1 Introduction
Differential equations, integral-equations or combi-
nations of them, integro-differential equations, are
obtained in the modeling of real-life engineering phe-
nomena that are inherently non-linear with variable
coefficients. Most of these types of equations do not
have an analytical solution.
Therefore, these problems should be solved by using
numerical or semi-analytical techniques. Computers
and more powerful treatments are required on digital
roads to achieve accurate results. Acceptable results are
obtained through semianalytical way that are computer
and more powerful treatments are required on digital
roads more suitable than numerical methods.
The main feature of semi-analytical roads, compared
to other methods depend on the fact that they can be
applied comfortably to solve various complex problems.
Integral equations of two kinds have obtained serious
content in mathematics, physics, biology, and another
problem in the theory of elasticity.
The author, [1], introduced integral equations with
some applications. Also, their results can be found an-
alytically in the last realizations. At the same time,
the meaning of numerical methods is taken a significant
place in solve of these equations.
The two kinds of integral equations:
f(x) = g(x) + λZx
aZb
a
K(r, t).
f(t)dtdr.
We will expand it to a multi-dimensional fractional
integral equation.
In our paper, we presented different methods, ho-
motopy perturbation and homotopy analysis way for
1
Using Homotopy Perturbation and Analysis Methods for Solving
Different-dimensions Fractional Analytical Equations
MARWA MOHAMED ISMAEEL, WASAN AJEEL AHMOOD
Department of Arabic Language
Department of Al-Quran Science
Al-Iraqia University
Faculty of Education for Women, Baghdad
IRAQ
Abstract: The aim of the research, we extended the one-dimensional to multi-dimensional, we applied the
homotopy perturbation and analysis methods to solve Volterra integral equations and to obtain approximate
analytical solutions of systems of the second kind multi-dimensional Volterra integral equations. We proved the
convergence of the homotopy analysis method (HAM). The HAM solutions contained an auxiliary parameter
that provides a convenient way of controlling the convergence region of series solutions. It is shown that the
solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions.
Several examples are given to illustrate the efficiency and implementation of the method. The results indicate
that this method is efficient for the linear and non - linear models with the dissipative terms.
Key-Words: perturbation method, analysis method and systems of multi-dimensional Volterra integral
equations.
Received: October 19, 2022. Revised: May 25, 2023. Accepted: June 18, 2023. Published: July 27, 2023.
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solving the two kinds of multi-dimensional integral
equations. The author, [2], studied a few problems
with or without small parameters with the homotopy
perturbation technique.
Also, in 2003 the author, [3], studied a new non-linear
analytical technique by the homotopy perturbation
method.
The authors, [4], extended the concept of homotopy
extension property in homotopy theory for topological
to its analogical structure in homotopy theory for
topological semigroups. In this extension, the authors
also gave some results concerning on absolutely retract
and its properties.
The author, [5], used the method of homotopy
perturbation by providing the numerical solutions going
out discretization to deal with the fifth-order boundary
value problems and the computation of the Adomian
polynomials with the coefficients of sixth-degree B-
spline functions.
The author, [6], presented the homotopy perturbation
method of the partial differential equations in many
dimensions with variable coefficients to find the exact
solutions. The results prove that this method is an
effective tool for solving partial differential equations
with variable coefficients.
Many analytical methods including linear crossing
technology: the authors, [7], variational iteration
methods. The authors, [8], a reliable approach for
higher-order integro-differential.
The authors, [9], used the homotopy perturbation
method for solving higher dimensional with initial
boundary value problems of variable coefficients.
The authors in [10] used the homotopy analysis
method for multiple solutions of nonlinear boundary
value problems.
The authors, [11], and the way of decomposing the
authors, [12], developed solving partial or non-linear
partial differential equations.
One of these semi-analytical solutions is the way to
analyze homotopy, the authors, [13], the laplace decom-
position way, the authors, [14], homotopy perturbation
method, the authors, [15], the matrix exponential way,
the authors, [16], the exp-Function.
The authors, [17], used the variational iteration way
and homotopy perturbation way to solve the fractional
Fredholm integral differential equations with constant
coefficients.
The authors, [18], proved the convergence of the
homotopy analysis and applied this method to obtain
approximative analytical solutions of systems of the
second kind integral equations.
The authors, [19], applied coupling of the two meth-
ods (variational iteration and homotopy perturbation)
to solve non-linear mixed integro - differential equations.
