One dimensional nonlinear integral operator with Newton-Kantorovich method

Z.K. Eshkuvatov; Hameed Husam Hameed; N.M.A. Nik Long

doi:10.1016/j.jksus.2015.10.004

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Original article

28 (

); 172-177

doi:

10.1016/j.jksus.2015.10.004

One dimensional nonlinear integral operator with Newton-Kantorovich method

Z.K. Eshkuvatov^{^a,^c,?}, Hameed Husam Hameed^{^b,^d}, N.M.A. Nik Long^{^b,^c}

a Faculty of Science and Technology, Universiti Saina Islam Malaysia (USIM), Malaysia

b Department of Mathematics, Faculty of Science, Universiti Putra Malaysia (UPM), Malaysia

c Institute for Mathematical Research, Universiti Putra Malaysia (UPM), Malaysia

d Technical Institute of Alsuwerah, The Middle Technical University, Iraq

?Corresponding author at: Institute for Mathematical Research, Universiti Putra Malaysia (UPM), Malaysia. zainidin@usim.edu.my (Z.K. Eshkuvatov)

Received: 2015-5-11, Accepted: 2015-10-19, Published: 2015-04-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

The Newton-Kantorovich method (NKM) is widely used to find approximate solutions for nonlinear problems that occur in many fields of applied mathematics. This method linearizes the problems and then attempts to solve the linear problems by generating a sequence of functions. In this study, we have applied NKM to Volterra-type nonlinear integral equations then the method of Nystrom type Gauss-Legendre quadrature formula (QF) was used to find the approximate solution of a linear Fredholm integral equation. New concept of determining the solution based on subcollocation points is proposed. The existence and uniqueness of the approximated method are proven. In addition, the convergence rate is established in Banach space. Finally illustrative examples are provided to validate the accuracy of the presented method.

Keywords

Newton-Kantorovich method

Nonlinear operator

Volterra integral equation

Gauss-Legendre quadrature formula

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1 Introduction

Nonlinear integral equations occur in many scientific fields, including fluid mechanics (Ladopoulos, 2003), physics (Agarwal and Khan, 2015), chemical kinetics (Tsokos and Padgett, 1974, pp.180), and economic systems (Boikov and Tynda, 2003). The difficulty lies in determining the exact solution for such equations. Therefore, an alternative option is to find an approximate solution to the problems. A well-known approximate method is the Newton-Kantorovich method (NKM), which reduces a nonlinear integral equation into a sequence of linear integral equations. The solution is then approximated by processing the convergent sequence.

Particularly, in Boikov and Tynda (2015), weakly singular Volterra integral equations of the different types are considered. The construction of accuracy-optimal numerical methods for one-dimensional and multidimensional equations is discussed. Since this question is closely related with the optimal approximation problem, the orders of the Babenko and Kolmogorov n-widths of compact sets from some classes of functions have been evaluated. Construction of complexity order optimal numerical methods for Volterra integral equations with different types of weakly singular kernels is shown in Tynda (2006) as well as it is shown that for Volterra equations (in contrast to Fredholm integral equations) using the "Block-by-Block" technique it is not necessary to employ the additional iterations to construct complexity optimal methods. The NKM is also used for nonlinear functional equations. For instance, the authors of Uko and Argyros (2008) proved a weak Kantorovich-type theorem that generates the same conclusions as obtained in Argyros (2004) by the combination of weak Lipschitz and center-Lipschitz conditions. A local convergence analysis is presented in Argyros and Hilout (2013) for a fast two-step Newton-like method to find the approximate solution of nonlinear equations in a Banach space. The authors in Argyros and Khattri (2015) developed sufficient convergence conditions of Newton's method based on the majorizing principle. The work (Argyros, 1998) presents results about polynomial equations as well as analyzes iterative methods for their numerical solution in various general space settings. A Kantorovich-type convergence criterion was established in Shena and Li (2009) for inexact Newton methods. This criterion assumed that the first derivative of an operator satisfies the Lipschitz condition. The inexact Newton method was proved in Ferreira and Svaiter (2012) given a fixed relative residual error tolerance that was Q-linearly convergent to a zero of the nonlinear operator. The authors in Saberi-Nadjafi and Heidari (2010) developed a new method that combines the NKM with quadrature methods to solve nonlinear integral equations in Urysohn form. Mixed Hammerstein-type nonlinear integral equations were also solved in Ezquerro et al. (2012) using the NKM based on the concept of sequence majorizing provided by Kantorovich. The authors of Ezquerro et al. (2013) studied the semilocal convergence of Newton's method in Banach spaces upon modifying the classic conditions of Kantorovich and applied to two Hammerstein integral equations of the second type. In Akyüz-Daşcioğlu and Yaslan (2006), Chebyshev collocation method has been presented to solve nonlinear integral equations. This method transforms the integral equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients. Finally, some examples are presented to illustrate the method and results discussed. The authors in Hameed et al. (2015) and Eshkuvatova et al. (2010) consider the system of nonlinear integral equations of different types and proved the existence and uniqueness of the solution together with the rate of convergence of the approximate solution as well as numerical examples provided to validate the proposed method.

