Abstract Integral equations occur naturally in many field of mechanics and mathematical physics. In this paper, an iterative scheme based on the homotopy analysis method (HAM) has been used for solving one of the most important cases in nonlinear integral equations which is called urysohn form. For this scope, we applied some examples with known exact solutions and the numerical solutions obtained confirm that the method is very effective and simple. Also, show the advantage of this method, the results obtained by (HAM) are compared with NewtonKantorovich-quadrature method. Convergence is also observed

A Jafarian

L Khoshbakhti

Z Esmailzadeh

CiteSeerX

Applied Mathematical Sciences, Vol. 7, 2013, no. 28, 1375 - 1385
HIKARI Ltd, www.m-hikari.com
A Numerical Method for Solving Nonlinear
Integral Equations in the Urysohn Form
A. Jafarian, Z. Esmailzadeh and L. Khoshbakhti
Department of Mathematics, Urmia Branch
Islamic Azad University, Urmia, Iran
jafarian5594@yahoo.com
zahra 1324@hotmail.com
leilakhoshbakhti@yahoo.com
Copyright c© 2013 A. Jafarian et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
Integral equations occur naturally in many field of mechanics and
mathematical physics. In this paper, an iterative scheme based on the
homotopy analysis method (HAM) has been used for solving one of the
most important cases in nonlinear integral equations which is called
urysohn form. For this scope, we applied some examples with known
exact solutions and the numerical solutions obtained confirm that the
method is very effective and simple. Also, show the advantage of this
method, the results obtained by (HAM) are compared with Newton-
Kantorovich-quadrature method. Convergence is also observed.
Keywords: Homotopy analysis; Nonlinear integral equation; Newton-
Kantorovich-quadrature method; Urysohn integral equation
1 Introduction
As a matter of fact, it might be said that many phenomena of almost all
practical engineering and applied science problems like physical applications,
potential theory and electrostatics are reduced to solve integral equations.
Since these equations usually can not be solved explicitly, so it is required to
obtain approximate solutions. Therefore, many different methods are used to
obtain the solution of the linear and nonlinear integral equations. The solu-
tion of nonlinear integral equations is a complicated problem of computational
1376 A. Jafarian, Z. Esmailzadeh and L. Khoshbakhti
mathematics, which is related to difficulties of both a principal and computa-
tional character. With the advent of computers, the use numerical methods
for solving these equations always been important in scientific investigations.
Of numerical methods has been popularized and more importantly, people are
now able to attack those problems. Which are fundamental to our understand-
ing of scientific phenomenon, but were so much more difficult to study in the
past. Several numerical methods for approximating the solution of nonlinear
integral equations are known. Several numerical methods for approximating
the solution of nonlinear integral equations are known. Variational iteration
method in the [6], also Homotopy perturbation method and Adomian decom-
position method are effective and convenient for solving integral equations.
The homotopy analysis method (HAM) in the [2-5] is a general analytic ap-
proach to get series solutions of various types of nonlinear equations, including
algebraic equations, ordinary differential equations, partial differential equa-
tions, differential-difference equation. More importantly, different from all
perturbation and traditional non-perturbation methods, the HAM provides
us a simple way to ensure the convergence of solution series, and therefore,
the (HAM) is valid even for strongly nonlinear problems. The HAM is based
on homotopy, a fundamental concept in topology and differential geometry.
Briefly speaking, by means of the HAM, one constructs a continuous mapping
of an initial guess approximation to the exact solution of considered equations
and the method enjoys great freedom in choosing initial approximations and
auxiliary linear operators [1]. Liao and Tan in [5] shown that by means of this
kind of freedom, a complicated nonlinear problem can be transferred into an
infinite number of simpler, linear sub-problems. Until recently, the applica-
tion of the homotopy analysis method in nonlinear problems has been devoted
by scientists and engineers, because this method is to continuously deform a
simple problem easy to solve into the difficult problem under study. When
solving nonlinear equations, the applicability domain of the method of simple
iterations is smaller, and if the process is still convergent. Then in many cases,
the rate of convergence can be very low. An effective method that makes
it possible to overcome the indicated complication is the Newton-kantorovich
method. In [7] Saber-Nadjafi and Heidari with this method solved nonlinear
(IE) of the urysohn form in systematic procedure. In the article, we use the
(HAM) for in the urysohn form such that:
u(x) = f(x) +
∫ b
a
k(x, t, u(t))dt, a ≤ x ≤ b.