The authors, [20], used the homotopy analysis
method to solve two-dimensional non-linear fuzzy
integral equations of two kinds. The authors, [21],
used a hybrid method to solve the analysis of the
fractional-order Navier Stokes equation. It is proven
that the hybrid method is reliable, efficient and easy to
apply for varied contact problems of engineering and
science. In recent years, the homotopy analysis method
has been used to get approximate solutions for a wide
category of differential, integrated, and integrated
equations.
The method provides the solution in a chain quickly
with the components that are elegantly calculated. The
main feature of the method is it can be used directly
without using assumptions or transformations. In this
work, we aim to implement this reliable technique
to solve multi-dimensional integral equation systems.
The authors, [22], presented work, the homotopy
perturbation way to solve the non-linear differential
fractional equation with the help of He’s Polynomials
provided as the transformation plays an essential role
in solving differential linear and non-linear equations.
The authors, [23], solved the Black-Scholes (B-S) model
for the European options pricing problem using a hybrid
way called fractional generalized homotopy analysis
way (FGHAM).
2 Basic Idea of HAM
We consider the following differential equation
N[u(τ)] = 0.Where N is a nonlinear operator,
denotes an independent variable, and u(τ) is an un-
known function, respectively.
For simplicity, we ignore all boundaries or initial con-
ditions, which can be treated similarly.
Using generalizing the traditional homotopy method,
Liao [2003] construct the so-called zero-order deforma-
tion equation:
(1 −p)L(ϕ(τ, p)−u0(τ)) =
phH(τ)N(ϕ(τ, p)).
2
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Where p∈[0,1] is the embedding parameter, h= 0 is a
non-zero auxiliary parameter, H(τ)= 0 is an auxiliary
function, u0(τ) is an initial guess of u(τ) and ϕ(τ;p) is
an unknown function and L an auxiliary linear operator
with the property L[f(τ)] = 0 when f(τ) = 0.
It is important, that one has great freedom to choose
auxiliary things in HAM. obviously, when p=0 and p=1,
it holds:
ϕ(τ; 0) = u0(τ), ϕ(τ; 1) = u(τ)
respectively.
Thus, as p increases from 0 to 1, the solution ϕ(τ, p)
varies from the initial guess u0(τ) to the solution u(τ).
Expanding ϕ(τ;p) in Taylor series with respect to p, we
have:
ϕ(τ;p) = u0(τ) +
+∞
X
m=1
um(τ)pm,
where,
um(τ) = 1
m!
∂mϕ(τ;p)
∂pmp=0
.
If the auxiliary linear operator, the initial guess, the
auxiliary parameter h, and the auxiliary function are so
properly chosen, the above series converges at p=1, then
we have:
u(t) = u0(τ) +
∞
X
m=1
um(τ).
Which must be one of solutions of original nonlinear
equation, as proved by, [24]. As h=-1 and H(τ) = 1,
becomes:
(1 −p)L(ϕ(τ, p)−u0(τ))
+pN(ϕ(τ, p))
Which is used mostly in the homotopy perturbation
method, [9], where as the solution obtained directly,
without using Taylor series, [7]. The governing equa-
tion can be deduced from the zero-order deformation.
Define the vector:
un=u0(τ), u1(τ), ..., un(τ).
Differentiating above equation m-time with respect to
the embedding parameter p and then setting p=0 and
finally dividing them by m!, we have the so-called mth-
order deformation equation:
L(um(τ)−xmum−1(τ)) =
¯hH(τ)Rm(u→
m−1) = 0.
Where
Rm(u→
m−1) =
1
(m−1)!
∂m−1ϕ(t, p)
∂pm−1
Where q=0, and
xm=0,m≤1
1, m > 1.
It should be emphasized that um(τ) for m≥1 is gov-
erned by the linear equation:
(1 −p)L(ϕ(τ, p)−u0(τ))
+pN(ϕ(τ, p))
under the linear boundary conditions that come from
the original problem which can be solved by symbolic
computation software such as Matlab.
For the convergence of the above method we refer the
reader to Liao’s work, [24].
If N[u(τ)] = 0,admits a unique solution, then this
method will produce the unique solution.
If the equation does not possess a unique solution, the
HAM will give a solution among many other (possible)
solutions.
3 The Solution Series Conver-
gent:
In this section, we will prove that, as long as the solution
series
u(t) = u0(τ) +
+∞
X
m=1
um(τ).
given by the homotopy analysis method is convergent,
it must be the solution of the considered nonlinear
problem.
