A New Approximation Method for the Systems of Nonlinear Fredholm Integral Equations
1Department of Mathematics, Faculty of Science and Arts, Celal Bayar University, Manisa, Turkey
In this paper, we present a new approximate method for solving systems of nonlinear Fredholm integral equation. This method is based on, first, differentiating both sides of integral equations n times and then substituting the Taylor series the unknown functions in the resulting equation and later, transforming to a matrix equation. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Numerical results and comparisons with the exact solution are included to demostrate the validity and applicability of the technique.
At a glance: Figures
Keywords: nonlinear Fredholm systems, Taylor polynomials and series
Applied Mathematics and Physics, 2014 2 (2),
Received February 17, 2014; Revised February 23, 2014; Accepted February 28, 2014Copyright: © 2014 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Yalçınbaş, Salih, and Kübra Erdem. "A New Approximation Method for the Systems of Nonlinear Fredholm Integral Equations." Applied Mathematics and Physics 2.2 (2014): 40-48.
- Yalçınbaş, S. , & Erdem, K. (2014). A New Approximation Method for the Systems of Nonlinear Fredholm Integral Equations. Applied Mathematics and Physics, 2(2), 40-48.
- Yalçınbaş, Salih, and Kübra Erdem. "A New Approximation Method for the Systems of Nonlinear Fredholm Integral Equations." Applied Mathematics and Physics 2, no. 2 (2014): 40-48.
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We are concerned with the systems of s nonlinear integral equations systems of Fredholm type in the form
where amj(x)(m,j=1,2,…,s),fm(x),Kmj(x,t) are functions having nth derivatives on an interval a≤x, c≤b; and the solutions is expresed in the form
which is a Taylor polynomial of degree N at x=c, where ym(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 . Conversely, the solution of integral equations system which occurs in physics , biology  and engineering [6, 7] is based on numerical integration methods such as Euler-Chebyshev  and Runge-Kutta  methods, and also in a recent research, the first-order linear Fredholm integral equations system is solved by using rationalized Haar functions method  and by Galerkin methods with hybrid functions .
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  and then it has been extended by Sezer to Volterra integral equations  and to differential equations . Similar approach has been used to solve linear Volterra-Fredholm integro-differential equations has been applied by Yalçınbaş and Sezer , nonlinear Volterra-Fredholm integral equations by Yalçınbaş , high-order linear differential equation system by [17, 18] and nonlinear systems of volterra integral equations by Yalçınbaş and Erdem . 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
Here the expression Em(x) and Im(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
and then analyse the expressions Em(x) and Im(x).2.1. Matrix Representation for the First Part
The expression Em(n)(x) can be more clearly written as
Using the Leibnitz’s rule (dealing with differentation of product of functions), simplifying x=c into the resulting relation, we have
Here the N+1 unknown coefficients yj(0)(c),yj(1)(c),…,yj(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
the elements of which are defined by
The values (Wnm)ij,(n,m=1,2,…1s;i,j=0,1,2,…,N) are defined by
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,
The expression can be more clearly written as
First, we put x=c in relation (10), thereby in expression (11), become
Thereby in expression (11) and then substitute the Taylor expansion of Y1(t),Y2(t),…,Ys(t) at t=c, i.e.
in the resulting relation. Thus, expression (10) become
The quantities [Yj(t)](k) (k=0,1,2,…) can be found from the relation
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),. . ., ys(0),ys(1),…,ys(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
where the matrices , , and are defined by
The matrices Tnk,(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
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
From this nonlinear system, the unknown Taylor coefficients are determined and substituted in (2); thus we get the Taylor series solution
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 .
If max(10-kri)=10-k (k is any positive integer) is prescribed, then the truncation limit N is increased until the difference at each of the points xr 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  with two unknown
and approximate the solution ym(x) by the Taylor polynomial
Then, we obtain the matrices , , and as
Using the matrices W, F, T1 and T2 we find coefficients ym(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:
and approximate the solution ym(x) by the Taylor polynomial
Then, we obtain the matrices W, F, T1 and T2 as
By substituting the obtained coefficients in (17), we find the unknown coefficients ym(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
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)=e2x,y2(x)=e-x of the problem is given below in Table 1, Table 2 and Figure 1, Figure 4.
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
Example 4. Our last example is the nonlinear Fredholm equation system
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)=ex2, y2(x)=e-x cos x of the problem is given in Table 3, Table 4 and Figure 3, Figure 4.
Table 4. Comparison of the error analysis of y1 and y2 which has been found for N=6,8,10 at Example 4
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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