Abstract and Applied Analysis
Volume 2015 (2015), Article ID 219616, 12 pages
http://dx.doi.org/10.1155/2015/219616

Research Article

On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation

Hameed Husam Hameed,1,2 Z. K. Eshkuvatov,3,4 Anvarjon Ahmedov,4,5 and N. M. A. Nik Long1,4

¹Department of Mathematics, Faculty of Science, Universiti Putra Malaysia (UPM), Selangor, Malaysia
²Technical Institute of Alsuwerah, The Middle Technical University, Baghdad, Iraq
³Faculty of Science and Technology, Universiti Sains Islam Malaysia (USIM), Negeri Sembilan, Malaysia
⁴Institute for Mathematical Research, Universiti Putra Malaysia (UPM), Selangor, Malaysia
⁵Department of Process and Food Engineering, Faculty of Engineering, Universiti Putra Malaysia (UPM), Selangor, Malaysia

Received 19 July 2014; Accepted 6 October 2014

Academic Editor: Gaohang Yu

Copyright © 2015 Hameed Husam Hameed 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.

We develop the Newton-Kantorovich method to solve the system of nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.

1. Introduction

Nonlinear phenomenon appears in many scientific areas such as physics, fluid mechanics, population models, chemical kinetics, economic systems, and medicine and can be modeled by system of nonlinear integral equations. The difficulty lies in finding the exact solution for such system. Alternatively, the approximate or numerical solutions can be sought. One of the well known approximate method is Newton-Kantorovich method which reduces the nonlinear into sequence of linear integral equations. The the approximate solution is then obtained by processing the convergent sequence. In 1939, Kantorovich [1] presented an iterative method for functional equation in Banach space and derived the convergence theorem for Newton method. In 1948, Kantorovich [2] proved a semilocal convergence theorem for Newton method in Banach space, later known as the Newton-Kantorovich method. Uko and Argyros [3] proved a weak Kantorovich-type theorem which gives the same conclusion under the weaker conditions. Shen and Li [4] have established the Kantorovich-type convergence criterion for inexact Newton methods, assuming that the first derivative of an operator satisfies the Lipschitz condition. Argyros [5] provided a sufficient condition for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear operator equation. Saberi-Nadjafi and Heidari [6] introduced a combination of the Newton-Kantorovich and quadrature methods to solve the nonlinear integral equation of Urysohn type in the systematic procedure. Ezquerro et al. [7] studied the nonlinear integral equation of mixed Hammerstein type using Newton-Kantorovich method with majorant principle. Ezquerro et al. [8] provided the semilocal convergence of Newton method in Banach space under a modification of the classic conditions of Kantorovich. There are many methods of solving the system of nonlinear integral equations, for example, product integration method [9], Adomian method [10], RBF network method [11], biorthogonal system method [12], Chebyshev wavelets method [13], analytical method [14], reproducing kernel method [15], step method [16], and single term Wlash series [17]. In 2003, Boikov and Tynda [18] implemented the Newton-Kantorovich method to the following system:

where , , and the functions , , , , and . In 2010, Eshkuvatov et al. [19] used the Newton-Kantorovich hypothesis to solve the system of nonlinear Volterra integral equation of the form

where and are unknown functions defined on , , and , . In 2010, Eshkuvatov et al. [20] developed the modified Newton-Kantorovich to obtain an approximate solution of system with the form

where , , and the functions , , and the unknown functions , , .

In this paper, we consider the systems of nonlinear integral equation of the form

where , , , and the unknown functions , to be determined, and .

The paper is organized as follows, in Section 2, Newton-Kantorovich method for the system of integral equations (4) is presented. Section 3 deals with mixed method followed by discretizations. In Section 4, the rate of convergence of the method is investigated. Lastly, Section 5 demonstrates the numerical example to verify the validity and accuracy of the proposed method, followed by the conclusion in Section 6.

2. Newton-Kantorovich Method for the System

Let us rewrite the system of nonlinear Volterra integral equation (4) in the operator form

where and

To solve (5) we use initial iteration of Newton-Kantorovich method which is of the form

where is the initial guess and and can be any continuous functions provided that and .

The Frechet derivative of at the point is defined as

Hence,

From (7) and (9) it follows that

where , , and is the initial given functions. To solve (10) with respect to and we need to compute all partial derivatives:

and in the same manner we obtain

So that from (10)–(12) it follows that

Equation (13) is a linear, and, by solving it for and , we obtain . By continuing this process, a sequence of approximate solution can be evaluated from

which is equivalent to the system

where and , .

Thus, one should solve a system of two linear Volterra integral equations to find each successive approximation. Let us eliminate from the system (13) by finding the expression of from the first equation of this system and substitute it in the second equation to yield

where and , and the second equation of (16) yields

where

In an analogous way, and can be written in the form

where

3. The Mixed Method (Simpson and Trapezoidal) for Approximate Solution

At each step of the iterative process we have to find the solution of (18) and (20) on the closed interval . To do this the grid () of points , , is introduced, and by the collocation method with mixed rule we require that the approximate solution satisfies (18) and (20). Hence

On the grid () we set , suct that

Consequently, the system (23) can be written in the form

By computing the integral in (26) using tapezoidal formula on the first integrals and Simpson formula on the second integral, we consider two cases.