Theorem [3.1]:-
As long as the series:
u0(τ) +
+∞
X
m=1
um(τ),
is convergent, where um(τ) is governed by the high-order
deformation equation
L[um(τ)−xmum−1(τ)] =
¯hH(τ)Rm(u→
m−1)=0,
under the definitions:
3
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Rm(u→
m−1) =
1
(m−1)!
∂m−1ϕ(t, p)
∂pm−1
where q=0 and
xm=0,m≤1
1, m > 1,
it must be a solution of equation
N[u(τ)] = 0.
Proof:-
Let
u(t) = u0(τ) +
+∞
X
m=1
um(τ).
Denote the convergent series by using the above defi-
nitions, we have:
¯hH(t)
+∞
X
m=1
ℜmum−1=
+∞
X
m=1
L(um(τ)−xmum−1(τ)) =
L
+∞
X
m=1
(um(τ)−xmum−1(τ))
=L((1 −x2)
+∞
X
m=1
um(τ))
=L((1 −x2)(s(t)−u0(t))),
which gives, since ¯h= 0, H(t) = 0,
+∞
X
m=1
ℜmum−1= 0.
On the other side, we have according to the above defi-
nition, that:
+∞
X
m=1
ℜmum−1=Rm(u→
m−1) =
1
(m−1)!
∂m−1ϕ(t, q)
∂qm−1
q=0
,
In general, Φ(t, q) does not satisfy the original
non-linear equation N[u(τ)] = 0.
Let E(t;q) = N[Φ(t, q)] = 0,denote the residual error
of equation
N[u(τ)] = 0.Clearly E(t;q) = 0.
Corresponds to the exact solution of the original
equation N[u(τ)] = 0.
According to above definition, the Maclaurin series of
the residual error E(t;q) about the embedding parame-
ter q is:
+∞
X
m=0
1
m!
∂mE(t, q)
∂qmqm=
+∞
X
m=0
1
m!
∂mN(t, q)
∂qmqm,
When q=1, the above expression gives
E(t;q) =
+∞
X
m=0
1
m!
∂mE(t, q)
∂qmqm= 0.
This means, according to the definition of E(t;q)
that we gain the exact solution of the original equation
N[u(τ)] = 0,when q.
Thus, as long as the series:
u0(τ) +
+∞
X
m=1
um(τ),
is convergent, it must be the solution of the original
equation N[u(τ)] = 0.
4 Applications:
In order to assess the advantages and the accuracy of
homotopy analysis way for solving system of multi-
dimensional integral equations of the second kind, we
will consider the following two examples.
Example[1]:-
Consider the following linear system of two-dimensional
Volterra integral equations:
f1(x, y) = ysiny −coshx
+Za
0Zb
0
[e−(s−y)f1(s1, s2)
+cos(s−x)f2(s1, s2)]ds1ds2
f2(x, y)=2siny +x(sin2x+ex)
−Za
0Zb
0
[e(s+y)f1(s1, s2)
+xcos(s)f2(s1, s2)]ds1ds2.
4
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Where a and b are any constant, the exact solutions to
above equations are given below:
f1(x, y) = ysiny −coshx,
f2(x, y) = 2siny +x(sin2x+ex).
The homotopy analysis way, the linear operators:
Li[ϕi(x, p)] = [ϕi(x, p)] Lj[ϕj(y, q)]
= [ϕj(y, q)] ,
for i=1,2 and j=1,2 We now define a non-linear operators
as
ℵ1[Φ1, ϕ2] = [ϕ1(x, y;p, q)]
−(ysiny −coshx)
+Za
0Zb
0he−(s−y)ϕ1(s1, s2;p, q)
+cos(s−x)Φ2(s1, s2;p, q)ds1ds2
ℵ2[Φ1, ϕ2]=[ϕ2(x, y;p, q)]
−2siny +x(sin2x+ex)
+Za
0Zb
0
e(s+y)Φ1(s1, s2;p, q)
+xcos(s)Φ2(s1, s2;p, q)ds1ds2.
by using the above definition, we construct the zeroth-
order deformation equations:
(1 −p)L11[Φ11(x;p)−f11,0(x)](1 −q)
L21[Φ21(y;q)−f21,0(y)] =
p¯h11H11(x)N11[Φ1,Φ2]
q¯h21H21(y)N21[Φ1,Φ2],
(1 −p)L21[Φ21(x;p)−f21,0(x)](1 −q)
L22[Φ22(y;q)f22,0(y)] =
p¯h21H21(x)N21[Φ1,Φ2]
q¯h22H22(y)N22[Φ1,Φ2].