where k(x, t, u(t)) is the kernel of the integral equation and u(x) is the un-
known function. some example are tested, and the obtained results suggest
that newly improvement technique introduces a promising tool and compare
with Newton-Kantorovich-quadrature method. This paper is arranged in the
Solving nonlinear integral equations 1377
following manner:
After an introduction to the present work, the homotopy analysis method
(HAM) is described in section 2. In section 3, Computational procedure
method is explained. we explain convergence analysis in section 4. In the
section 5, some numerical examples is presented which show efficiency and ac-
curacy of the proposed method and comparison of the obtained results between
(HAM) an Newton-Kantorovich-quadrature method are discussed. finally con-
clusion is drown in section 6.
2 Description Of The Method
The general form of nonlinear Fredholm integral equations of the urysohn form
is as follows:
u(x) = f(x) +
∫ b
a
k(x, t, u(t))dt, a ≤ x ≤ b,
and
N [u] = u(x) − f(x) −
∫ b
a
k(x, t, u(t))dt = 0. (1)
Where N is an operator, u(x) is unknown function and x the independent
variable. An auxiliary linear operator is chosen to construct such kind of con-
tinuous mapping, and an auxiliary parameter is used to ensure the convergence
of solution series [1]. Let u0(x) denote an initial guess of the exact solution
u(x), h = 0 an auxiliary parameter, H(x) = 0 an auxiliary function, and L an
auxiliary linear operator with the property L[r(x)] = 0 when r(x) = 0. Then
using q ∈ [0, 1] as an embedding parameter, we construct such a homotopy
(1 − q)L[φ(x; q) − u0(x)] − qhH(x)N [φ(x; q)] = Ĥ[φ(x; q); u0(x), H(x), h, q].
(2)
It should be emphasized that we have great freedom to choose the initial
guess u0(x), the auxiliary linear operator L , the non-zero auxiliary parameter
h, and the auxiliary function H(x).
Enforcing the homotopy (2) to be zero, i.e;
Ĥ[φ(x; q); u0(x), H(x), h, q] = 0
we have the so-called zero-order deformation equation
(1 − q)L[φ(x; q) − u0(x)] = qhH(x)N [φ(x; q)]. (3)
1378 A. Jafarian, Z. Esmailzadeh and L. Khoshbakhti
When q = 0 and q = 1, since h = 0 and H(x) = 0, the zero-order deformation
equation (3) is equivalent to
φ(x; 0) = u0(x), (4)
φ(x; 1) = u(x). (5)
Thus, as q increase from 0 to 1, the solution φ(x; q) varies from initial guess
u0(x) to the solution u(x). By Taylor’s theorem, φ(x; q) can be expanded in a
power series of q as follows
φ(x; q) = u0(x) +
∞∑
m=1
um(x)q
m (6)
where
um(x) =
1
m!
∂mφ(x; q)
∂qm
| q = 0. (7)
If the initial guess u0(x), the auxiliary linear parameter L , the nonzero auxil-
iary function H(x) are properly chosen so that the power series (6) of φ(x; q)
convergence at q = 1. Then, we have under these assumptions the solution
series
u(x) = φ(x; 1) = u0(x) +
∞∑
m=1
um(x). (8)
For brevity, define the vector
−→u n(x) = {u0(x), u1(x), u2(x), u3(x), . . . , un(x)}. (9)
According to the definition (7), the governing equation of um(x) can be derived
from the zero-order deformation equation (3). Differentiating the zero-order
deformation equation (3) m times with respective to q and then dividing by
m! and finally setting q = 0 , we have the so-called mth-order deformation
equation
L[um(x) − χmum−1(x)] = hH(x)Rm(−→u m−1(x)), um(0) = 0. (10)
where
Rm(−→u m−1(x)) = 1
(m − 1)!