Case 1. When , , then

where

Case 2. When , , then    

where

Also, to compute on the grid , (18) can be represented in the form

Let us set and and

Then (30) can be written as

and by applying mixed formula for (32) we obtain the following four cases.

Case 1. When and , we have

Case 2. If and , then

Case 3. When and , we get

Case 4. If and , then

Thus, (32) can be computed by one of (33)–(36) according to the cases.

4. The Convergence Analysis of the Method

On the basis of general theorems of Newton-Kantorovich method [21, Chapter XVIII] for the convergence, we state the following theorem regarding the successive approximations described by (18)–(20).

First, consider the following classes of functions:

(i) the set of all continuous functions defined on the interval ,(ii) the set of all continuous functions defined on the region ,(iii),(iv).

And define the following norms

Let

Let us consider real valued function

where and are nonnegative real coefficients.

Theorem 1. Assume that the operator in (5) is defined in and has continuous second derivative in closed ball where . Suppose the following conditions are satisfied:

(1),(2), when ,

where and as in (39). Then the function defined by (39) majorizes the operator .

Proof. Let us rewrite (5) and (39) in the form

where and .
Let us show that (40) and (41) satisfy the majorizing conditions [21, Theorem 1, page 525]. In fact

and for the with the Remark in [21, Remark 1, page 504] we have

Hence is a majorant function of .

Theorem 2. Let the functions , , , , and the kernels and ; then

(1)the system (7) has unique solution in the interval ; that is, there exists , and ,(2),(3),(4) and ,

where and as in (39). Then the system (4) has unique solution in the closed ball and the sequence , of successive approximations

where and , , and converge to the solution . The rate of convergence is given by

Proof. It is shown that (7) is reduced to (17). Since (17) is a linear Volterra integral equation of 2nd kind with respect to and since , which implies that the kernel defined by (18) is continues it follows that (17) has a unique solution which can be obtained by the method of successive approximations. Then the function is uniquely determined from (16). Hence the existence of is archived.
To verify that is bounded we need to establish the resolvent kernel of (17), so we assume the integral operator from is given by

where , and is defined in (18).
Due to (46), (17) can be written as

The solution of (47) is expressed in terms of by means of the formula

where is an integral operator and can be expanded as a series in powers of [21, Theorem 1, page 378]:

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

where is the iterated kernel.
Substituting (50) into (48) we obtain an expression for the solution of (47):

Next, we show that the series in (51) is convergent uniformly for all . Since

Let ; then by mathematical induction we get

then

Therefore the th root test of the sequence yields

Hence and a Volterra integral equations (17) has no characteristic values. Since the series in (51) converges uniformly (48) can be written in terms of resolvent kernel of (17):

where

Since the series in (57) is convergent we obtain

To establish the validity of second condition, let us represent operator equation

as in (41) and its the successive approximations is

For initial guess we have

From second condition of (Theorem 1) we have

In addition, we need to show that for all where is defined in (38). It is known that the second derivative of the nonlinear operator is described by 3-dimensional array , which is called bilinear operator; that is, where

where , so we have

where

Then the norms of every components of and have the estimate

Therefore, all the second derivatives exist and are bounded:

Since majorizes operator and utilizing the second condition of (Theorem 1) we get

Let us consider the discriminant of equation :

and the two roots of are and ; therefore, when implies

then under the assumption of the fourth condition, that is, is the unique solution of in and the condition in (70) [21, Theorem 4, page 530] implies that is the unique solution of operator equation (5) [21, Theorem 6, page 532] and

where is the unique solution of in .
To show the rate of convergence let us write the equation in a same form as in (40) then its successive approximation is

To estimate the difference between and successive approximation :

where and

therefore

then

consequently,

it implies

5. Numerical Example

Consider the system of nonlinear equation

The exact solution is

and the initial guesses are

Table 1 shows that coincides with the exact from the first iteration whereas only six iterations are needed for to be very close to . Notations used here are as follows: is the number of nodes, is the number of iterations, and and .

Table 1: Numerical results for (79).

1
0.00
0.0029

2
0.00

3
0.00

4
0.00

5
0.00

6
0.00

6. Conclusion

In this paper, the Newton-Kantorovich method is developed to solve the system of nonlinear Volterra integral equations which contains logarithmic function. We have introduced a new majorant function that leads to the increment of range of convergence of successive approximation process. A new theorem is stated based on the general theorems of Kantorovich. Numerical example is given to show the validation of the method. Table 1 shows that the proposed method is in good agreement with the theoretical findings.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by University Putra Malaysia under Fundamental Research Grant Scheme (FRGS). Project code is 01-12-10-989FR.

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