In this note we consider the Volterra-type nonlinear integral equation of the form.

(1)

x (t) - ? ?_{y (t)}^{t} H (t, ?) F (x (?)) d ? = f (t),

where

0 < t_{0} ? t ? T, ?

is a real or complex number, the known function

y (t) ? C_{[t_{0}, T]}^{1}

provide that

y (t) < t

and

f (t) ? C_{[t_{0}, T]}

. The given kernel

H (t, ?) ? C_{[t_{0}, T] \times [t_{0}, T]}

F (?)

is a differentiable continuous function.

The current paper is structured as follows. In Section 2, we describe (NKM). In Section 3, we solve the system of algebraic linear Fredholm integral equation by Nystrom type Gauss-Legendre quadrature formula (QF). Section 4 discusses the convergence rate of the approximate method and the error estimation. In Section 5, we apply the proposed method to three examples to demonstrate the accuracy and efficiency of the method. Finally, Section 6 summarizes the main concepts of the approximation method.

2 Newton-Kantorovich approach for nonlinear integral equation

Rewrite Eq. (1) in the operator equation

(2)

P (x) = x (t) - f (t) - ? ?_{y (t)}^{t} H (t, ?) F (x (?)) d ? = 0 .

Consider the initial iteration of NKM which is of the form

(3)

P^{'} (x_{0} (t)) (x (t) - x_{0} (t)) + P (x_{0} (t)) = 0,

where

x_{0} (t)

is the initial guess that might be any continuous function. The Frechet derivative of

P (x (t))

at the initial condition

x_{0} (t)

is defined as

(4)

\begin{matrix} P^{'} (x_{0}) x = & \lim_{s ? 0} \frac{1}{s} [P (x_{0} + sx) - P (x_{0})] \\ = & \lim_{s ? 0} \frac{1}{s} [\frac{dP (x_{0})}{dx} sx + \frac{1}{2} \frac{d^{2} P}{{dx}^{2}} (x_{0} + ? sx) s^{2} x^{2}], ? ? (0, 1) \\ = & \frac{dP (x_{0})}{dx} x . \end{matrix}

From Eqs. (3) and (4) we obtain

(5)

{\frac{dP}{dx}|}_{x_{0}} (? x (t)) = - P (x_{0} (t)),

where

? x (t) = x_{1} (t) - x_{0} (t)

, and

x_{0} (t)

is the initial guess. To solve Eq. (5) for

? x (t)

we need to compute the derivative

(6)

\begin{matrix} {\frac{dP}{dx}|}_{x_{0}} = & \lim_{s ? 0} \frac{1}{s} [P (x_{0} + sx) - P (x_{0})] \\ = & \lim_{s ? 0} \frac{1}{s} [sx - ? ?_{y (t)}^{t} H (t, ?) [F (x_{0} (?) + sx (?)) - F (x_{0} (?))] d ?], \end{matrix}

= x (t) - ? ?_{y (t)}^{t} H (t, ?) F^{'} (x_{0} (t)) x (?) d ?,

From Eqs. (5) and (6) we obtain

(7)

? x (t) - ? ?_{y (t)}^{t} H_{0} (t, ?) ? x (?) d ? = G_{0} (t),

where

(8)

H_{0} (t, ?) = H (t, ?) F^{'} (x_{0} (?))

