξ) =
wξ (τ, ξ)

ψ (ξ)
=

ξ

1 + τ
, |τ | < 1

is the exact solution of the nonlocal IBVP (37) compatible with ADM.

4. Conclusion

To obtain approximate-exact solutions, the HPM was effectively employed to resolve
nonlocal IBVPs for linear/non-linear parabolic and hyperbolic PDEs subjecting to initial
and nonlocal BCs of integral type. The presented nonlocal integral IBVPs for linear/non-
linear parabolic and hyperbolic PDEs have been turned into local Dirichlet IBVPs. The
HPM has proven to be useful in dealing with these models, broadening its applicability.
The method was put to the test by using it on five different Problems. The results obtained
in each Problem show that this strategy is reliable and efficient for handling this type of
nonlocal IBVPs.

References

[1] Rasha F Ahmed, Waleed Mohammed Al-Hayani, and Abbas Y Al-Bayati. The ho-
motopy analysis method to solve the nonlinear system of volterra integral equations
and applying the genetic algorithm to enhance the solutions. European Journal of
Pure and Applied Mathematics, 16(2):864–892, 2023.

[2] Waleed Mohammed Al-Hayani and Mahasin Thabet Younis. Solving fuzzy system of
boundary value problems by homotopy perturbation method with green’s function.
European Journal of Pure and Applied Mathematics, 16(2):1236–1259, 2023.

[3] Gérard Belmont, Laurence Rezeau, Caterina Riconda, and Arnaud Zaslavsky. Intro-
duction to Plasma Physics. Elsevier, 2019.

[4] Lazhar Bougoffa and Randolph C Rach. Solving nonlocal initial-boundary value
problems for linear and nonlinear parabolic and hyperbolic partial differential equa-
tions by the adomian decomposition method. Applied Mathematics and Computation,
225:50–61, 2013.

[5] Lafta Dawood, Abdulrahman Sharif, and Ahmed Hamoud. Solving higher-order in-
tegro differential equations by vim and mhpm. International Journal of Applied
Mathematics, 33(2):253, 2020.

[6] Mehreen Fiza, Hakeem Ullah, Saeed Islam, Qayum Shah, Farkhanda Inayat Chohan,
and Mustafa Bin Mamat. Modifications of the multistep optimal homotopy asymp-
totic method to some nonlinear kdv-equations. European Journal of Pure and Applied
Mathematics, 11(2):537–552, 2018.



REFERENCES 1566

[7] DD Ganji and A Sadighi. Application of he’s homotopy-perturbation method to
nonlinear coupled systems of reaction-diffusion equations. International Journal of
Nonlinear Sciences and Numerical Simulation, 7(4):411–418, 2006.

[8] Asghar Ghorbani. Beyond adomian polynomials: he polynomials. Chaos, Solitons &
Fractals, 39(3):1486–1492, 2009.

[9] Ahmed A Hamoud and K Ghadle. Homotopy analysis method for the first order
fuzzy volterra-fredholm integro-differential equations. Indonesian Journal of Electri-
cal Engineering and Computer Science, 11(3):857–867, 2018.

[10] Ahmen Hamoud and Kirtiwant Ghadle. Usage of the homotopy analysis method for
solving fractional volterra-fredholm integro-differential equation of the second kind.
Tamkang Journal of Mathematics, 49(4):301–315, 2018.

[11] Ji-Huan He. Homotopy perturbation technique. Computer methods in applied me-
chanics and engineering, 178(3-4):257–262, 1999.

[12] Ji-Huan He. A coupling method of a homotopy technique and a perturbation
technique for non-linear problems. International journal of non-linear mechanics,
35(1):37–43, 2000.

[13] Ji-Huan He. Addendum: New interpretation of homotopy perturbation method.
International journal of modern physics B, 20(18):2561–2568, 2006.

[14] Sadık Kakaç, Yaman Yener, and Carolina P Naveira-Cotta. Heat conduction. CRC
press, 2018.

[15] S Tauseef Mohyud-Din, Muhammad Aslam Noor, and Khalida Inayat Noor. Traveling
wave solutions of seventh-order generalized kdv equations using he’s polynomials.
International Journal of Nonlinear Sciences and Numerical Simulation, 10(2):227–
234, 2009.

[16] Lihua Mu and Hong Du. The solution of a parabolic differential equation with non-
local boundary conditions in the reproducing kernel space. Applied mathematics and
computation, 202(2):708–714, 2008.

[17] Witold Nowacki. Thermoelasticity. Elsevier Science, 2013.

[18] Turgut Öziş and Ahmet Yıldırım. Traveling wave solution of korteweg-de vries equa-
tion using he’s homotopy perturbation method. International Journal of Nonlinear
Sciences and Numerical Simulation, 8(2):239–242, 2007.

[19] M Rafei and DD Ganji. Explicit solutions of helmholtz equation and fifth-order kdv
equation using homotopy perturbation method. International Journal of Nonlinear
Sciences and Numerical Simulation, 7(3):321–328, 2006.



REFERENCES 1567

[20] M Tadi and Miloje Radenkovic. A numerical method for 1-d parabolic equation with
nonlocal boundary conditions. International Journal of Computational Mathematics,
2014, 2014.

[21] E Tohidi and A Kılıçman. An efficient spectral approximation for solving several
types of parabolic pdes with nonlocal boundary conditions. Mathematical Problems
in Engineering, 2014, 2014.

[22] M Turkyilmazoglu. Parabolic partial differential equations with nonlocal initial and
boundary values. International Journal of Computational Methods, 12(05):1550024,
2015.

[23] Mustafa Turkyilmazoglu. Hyperbolic partial differential equations with nonlocal
mixed boundary values and their analytic approximate solutions. International Jour-
nal of Computational Methods, 15(02):1850003, 2018.

[24] Mustafa Turkyilmazoglu. Accelerating the convergence of adomian decomposition
method (adm). Journal of Computational Science, 31:54–59, 2019.

[25] Mahasin Thabet Younis and Waleed Mohammed Al-Hayani. Solving fuzzy system
of volterra integro-differential equations by using adomian decomposition method.
European Journal of Pure and Applied Mathematics, 15(1):290–313, 2022.

[26] Reza Zolfaghari and Abdollah Shidfar. Solving a parabolic pde with nonlocal bound-
ary conditions using the sinc method. Numerical Algorithms, 62:411–427, 2013.