Thus, we obtain the mth-order (m≥1) deformation
equations
L1f11,m11 (x)−x11f11,(m11 −1)
[f21,m21 (y)−f21,m21 −1(y)]
= ¯h1H1(x)ℜ1,m
[f11,m11 −1, f12,m12 −1],
L2[f21,m21 (y)−y21f21,m21 −1(y)]
f22,m22 (y)−y22f22,(m22 −1)(y)=
¯h22H22(y)ℜ22,m[f21,m−1, f22,m−1].
Where
ℜ1,m[f11,m−1, f12,m−1] = f1,m−1(x, y)
−Za
0Zb
0
[e−(s−y)f1(s1, s2)
+cos(s−x)f2(s1, s2)]ds1ds2
ℜ2,m[f21,m−1, f22,m−1] = f2,m−1(x, y)
+Za
0Zb
0
[e(s+y)f1(s1, s2)
+xcos(s)f2(s1, s2)]ds1ds2.
Now, the solution of the mth-order m≥1 zeroth-order
deformation equations becomes:
f11,m11 (x) = x11f11,(m11 −1)
[f21,m21 (y)−f21,m21 −1(y)]
+¯h11H11(x)ℜ11,m
[f21,m21 −1, f22,m22 −1],
f21,m21 (y) = y21f21,m21 (y)
f22,m22 (y)−y22f22,(m22 −1)(y)
+¯h22H22(y)ℜ22,m[f21,m−1, f22,m−2].
By start with an initial approximations:
f1,0(x, y) = ysiny −coshx,
f2,0(x, y)=2siny +x(sin2x+ex).
and by choose Hi= 1, i = 1,2,we suppose:
f1(x, y)≈
m=0
X
5
f1,m,
f2(x, y)≈
m=0
X
5
f2,m.
The comparison of the results of the HAM and the
HPM, [8], are presented in Table 1.
Table 1, shows the absolute error between the HAM
and HPM when (h =-1)by the comparison and the
exact solution.
If e1,2(f1(HAM =HP M))
x,y=0.0 0
x,y=0.1 1.4E−07
x,y=0.2 3.5E−06
x,y=0.3 5.5E−05
x,y=0.4 3.8E−04
x,y=0.5 1.6E−03
5
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e1,2(f2(HAM)) e1,2(f2(HP M ))
5.0e-8 5.0e−8
3.2E-7 3.2E−7
1.1E-5 1.1E−5
1.2E-4 1.2E−4
6.3E-4 6.3E−4
2.2E-3 2.2E−3
Example[2]:-
Let us solve the following non-linear system of two-
dimensional Volterra integral equations:
f1(x, y) = sinx −y
+Zx
0Zy
0f2
1(s1, s2)
+f2
2(s1, s2)ds1ds2
f2(x, y) = cosx −1
2sin2y
+Zx
0Zy
0
f1(s1, s2)
f2(s1, s2)ds1ds2
With the exact solutions:
f1(x, y) = sinx −y,
f2(x, y) = cosx −1
2sin2y.
To solve the system (deformation equation) by means of
homotopy analysis method, we choose the linear opera-
tors:
Li[ϕi(x, p)] = [ϕi(x, p)] Lj[ϕj(y, q)]
= [ϕj(y, q)] ,
for i=1,2 and j=1,2.
We now define a nonlinear operators as:
ℵ1[Φ1, ϕ2]=[ϕ1(x, y;p, q)]
−(sinx −y)
+Zx
0Zy
0ϕ2
1(s1, s2;p, q)
+ Φ2
2(s1, s2;p, q)ds1ds2
ℵ2[Φ1, ϕ2]=[ϕ2(x, y;p, q)]
−cosx −1
2sin2y
+Zx
0Zy
0
Φ1(s1, s2;p, q)
+ Φ2(s1, s2;p, q)ds1ds2
Using the above definition, we construct the zeroth-
order deformation equations:
(1 −p)L11[Φ11(x;p)−f11,0(x)](1 −q)
L21[Φ21(y;q)−f21,0(y)] =
p¯h11H11(x)N11[Φ1,Φ2]
q¯h21H21(y)N21[Φ1,Φ2],
(1 −p)L21[Φ21(x;p)−f21,0(x)](1 −q)
L22[Φ22(y;q)f22,0(y)] =
p¯h21H21(x)N21[Φ1,Φ2]
q¯h22H22(y)N22[Φ1,Φ2].