(9)

G_{0} (t) = f (t) + ? ?_{y (t)}^{t} H (t, ?) F (x_{0} (?)) d ? - x_{0} (t) .

Eq. (7) is linear with respect to

? x (t)

, and it is easy to find

x_{1} (t) = x_{0} (t) + ? x (t)

By continuing this process, a sequence of approximate solution $x_{m} (t), (m = 2, 3, .)$ can be evaluated from the equation

(10)

P^{'} (x_{0}) ? x_{m} + P (x_{m}) = 0

which is equivalent to the equation

(11)

? x_{m} (t) - ? ?_{y (t)}^{t} H_{0} (t, ?) ? x_{m} (?) d ? = G_{m - 1} (t),

where

(12)

? x_{m} (t) = x_{m} (t) - x_{m - 1} (t), m = 1, 2, .,

and

G_{m - 1} (t) = f (t) + ? ?_{y (t)}^{t} H (t, ?) F (x_{m - 1} (?)) d ? - x_{m - 1} (t) .

Solving Eq. (11) for

? x_{m} (t)

gives a sequence of approximate solution

x_{m} (t)

3 Gauss-Legendre quadrature formula for a numerical solution

Introducing a uniform grid $?_{1} = \{t_{i} : t_{i} = t_{0} + ih, h = \frac{T - t_{0}}{n}, i = 1, 2, ., n\}$ , where n refers to the number of partitions in $[t_{0}, T]$ , Eq. (11) becomes

(13)

? x_{m} (t_{i}) - ? ?_{y (t_{i})}^{t_{i}} H_{0} (t_{i}, ?) ? x_{m} (?) d ? = G_{m - 1} (t_{i}),

where

G_{m - 1} (t_{i}) = f (t_{i}) + ? ?_{y (t_{i})}^{t_{i}} H (t_{i}, ?) F (x_{m - 1} (?)) d ? - x_{m - 1} (t_{i}) .

The robust way to approximate the integration in the system (13) is Gauss-Legendre QF. It is known that the Legendre polynomials

P_{n} (t)

are orthogonal on

[- 1, 1]

with weight

w = 1

. Consider the Gauss-Legendre QF (Jeffrey, 2000, pp. 318)

(14)

?_{- 1}^{1} f (x) dx = ?_{i = 1}^{n} ?_{i} f (s_{i}) + R_{i} (f),

where

?_{i} = \frac{2}{(1 - s_{i}^{2}) {[P_{n}^{'} (s_{i})]}^{2}}, ?_{i = 1}^{n} ?_{i} = 2, P_{n} (s_{i}) ? 0, i = 1, 2, ., n,

s_{i}, i = 1, 2, ., n

are roots of the Legendre polynomial

P_{n} (t)

on the interval

[- 1, 1]

. The error term of Gauss-Legendre QF is

R_{n} (f) = \frac{2^{2 n + 1} {(n!)}^{4}}{(2 n + 1) {[(2 n)!]}^{3}} f^{2 n} (?), - 1 < ? < 1 .

The Gauss-Legendre (QF) formula for arbitrary interval

[a, b]

has the form

(15)

?_{a}^{b} f (x) dx = \frac{b - a}{2} ?_{i = 1}^{n} ?_{i} f (t_{i}) + R_{i} (f),

where the nodes

t_{i} = (\frac{b - a}{2}) s_{i} + (\frac{b + a}{2})

Let us describe the new idea to solve the Eq. (13). Firstly, we introduce a subgrid $?_{2} = \{?_{i}^{j}\}$ of $?_{1}$ at each subinterval $[y (t_{i}), t_{i}] ? [t_{0}, T]$ , where

(16)

?_{i}^{j} = \frac{t_{i} - y (t_{i})}{2} s_{j} + \frac{t_{i} + y (t_{i})}{2}, i = 1, 2, ., n, j = 1, 2, ., ?

where

?_{i}^{j} ? t_{i}

and

?

refers to the number of sub partitions of

[y (t_{i}), t_{i}]

. Secondly, we apply Gauss-Legendre quadrature formula to the kernel integral of (13) at each subinterval

[y (t_{i}), t_{i}]

which yields

(17)

?_{y (t_{i})}^{t_{i}} H (t_{i}, ?) F (x_{m - 1} (?)) d ? ? \frac{t_{i} - y (t_{i})}{2} ?_{j = 1}^{?} H_{0} (?_{i}^{k}, ?_{i}^{j}) ? x_{m} (?_{i}^{j}) w_{j},

where

H_{0}

is defined by (8).