∂m−1N [φ(x; q)]
∂qm
, (11)
Solving nonlinear integral equations 1379
and
χm =
⎧⎨
⎩
0, m ≤ 1
1, m > 1
Note that the high-order deformation equation (10) is governing by the linear
operator L, and the term Rm(−→u m−1(x)) can be expressed simply by (11) for
any nonlinear operator N.
According to the definition (11), the right-hand side of equation (10) is only
dependent upon um−1(x) . Therefore um(x) can be easily gained, especially
by means of computational software such as MATLAB. Here, we rigorous def-
initions and then give some properties of the homotopy derivative. These
properties are useful to deduce the high-order deformation equations and pro-
vide us with a simple and convenient way to apply the HAM to nonlinear
problems. Let φ be a function of the homotopy-parameter q , then
Dm(φ) =
1
(m)!
∂m[φ(x; q)]
∂qm
| q = 0,
is called the mth-order homotopy-derivative of φ, where m ≥ 0 is an integer.
According to the Leibnitz’s rule for derivatives and using the induction, one
can show the following properties of the homotopy-derivative. For homotopy
series we have:
φ(x; q) =
∞∑
m=0
um(x)q
m,
where m is positive integers, then (1) For p ≥ 0, a positive integer, it holds
Dm(φ
p) =
m∑
r1=0
um−r1
r1∑
r2=0
ur1−r2
r2∑
r3=0
ur2−r3 . . .
rpij−3∑
rpij−2=0
urp−3−rp−2
rpij−2∑
rpij−1=0
urp−2−rp−1urp−1.
3 Computational procedure
One kind of the nonlinear integral equation is the nonlinear integral equation
in the urysohn form. This kind of integral equation is defined in the following
general form:
u(x) = f(x) +
∫ b
a
k(x, t, u(t))dt a ≤ x ≤ b. (12)
1380 A. Jafarian, Z. Esmailzadeh and L. Khoshbakhti
Where k(x, t, u(t)) is the kernel of the integral equation and u(x) is the un-
known function. All function in (12) are usually assumed to be continuous.
Also, k(x, t, u(t)) equal with k(x, t, Fu(t)) synonym. Where the unknown func-
tion u(x) occurs inside and outside the integral sign. For this type of equations
the kernel k(x, t) and function f(x) are given real-valued function, and F (u(x))
is a nonlinear function of u(x) such as u2(x), sin(u(x)). Whit hypothesis
F (u(t)) = [u(t)]p where, p is a positive integer. We have:
u(x) = f(x) +
∫ b
a
k(x, t)[u(t)]pdt.
In this section we will use the HAM approach to consider above equation.
N [u] = u(x) − f(x) −
∫ b
a
k(x, t)[u(t)]pdt.
The corresponding mth-order deformation equation (10) reads
L[um(x) − χmum−1(x)] = hH(x)Rm−1(−→u m−1(x)) , um(0) = 0 (13)
where
Rm−1(−→u m−1(x)) = um−1 − (1 − χm)f −
∫ b
a
k(x, t)Rm−1(φp)dt
and
Rm(φ
p) =
m∑
r1=0
um−r1
r1∑
r2=0
ur1−r2
r2∑
r3=0
ur2−r3 . . .
rp−3∑
rp−2=0
urp−3−rp−2
rp−2∑
rp−1=0
urp−2−rp−1urp−1.
To obtain a simple iteration formula for um(x), choose Lu = u as an auxiliary
linear operator, as a zero-order approximation to the desired function u(x),
the solution u0(x) = f(x), is taken, the nonzero auxiliary parameter h and
the auxiliary function H(x), can be taken as h = −1 and H(x) = 1. This is
substituted into (13) to obtain
u0(x) = f(x), um(x) =
∫ b
a
k(x, t)Rm−1(φp)dt, m = 1, 2, 3, . . .
