1. Introduction

The subject of integral equations is one of the most important mathematical tools in both pure and applied mathematics. Integral equations play a very important role in modern science such as numerous problems in engineering and mechanics, for more details see ^{[1, 2, 3]}. For such equations as well as a system of such equations, various techniques such as iterative, extrapolation, Galerkin, collocation, quadrature, projection, spline, orthogonal polynomial, and multiple grid methods have been presented to determine desired solutions (see e.g. ^{[1, 2, 3]} and the references quoted there).

Let us consider the system of non linear Fredholm integral equations of the form:

(1)

where,

In system (1) the known kernel K (x, t) is continuous, the function G (x) is given, and F (x) is the solution to be determined ^[4].

There have been considerable interests in solving integral equation (1). In addition to the well-known techniques, there are several new techniques for solving integral equation systems [5-17]^[5]. As we know, it is important to select a suitable basis function in numerical methods for system of integral equations. One of the most attractive proposals made in the recent years was an idea connected to the application of wavelets as basis functions in the method of moments ^[18]. The wavelet technique allows the creation of very fast algorithms when compared to the algorithms ordinarily used and the main advantage of the wavelet technique is its ability to transform complex problems into a system of algebraic equations. Various wavelet basis are applied. In addition to the conventional Duabechies wavelets, Haar wavelets ^[19], linear B-splines ^[18], Walsh functions ^[20] have been used.

In this paper, we propose the application of the linear Legendre multi-wavelets as basis functions in collocation’s method for numerical solution of the system of nonlinear Fredholm integral equations (1). The method is tested with the numerical examples.

2. Legendre Multi-Wavelets

Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets as ^[21].

If we restrict the parameters a and b to discrete values as , then

form an orthogonal basis ^[21].

The linear Legendre multi-wavelets are described in ^[22] and applied in ^{[23, 24, 25, 26]}. For constructing the linear Legendre multi-wavelets, at first we describe the following scaling functions;

Now let and be the corresponding mother wavelets, then by Multiresolution of analysis (MRA) and applying suitable conditions ^[22] on and the explicit formula for linear Legendre mother wavelets will obtain as;

(2)

(3)

and the family , k is any nonnegative integer, and j = 0, 1, forms an orthonormal basis for .

3. Function Approximation

A function f (t) defined over [0, 1) can be expanded as;

(4)

where,

(5)

In Eq.(5), denoting the inner product. If the infinite series of Eq.(4) is truncated, then it can be written as;

(6)

where,

and M is a nonnegative integer.

4. System of Nonlinear Fredholm Integral Equations

In this section we apply collocation method to convert Eq.(1) to algebraic system of nonlinear equations and then solve this system by well known solver. We assume that Eq. (1) has a unique solution. However, the necessary and sufficient conditions for existence and uniqueness of the solution of system Eq.(1) could be found in ^[4]. We approximate f_i (x)’s, such that;

(7)

where defined in Eq. (6) and ’s are unknown coefficients which are determined by solving algebraic system.

By substituting Eq. (7) in Eq. (1) we have;

Now, we choose some collocation points such as;

which are equidistant, also define system of residual equations by:

Then, by imposing the conditions;

we can conclude algebraic system of nonlinear equations.

Next, we present the convergence analysis of our method.

Let us consider the following norms;

Lemma 4.1. Let (Sobolev space) and be the best approximation polynomial of in L₂-norm. Then we have,

where is a positive constant, which depends on the selected norm and is independent of and .

Proof. ^[26].

Theorem 4.1. Le and be the numerical solution of the Eq. (1). Then we have;

where, is a positive constant.

Proof. By Eq.(1), we get;

Since ,

Furthermore, by Lemma 4.1, we obtain;

and therefore,

and by choosing the proof is completed.

5. Numerical Experiments

In this section, we give some numerical experiments to illustrate the results obtained in previous sections. All the numerical experiments presented in this section were computed by a Maple 16 on a PC with a 1.86GHz 32-bit processor and 1GB memory. Moreover, these examples are solved by M = 1.

Example 5.1. Consider the following nonlinear system of Fredholm integral equations:

With the exact solutions u (x) = x and v (x) = x².

Table 1 is the numerical results for Example 5.1.

Table 1. Shows the results of example 5.1

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Example 5.2. Consider the following nonlinear system of Fredholm integral equations:

Where;

With the exact solutions u (x) = sin(x) and v (x) = x+ex.

Table 2 is the numerical results for Example 5.2.

Table 2. Shows the results of example 5.2

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6. Conclusions

In this paper, the systems of nonlinear Fredholm integral equations are investigated and a practical projection method known as collocation method based on Legendre multi-wavelets is established. Finally, illustrative examples are included to demonstrate the validity and applicability of the technique.

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