Thirdly, from (13) and (17) it follows that

(18)

? x_{m} (?_{i}^{k}) - ? \frac{t_{i} - y (t_{i})}{2} ?_{j = 1}^{?} H_{0} (?_{i}^{k}, ?_{i}^{j}) ? x_{m} (?_{i}^{j}) w_{j} = G_{m - 1} (?_{i}^{k}),

i = 1, 2, ., n; k = 1, 2, ., ?,

where

G_{m - 1} (?_{i}^{k}) = f (?_{i}^{j}) + ? \frac{t_{i} - y (t_{i})}{2} ?_{j = 1}^{?} H (?_{i}^{k}, ?_{i}^{j}) F (x_{m - 1} (?_{i (m - 1)}^{j})) w_{j} - x_{m - 1} (?_{i}^{k}) .

Eq. (18) is a linear algebraic system of

n \times ?

equations and

n \times ?

unknowns. If its matrix is non singular then it has a unique solution in terms of

? x_{m} (?_{i}^{k}), i = 1, 2, ., n, k = 1, 2, ., ?

, then

x_{m} (?_{i}^{k})

can be evaluated as

(19)

x_{m} (?_{i}^{k}) = ? x_{m} (?_{i}^{k}) + x_{m - 1} (?_{i}^{k}), m = 2, 3, . .

Since the values of the functions

x_{m} (?_{i}^{k})

is known at

?

Legendre grid points in each subinterval

(y (t_{i}), t_{i})

for each m iteration, the values of unknown function

x_{m} (t_{i})

can be found by using Newton forward interpolation formula given below

(20)

x_{m} (t) ? P_{?} (t) = x_{m} (?_{i}^{?}) + x_{m} (?_{i}^{?}, ?_{i}^{? - 1}) (t - ?_{i}^{?}) + x_{m} (?_{i}^{?}, ?_{i}^{? - 1}, ?_{i}^{? - 2}) (t - ?_{i}^{?}) (t - ?_{i}^{? - 1}) + x_{m} (?_{i}^{?}, ?_{i}^{? - 1}, ?_{i}^{? - 2}, ., ?_{i}^{1}) (t - ?_{i}^{?}) (t - ?_{i}^{? - 1}) ? (t - ?_{i}^{1}),

with the error (Atkinson, 1997, pp.110)

? x_{m} (t) - P_{?} (t) ? ? \frac{M}{(? + 1)!},

where

M = \max \{| x_{m}^{? + 1} (?) | | (t - ?_{i}^{?}), ., (t - ?_{i}^{1}) |\} .

4 Convergence analysis

Based on the general theorems of (NKM) and their applications to functional equations, we state the following theorem with respect to the successive approximations which are characterized by Eq. (11).

First, since $f (t), x_{0} (t), H (t, ?), F (?), F^{'} (?)$ and $F^{?} (?)$ are continuous functions in their domains of definitions, then they are bounded (Zeidler, 1995, pp.33), i.e. $\begin{matrix} | f (t) | ? M_{1}, | x_{0} (t) | ? M_{2}, | H (t, ?) | ? M_{3}, | F (x_{0} (t)) | ? M_{4}, \\ | F^{'} (x_{0} (t)) | ? M_{5}, | F^{?} (x_{0} (t)) | ? M_{6}, M_{7} = \min_{t ? [t - 0, T]} | y (t) | . \end{matrix}$ Then, we use the majorant function (Kantorovich and Akilov, 1982, pp.533)

(21)

? (t) = {Kt}^{2} - 2 t + 2 ?,

where

K = M_{3} M_{6} (T - M_{7})

and

?

to be nonnegative real number.