The corresponding homotopy-series solution is given by
u(x) =
∞∑
m=0
um(x). (14)
Solving nonlinear integral equations 1381
4 Convergence Analysis
Theorem 1. The urysohn Integral Equation
u(x) = f(x) +
∫ b
a
k(x, t)F (u(t))dt. (15)
With the kernel k(x, t) satisfies |k(x, t)| < M for all (x, t)∈ [a, b] × [a, b], also
f(x) is a given continuous function defined on [a, b] and f(u) is Libschitz
continuous with |f(u) − f(w)| ≤ L|u − w|, has a unique solution whenever 0 <
α < 1, where, α = LM(b − a). As the function f(x) = xp is Lipschitz
continuous, the integral equation (12) has a unique solution. If the kernel is
separable , i.e. k(x, t) = g(x)h(t), then the following condition
|1 −
∫ b
a
K(t, t)dt| < 1,
must be justified for convergence.
Theorem 2. Let S(x) =
∑∞
n=0 un(x) then for k ≥ 2, where k is an integer,
∞∑
m=0
Rm(φ
k) = Sk(x).
See the proof Theorem 1 , 2 in [1].
Theorem 3. As long as the series (14) convergence, it must be the exact
solution of the integral equation (12).
Proof. If the series (14) convergence, we can write
S(x) =
∞∑
n=0
um(x),
and it hold that
lim
m→∞
um(x) = 0.
We can verify that
n∑
m=1
[um(x) − χmum−1(x)] = u1 + (u2 + u1) + . . . + (un + un−1) = un(x),
which gives us, according to (15),
∞∑
m=1
[um(x) − χmum−1(x)] = lim
n→∞
un(x) = 0. (16)
1382 A. Jafarian, Z. Esmailzadeh and L. Khoshbakhti
Furthermore, using (16) and the definition of the linear operator L, we have
∞∑
m=1
L[um(x) − χmum−1(x)] = L[
∞∑
m=1
L[um(x) − χmum−1(x)]] = 0,
In this line, we can obtain that
∞∑
m=1
L[um(x) − χmum−1(x)] = hH(x)
∞∑
m=1
Rm−1(−→u m−1(x)) = 0,
which gives, since h = 0and H = 0, that
∞∑
m=1
Rm−1(−→u m−1(x)) = 0. (17)
substituting Rm−1(−→u m−1(x)) into the above expression, recall Theorem 3, and
simplifying it, we have
∞∑
m=1
Rm−1(−→u m−1(x)) =
∞∑
m=1
[um−1 − (1 − χm)f −
∫ b
a
k(x, t)Rm−1(φp)dt]
=
∞∑
m=0
um(x) − f(x) − (1 − χm)f −
∫ b
a
k(x, t)
∞∑
m=1
Rm−1(φp)dt
=
∞∑
m=0
um(x) − f(x) − (1 − χm)f −
∫ b
a
k(x, t)[
∞∑
m=0
um(t)
p]dt
= S(x) − f(x) −
∫ b
a
k(x, t)S(t)pdt (18)
and so, S(x) must be the exact solution of Eq.(12).
5 Numerical Result and Discussion
The HAM provides an analytical solution terms of an infinite power series.
However, there is a particle need to evaluate this solution. The consequent
series truncation, and the particle procedure conducted to accomplish this
task, together transform the analytical results into an exact solution, which is
evaluated to a finite degree of accuracy. In order to investigated the accuracy
of the HAM solution with a finite number of terms, two examples were solved.
Solving nonlinear integral equations 1383
For efficiency of the present method for our problem in comparison with those
obtained by the Newton-Kantorovich-quadrature method.
Example 6.1. Consider the following nonlinear Fredholm integral equation
urysohn form:
u(x) = sin(πx) +
1
5
∫ 1
0
cos(πx)sin(πt)u3(t)dt, 0 ≤ x ≤ 1.
The exact solution to this integral equation is u(x) = sin(πx)+ 20−
√
391
3
cos(πx).
The formulas corresponding to this problem are
u0(x) = sin(πx),
um(x) =
1
5
∫ 1
0
[cos(πx)sin(πt)
m−1∑
i=0
um−i−1
i∑
j=0
ujui−j]dt m = 1, 2 . . .
