1. Introduction

We are concerned with the systems of s nonlinear integral equations systems of Fredholm type in the form

(1a)

where a_mj(x)(m,j=1,2,…,s),f_m(x),K_mj(x,t) are functions having nth derivatives on an interval a≤x, c≤b; and the solutions is expresed in the form

(2)

which is a Taylor polynomial of degree N at x=c, where y_m⁽ⁿ⁾(c),n=0,1,…,N are the coefficients to be determined.

The literatures of systems of nonlinear integral equation contain few numerical method. So, many different methods have been used to approximate the solution of the integral equation system. On the grounds that few of these equations can be solved explicitly, it is often necessary to resort to numerical techniques which are appropriate combinations of numerical integration and interpolation ^{[1, 2]}. Furthermore, there are also expansion methods for integral equations such as El-gendi’s, Wolfe’s and Galerkin methods ^[3]. Conversely, the solution of integral equations system which occurs in physics ^[4], biology ^[5] and engineering ^{[6, 7]} is based on numerical integration methods such as Euler-Chebyshev ^[8] and Runge-Kutta ^[9] methods, and also in a recent research, the first-order linear Fredholm integral equations system is solved by using rationalized Haar functions method ^[10] and by Galerkin methods with hybrid functions ^[11].

Additionally, application of HPM (Homotopy Perturbation Method) and ADM (Adomian Decompositon Method) in nonlinear problems has been undertaken by scientists and engineers. In this paper, we consider Taylor method for solving nonlinear systems of Fredholm equations. This method has been presented by Kanwal and Liu ^[12] and then it has been extended by Sezer to Volterra integral equations ^[13] and to differential equations ^[14]. Similar approach has been used to solve linear Volterra-Fredholm integro-differential equations has been applied by Yalçınbaş and Sezer ^[15], nonlinear Volterra-Fredholm integral equations by Yalçınbaş ^[16], high-order linear differential equation system by ^{[17, 18]} and nonlinear systems of volterra integral equations by Yalçınbaş and Erdem ^[21]. Thus, the presented method which is an expansion method has been proposed to obtain approximate solution and also analytical solution of systems of higher-order nonlinear integral equations.

2. Fundamental Relations and Solution Method

Let us first write the Eq. (1a) in the form

(1b)

so that

Here the expression E_m(x) and I_m(x), respectively, are called as the first part and second part (or integral part) of equation (1b). To obtain the solution of the given problem in the form of expression (2) we first differentiate equation (1a) equations n times with respect to x to obtain

(3)

and then analyse the expressions E_m(x) and I_m(x).

The expression E_m⁽ⁿ⁾(x) can be more clearly written as

(4)

Using the Leibnitz’s rule (dealing with differentation of product of functions), simplifying x=c into the resulting relation, we have

(5)

Here the N+1 unknown coefficients y_j⁽⁰⁾(c),y_j⁽¹⁾(c),…,y_j^(N)(c);(j=1,2,…,s) are Taylor coefficients to be determined and ; , respectively, denote the values of the ith derivatives of the functions at .

We now write the matrix form of expression (5) as

(6)

where

and

the elements of which are defined by

(7)

The values (W_nm)_ij,(n,m=1,2,…1s;i,j=0,1,2,…,N) are defined by

(8)

Note in Eq. (8) that for

and for and , , where and are integers. In this case, in Eq. (8), for

Hence, the matrix W becomes, clearly,

(9)

The expression can be more clearly written as

(10)

or

First, we put x=c in relation (10), thereby in expression (11), become

(11)

Thereby in expression (11) and then substitute the Taylor expansion of Y₁(t),Y₂(t),…,Y_s(t) at t=c, i.e.

(12)

in the resulting relation. Thus, expression (10) become

(13)

where

The quantities [Y_j(t)]^(k) (k=0,1,2,…) can be found from the relation

(14)

where .

Note that the relation (14) can be obtained from the generalised Leibnitz’s rule (dealing with differentiation of products of p-functions)

If we take n,k=0,1,2,…,N, then equation (13) becomes a system of equation which is a system of s(N+1) nonlinear equations for the s(N+1) unknowns y₁⁽⁰⁾,y₁⁽¹⁾,…,y₁^(N), y₂⁽⁰⁾,y₂⁽¹⁾,…,y₂^(N),. . ., y_s⁽⁰⁾,y_s⁽¹⁾,…,y_s^(N), where j=0,1,…,s. These can be solved numerically by standard methods.