Theorem 1

Let the operator $P (x) = 0$ in (2) is defined in $? = \{x ? C_{[t_{0}, T]} : | x - x_{0} | ? R\}$ and has a continuous second derivative in $?_{0} = \{x ? C_{[t_{0}, T]} : | x - x_{0} | ? r\}$ , where $T = t_{0} + r ? t_{0} + R$ . Moreover, let the functions $f (t) ? C_{[t_{0}, T]}, x_{0} (t) ? C_{[t_{0}, T]}^{1}, F (?) ? C_{(- ?, ?)}$ , $F^{'} (?) ? C_{(- ?, ?)}$ and the kernel $H (t, ?) ? C_{[t_{0}, T] \times [t_{0}, T]}$ , then if

The linear Volterra integral equation in Eq. (10) has a resolvent kernel $? (t, ?; ?)$ where $? ? ? ? M_{3} M_{5} e^{? M_{3} M_{5} (T - M_{6})}$ ,
$| ? x | ? ?$ ,
$| P^{?} (x) | ? K$ .

Then Eq. (1) has a unique solution

x^{?}

in the closed ball

?_{0}

and the sequence

x_{m} (t), m ? 0

of successive approximations

(22)

? x_{m} (t) - ? ?_{y (t)}^{t} H_{0} (t, ?) ? x_{m} (?) d ? = G_{m - 1} (t),

where

? x_{m} (t) = x_{m} (t) - x_{m} (t)

converges to the solution

x^{?} (t)

. The rate of convergence is

(23)

? x^{?} - x_{m} ? ? \frac{1}{K} {(1 - \sqrt{1 - 2 K ?})}^{m + 1}, m = 1, 2, .

Proof

Since Eq. (7) is a linear integral equation of the second kind, it has a unique solution in term of $? x (t)$ provided that its kernel $H_{0} (t, ?)$ is a continuous function. Hence the existence of $?_{0}$ is accomplished.

To prove $?_{0}$ is bounded we need to establish the resolvent kernel $?_{0} (t, ?; ?)$ of Eq. (7). Assume the integral operator U from $C [t_{0}, T] ? [t_{0}, T]$ is given by

(24)

Z = U (? x), Z (t) = ?_{y (t)}^{t} H_{0} (t, ?) ? x (?) d ?,

where

H_{0} (t, ?)

is defined in Eq. (8). Due to Eq. (24), Eq. (7) can be written as

(25)

? x - ? U (? x) = G_{0} (t) .

The solution

? x^{?}

of Eq. (25) is written in terms of

G_{0}

(26)

? x^{?} = G_{0} + B (G_{0}),

where B is an integral operator and can be expressed as a power series of U (Atkinson, 1997, Theorem 1, pp.378)

(27)

B (G_{0}) = I + ? U (G_{0}) + ?^{2} U^{2} (G_{0}) + ? + ?^{n} U^{n} (G_{0}) + ?,

and it is well known that the powers of U are also integral operators. In fact

(28)

Z_{n} = U^{n}, Z_{n} (t) = ?_{y (t)}^{t} H_{0}^{(n)} (t, ?) ? x (?) d ?, (n = 1, 2, .),

where

H_{0}^{(n)}

is the iterated kernel. Substituting (28) into (26) we obtain the solution of Eq. (25) which is of the form

(29)

? x^{?} (t) = G_{0} (t) + ? ?_{y (t)}^{t} ?_{0} (t, ?; ?) G_{0} (?) d ?,

where

(30)

?_{0} (t, ?; ?) = ?_{j = 0}^{?} ?^{j} H_{0}^{j + 1} (t, ?),

and

?_{0} (t, ?, ?)

is the resolvent kernel. Next, we elucidate that the series in Eq. (29) is convergent uniformly for all

t ? [t_{0}, T]

. Since

(31)

| H_{0} (t, ?) | = | H (t, ?) F (x_{0} (?)) | ? | H (t, ?) | | F^{'} (x_{0} (?)) | ? M_{3} M_{5} .