Table 1. shows numerical results calculated according the presented method
and Newton-Kantorovich-quadrature.
xi uexact u
15
HAM u(Newoton−Kantorovich)
0 0.0754266889 0.0754266889 0.0255947861
0.1 0.3807520383 0.3807520383 0.3333590824
0.2 0.6488067254 0.6488067254 0.6084918692
0.3 0.8533516897 0.8533516897 0.8240612321
0.4 0.9743646449 0.9743646449 0.9589657401
0.5 1.0000000000 1.0000000000 1.0000000000
0.6 0.9277483875 0.9277483875 0.9431472924
0.7 0.7646822990 0.7646822990 0.7939727565
0.8 0.5267637791 0.5267637791 0.5670786353
0.9 0.2372819503 0.2372819503 0.2446749062
1 -0.0754266889 -0.0754266889 -0.025597861
Table 1 solution to the example6.1.
Example 6.2. The presented HAM iterative scheme is applied for solving
urysohn integral equation, that the exact solution is u(x) = ln(x + 1).
u(x) = ln(x + 1) + 2 ln 2(1 − x ln 2 + x) − 2x − 5
4
+
∫ 1
0
(x − t)u2(t)dt, 0 ≤ x ≤ 1.
We begin with
u0(x) = ln(x + 1) + 2 ln 2(1 − x ln 2 + x − 2x − 5
4
.
1384 A. Jafarian, Z. Esmailzadeh and L. Khoshbakhti
Its iteration formulation reads
um(x) =
∫ 1
0
[(x − t)
m−1∑
j=0
uj(t)um−j−1(t)]dt, m = 1, 2, . . .
Some numerical results of these solutions are presented and compare results
calculated by Newton-Kantorovich method in follow table.
xi uexact u
15
HAM u(Newoton−Kantorovich)
0 0 0.000000026768 0.000000026768
0.1 0.095310179804 0.095310206285 0.080610206788
0.2 0.182321556793 0.182321579887 0.195465677887
0.3 0.262364264467 0.262364280170 0.223454563210
0.4 0.336472236621 0.336472248418 0.315434533211
0.5 0.405465108108 0.405465120209 0.374545130278
0.6 0.470003629245 0.470003629794 0.421234567300
0.7 0.530628251062 0.530628259053 0.500678789650
0.8 0.587786664902 0.587786677053 0.564455564053
0.9 0.641853886172 0.641853890141 0.621756891141
1 0.693147180559 0.693147181293 0.682314518129
6 Conclusions
The proposed method is a powerful procedure for solving nonlinear urysohn
integral equations. The examples analyzed illustrate the ability and reliability
of the method presented in this paper and reveals that this one is very simple
and effective. The obtained solutions, are compare with Newton-Kantorovich
method. Results indicate that the convergence rate is very fast, and lower
approximations can achieve high accuracy.
Acknowledgment. This paper is based on the Masters project at the
Department of Mathematics, Urmia Branch Islamic Azad University of Iran,
under graduate research fellowship and funda-mental research grant scheme.
The respected anonymous referees have carefully reviewed this paper. As a
result of their careful analysis, our paper has been improved. The authors
would like to express their thankfulness to them for their helpful constructive
comments.
References
[1] F. Awawdeh, A. Adawi A numerical method for solving nonlinear integral
equations, International Mathematical Forum. 17 (2009), 805-817.
Solving nonlinear integral equations 1385
[2] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis
Method, Chapman Hall/CRC Press, Boca Raton, 2003.
[3] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl.
Math. Comput. 147 (2004) 499-513.
[4] S.J. Liao, Notes on the homotopy analysis method: some definitions and
theorems, Communications in Nonlinear Science and Numerical Simula-
tion (2008), doi: 10.1016/j.cnsns.2008.04.013.
[5] S.J. Liao and Y. Tan, A general approach to obtain series solutions of non-
linear differential equations. Studies in Applied Mathematics, 119 (2007)
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[6] X. Lan, Variational iteration method for solving integral equations, Com-
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[7] J. Saber-Nadjafi, M. Heidaryi, Solving nonlinera integral equations in the
Urysohn by Newoton-Kantorovich-quadrature method, journal of com-
puters and Mathematics with application. 60 (2010) 2058-2065.
Received: December, 2012


A numerical method for solving nonlinear integral equations

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A numerical method for solving nonlinear integral equations

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