In this situation matrix representation for the integral part can be put in a matrix form as

(15)

where the matrices , , and are defined by

The matrices T_nk,(n,k=1,2,…,s) are defined by

Substituting the matrix forms (6) and (15) in expression of Eq. (3) at the point x=c we get the matrix form of Eq. (3) as

(16)

which is a fundemental equation for the ingeral equations system (1a).

Also, if some equations are linear in the system of (1a), the T matrix transforms matrix and matrix for respectively linear part of the system and the nonlinear part of the system.

Similarly, the W matrix transforms matrix and matrix for respectively linear part of the system and the nonlinear part of the system.

So (16) becomes

(17)

From this nonlinear system, the unknown Taylor coefficients are determined and substituted in (2); thus we get the Taylor series solution

(18)

3. Accuracy of Solution

We can easily check the accuracy of the solution obtained in the fom (18) as follows. Since the truncated Taylor series (18) or the corresponding polynomial expansion is an approximate solution of Eqs. (1a) and (1b), when the solution are substituted Eqs. (1a) and (1b), resulting equation must be satisfied approximately; that is, for .

or

If max(10^-k_ri)=10^-k (k is any positive integer) is prescribed, then the truncation limit N is increased until the difference at each of the points x_r becomes smaller than the prescribed10^-k.

On the other hand, the error function can be estimated by

4. Numerical Illustrations

In this section we consider four examples of systems of Fredholm type to illustrate the use of presented method.

Example 1. Let us first consider the nonlinear Fredholm equation system ^[20] with two unknown

(19)

and approximate the solution y_m(x) by the Taylor polynomial

where a=0,b=1,c=0,N=3.

Then, we obtain the matrices , , and as

Using the matrices W, F, T₁ and T₂ we find coefficients y_m⁽ⁿ⁾(0)are uniquely determined as

By substituting the obtained coefficients in (18) the solution of (19) becomes

which are the exact solutions.

Example 2. Consider the nonlinear Fredholm equation system:

(20)

where

and approximate the solution y_m(x) by the Taylor polynomial

where a=0,b=1,c=0,N=6.

Then, we obtain the matrices W, F, T₁ and T₂ as

By substituting the obtained coefficients in (17), we find the unknown coefficients y_m⁽ⁿ⁾(0) as

We get the aproximate solution of problem (20) for c=0,N=6 as

Example 3. Consider the system of linear Fredholm integral equations

(21)

Following the previous procedures, we find the unknown coefficients as

We get the aproximate solution of problem (21) for c=0, N=6 as

Now let us find the solution of problem (21) taking c=0; N=6,8,10,12,14. The comparison of the solutions given above with exact solutions y₁(x)=e^2x,y₂(x)=e^-x of the problem is given below in Table 1, Table 2 and Figure 1, Figure 4.

Table 1. Comparing the solutions of y₁ and y₂ which has been found for N=6,8,10,12,14 at Example 3

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Table 2. Comparison of the error analysis of y₁ and y₂ which has been found for N=6,8,10,12,14 at Example 3

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Figure 1. Comparison of the error analysis of y₁ for N=6,8,10,12,14

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Figure 2. Comparison of the error analysis of y₂ for 6,8,10,12,14

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Example 4. Our last example is the nonlinear Fredholm equation system

(22)

where

Applying the previous procedures, we get the aproximate solution of problem (22) for c=0,N=6,8,10. The comparison of the solutions (for c=0,N=6,8,10) with exact solutions y₁(x)=e^x2, y₂(x)=e^-x cos x of the problem is given in Table 3, Table 4 and Figure 3, Figure 4.

Table 3. Comparing the solutions of y₁ and y₂ which has been found for N=6,8,10 at Example 4

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Table 4. Comparison of the error analysis of y₁ and y₂ which has been found for N=6,8,10 at Example 4

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Figure 3. Comparison of the error analysis of y₁ for N=6,8,10

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Figure 4. Comparison of the error analysis of y₂ for N=6,8,10.

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5. Conclusions

In this paper a new approximate method is used to solve nonlinear systems of Fredholm integral equations. We described the method, used in four test problems, and compared the results with their exact solutions in order to demonstrate the validity and applicability of the method. The method can be developed and applied to another high-order linear and nonlinear integro- differential equation systems with variable coefficients.

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