Let

M = M_{3} M_{5}

, then by mathematical induction we obtain

\begin{matrix} |H_{0}^{(2)} (t, ?)| ? ?_{y_{0} (t)}^{t} |H_{0} (t, u) H_{0} (u, ?)| du ? \frac{M^{2} (t - M_{7})}{(1)!}, \\ |H_{0}^{(3)} (t, ?)| ? ?_{y_{0} (t)}^{t} |H_{0} (t, u) H_{0}^{(2)} (u, ?)| du ? \frac{M^{3} {(t - M_{7})}^{2}}{(2)!}, \\ ? \\ |H_{0}^{(n)} (t, ?)| ? ?_{y_{0} (t)}^{t} |H_{0} (t, u) H_{0}^{(n - 1)} (u, ?)| du ? \frac{M^{n} {(t - M_{7})}^{n - 1}}{(n - 1)!}, (n = 1, 2 .), \end{matrix}

then

(32)

\begin{matrix} ? ?_{0} ? = ? B (G_{0}) ? ? & ?_{j = 0}^{N} | ? |^{j} | H_{0}^{j + 1} (t, ?), | \\ ? & ?_{j = 0}^{?} | ? |^{j} M^{j + 1} \frac{{(T - M_{7})}^{j}}{j!}, \\ = & M ?_{j = 0}^{?} M^{j} \frac{{(T - M_{7})}^{j}}{j!}, \\ = & {Me}^{| ? | M (T - M_{7})} . \end{matrix}

Therefore, the infinite series in Eq. (30) for

?_{0} (t, ?; ?)

is absolutely and uniformly convergent for all values of

?

in the case of continuous Volterra kernel. Furthermore, we state that

? P^{?} (x) ? ? K

for all

x ? ?_{0}

. The second derivative

P^{?} (x_{0}) (x)

of nonlinear operator

P (x)

is represented as

\begin{matrix} P^{?} (x_{0}) x = & \lim_{s ? 0} \frac{1}{s} [P^{'} (x_{0} + sx) - P^{'} (x_{0})], \\ = & \lim_{s ? 0} \frac{1}{s} (\frac{d^{2} P}{{dx}^{2}} (x_{0}) s \overset{?}{x} + \frac{1}{2} \frac{d^{3} P}{{dx}^{3}} (x_{0} + ? s \overset{?}{x}) s^{2} {\overset{?}{x}}^{2}), \\ = & {\frac{d^{2} P}{{dx}^{2}}|}_{x_{0}} \overset{?}{x}, \end{matrix}

then the norm of

?\frac{{dp}^{2}}{{dx}^{2}}?

has the estimate

?\frac{{dp}^{2}}{{dx}^{2}}? = \max_{| x | ? 1, | \overset{?}{x} | ? 1} |?_{y (t)}^{t} H (t, ?) F^{?} (x_{0} (?)) x (?) \overset{?}{x} (?) d ?| ? M_{3} M_{6} (T - M_{7}) .

Therefore, the second derivative is bounded and by using (Kantorovich and Akilov, 1982, Theorem 6, pp.532) implies that $x^{?} (t)$ is the unique solution of operator Eq. (2) and

(33)

? x^{?} - x_{m} ? ? \frac{1}{K} {(1 - \sqrt{1 - 2 K ?})}^{m + 1}, m = 1, 2, .

5 Numerical result

Example 1

Consider the following integral equation

(34)

x (t) - ?_{y (t)}^{t} t ? x^{3} (?) d ? = t - \frac{13446}{1000000} t^{6},

where

t ? [0, 1]

and

y (t) = \frac{4}{5} t

The exact solution is $x^{?} (t) = t .$ Consider the initial guess as $x_{0} (t) = \frac{t}{2},$

Example 2

Consider the following integral equation

(35)

x (t) - ?_{y (t)}^{t} t ? x^{2} (?) d ? = \frac{t^{2}}{2} + \frac{t^{3}}{3} - \frac{229 t^{6}}{2560} - \frac{137 t^{7}}{1344} - \frac{3817 t^{8}}{129024},

where

t ? [0, 1]

, and

y (t) = \frac{t}{2}

The exact solution and initial guess are $x^{?} (t) = \frac{t^{2}}{2} + \frac{t^{3}}{3},$ $x_{0} (t) = \frac{t^{2}}{4} + \frac{t^{3}}{6} .$