Thus, we obtain the mth-order (m≥1) deformation
equations:
L1f11,m11 (x)−x11f11,(m11 −1)
[f21,m21 (y)−f21,m21 −1(y)] =
¯h1H1(x)ℜ1,m[f11,m11 −1, f12,m12 −1],
L2[f21,m21 (y)−y21f21,m21 −1(y)]
f22,m22 (y)−y22f22,(m22 −1)(y)=
¯h22H22(y)ℜ22,m[f21,m−1, f22,m−1].
Where
ℜ1,m[f11,m−1, f12,m−1] = f1,m−1(x, y)
−Zx
0Zy
0f2
1(s1, s2)
+f2
2(s1, s2)ds1ds2
ℜ2,m[f21,m−1, f22,m−1] = f2,m−1(x, y)
Zx
0Zy
0
f1(s1, s2)f2(s1, s2)ds1ds2
Now, the solution of the mth - order m≥1 zeroth-order
deformation equations becomes:
f11,m11 (x) = x11f11,(m11 −1)
[f21,m21 (y)−f21,m21 −1(y)] + ¯h11
H11(x)ℜ11,m[f21,m21 −1, f22,m22 −1],
f21,m21 (y) = y21f21,m21 (y)
[f22,m22 (y)−f22,(m22 −1)(y)] + ¯h22
H22(y)ℜ22,m[f21,m−1, f22,m−2].
6
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By start with an initial approximations:
f1,0(x, y) = sinx −y
f2,0(x, y) = cosx −1
2sin2y,
and by choose Hi= 1, i = 1,2,
we suppose:
f1(x, y)≈
m=0
X
5
f1,m,
f2(x, y)≈
m=0
X
5
f2,m.
The comparison of the results of the HAM and the
HPM, [8], are presented in Table 2.
Table 2, shows the absolute error between the HAM
(h=-0.98), the HPM (h=-1) and the exact solution:
E1,2(HAM)E1,2(HP M )
ifx, y = (0,0),0 8.8818e−16
(0,1), 2.0163E-09 2.3210E−08
(0,2),1.1357E−07 1.9188E−06
(0,3),6.2312E−06 2.8336E−05
(0,4),7.3007E−05 2.0695E−04
(0,5),4.6483E-04 1.0278E−03
E1,2(HAM)E1,2(HP M )
1.3878e−17 4.7878e−16
3.1167E−09 1.7048E−08
4.2050E−08 1.6652E−06
6.0365E−06 2.8771E−05
8.5015E−05 2.4354E−04
6.2664E−04 1.3895E−03
5 Conclusion:
In this paper, the HAM was used to obtain analytical
solutions for systems of linear and non-linear Volterra
integral equations of the second kind.
We studied multi-dimensional integral equations
of the second kind by using the homotopy analysis
method, and the homotopy perturbation way, which
can degenerate to an approximate solution in the limit
case. A comparison was made between HAM and HPM
and found that HAM is more effective than HPM. In
addition, the advantage of this method is the rapid
convergence of solutions by the additional parameter
H and the freedom to choose ¯hfor HAM a smoothness
gives us more accuracy from HPM. Hence, it may be
concluded that this method is a powerful and efficient
technique for finding the analytic solutions for wide
classes of problems.
Our results show that homotopy perturbation and
analysis methods are applicable to solve the Volterra in-
tegral equations, how to apply this method to different-
dimensions fractional analytical equations remains to be
further studied.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
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Wasan Ajeel: Theorems, examples, and methodology
Marwa Mohamed: Investigation and writing
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Conflicts of Interest
we applied the homotopy perturbation and
analysis methods to solve Volterra integral
equations and to obtain approximate analytical
solutions of systems of the second kind
multi-dimensional Volterra integral equations
Alternatively, in case of no conflicts of interest
the following text will be published:
The authors have no conflicts of interest to
declare that are relevant to the content of this
article.
8
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.69
Marwa Mohamed Ismaeel, Wasan Ajeel Ahmood
E-ISSN: 2224-2678
691
Volume 22, 2023
Creative Commons Attribution License 4.0
(Attribution 4.0 International , CC BY 4.0)
This article is published under the terms of the
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US
9
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.69
Marwa Mohamed Ismaeel, Wasan Ajeel Ahmood
E-ISSN: 2224-2678
692
Volume 22, 2023