Example 3

Consider the following integral equation

(36)

y (t) = e^{t} - 0.5 (e^{t} - 1) + ?_{0}^{t} y^{2} (?) d ?,

where

t ? [0, 1]

, and

y (t) = 0

The exact solution is $x^{?} (t) = e^{t} .$ Consider the initial guess as $x_{0} (t) = 0.5 + 2 t,$ Taken $h = 0.1, ? = 4, n = 10$ , and $m = 20$ . In Table 3, the absolute errors of NKM are compared with the errors given by Korobov's polynomial transformation, Sidi's trigonometric transformation, and Laurie's special polynomial type transformation (Galperin et al., 2000) and Chebyshev collocation method (Akyüz-Daşcioğlu and Yaslan, 2006).

It is noted from the Table 1 and Table 2 that only a few iterations are needed for $x_{m} (t)$ to be very close to the exact solution $x^{?} (t)$ . Furthermore, Table 3 shows that the results obtained by NKM are more accurate than other methods for different nodes $t_{i} ? [0, 1]$ , except the last point $t = 1$ . For this point the result of Chebyshev's is better than the NKM result.

Notations used here are: n is the number of partitions on $[t_{0}, T], ?$ is the number of subpartitions on $(y (t_{i}), t_{i})$ , $i = 1, 2, ., n$ , where m is the number of iterations, and $?_{x} = \max_{t ? (0, 1]} |x_{m} (t) - x^{?} (t)|,$

Table 1 Numerical results for Eq. (34).

$n = 2$ , $? = 4, h = 0.5$
m	$?_{x}$
1	0.07407
2	0.02250
3	0.00734
4	0.00245
5	8.20482E?004
10	3.53286E?006
20	6.57169E?011

Table 2 Numerical results for Eq. (35).

$n = 2$ , $? = 4, h = 0.5$
m	$?_{x}$
1	0.08227
2	0.04207
3	0.00861
4	0.00355
5	10.27756?004
10	5.22045E?005
20	1.25811E?010

Table 3 Error analysis of Example 3.

t	Korobov's	Sidi's	Laurie's	Chebyshev's	NKM
0.1	0.12E?004	0.66E?008	0.12E?006	0.36E?007	0.28E?016
0.2	0.31E?004	0.34E?007	0.27E?006	0.63E?007	0.56E?016
0.3	0.60E?004	0.11E?006	0.47E?006	0.35E?007	0.11E ?015
0.4	0.11E?003	0.30E?006	0.71E?006	0.88E?007	0.00
0.5	0.18E?003	0.71E?006	0.95E?006	0.23E?007	0.00
0.6	0.29E?003	0.15E?005	0.12E?005	0.70E?007	0.11E?015
0.7	0.49E?003	0.31E?005	0.13E?005	0.69E?007	0.44E?015
0.8	0.82E?003	0.62E?005	0.11E?005	0.14E?007	0.33E?015
0.9	0.14E?002	0.12E?004	0.27E?006	0.12E?007	0.11E?010
1	0.25E?002	0.23E?004	0.20E?005	0.86E?007	0.92E?006

6 Conclusion

In this note, the NKM is presented to solve the nonlinear integral equations of Volterra type. We have proposed a new idea by introducing subgrid collocation points $?_{i}^{k}, i = 1, 2, ., n, k = 1, 2, ., ?$ which lie in the intervals $(y_{0} (t_{i}), t_{i})$ and $(y_{m - 1} (t_{i}), t_{i})$ . Gauss-Legendre QF is used for each subgrid interval. The theorem of existence and uniqueness of the approximate solution are established based on the general theorems of Kantorovich. Numerical examples revealed that the accuracy of the NKM can be achieved by a few number of iterations.

Acknowledgment

This work was supported by University Putra Malaysia under Research Grand Universiti Putra Malaysia (Geran Putra, 2014, Project code is GP-i/2014/9442300) and Universiti Sains Islam Malaysia (USIM) under Research Grand (Project code is PPP/GP/FST/30/14915). Authors are grateful for sponsor and financial support of the Research Management Center, Universiti Putra Malaysia (UPM) and Universiti Sains Islam Malaysia (USIM).

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