Abstract and Applied Analysis

Volume 2014 (2014), Article ID 418909, 9 pages

http://dx.doi.org/10.1155/2014/418909

Research Article

Haar Wavelet Method for the System of Integral Equations

Hassan A. Zedan1,2 and Eman Alaidarous¹

¹Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Mathematics Department, Faculty of Science, Kafrelsheikh University, Kafrelsheikh, Egypt

Received 15 March 2014; Revised 13 May 2014; Accepted 20 May 2014; Published 21 July 2014

Academic Editor: Imran Naeem

Copyright © 2014 Hassan A. Zedan and Eman Alaidarous. 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 employed the Haar wavelet method to find numerical solution of the system of Fredholm integral equations (SFIEs) and the system of Volterra integral equations (SVIEs). Five test problems, for which the exact solution is known, are considered. Comparison of the results is obtained by the Haar wavelet method with the exact solution.

1. Introduction

The Haar wavelet was initiated and independently developed by Haar [1] in the early nineteen tens. In recent years, many different methods and different basis functions have been used to estimate the solution of the system of integral equations, such as Adomian decomposition method [1, 2], Taylor’s expansion method [3, 4], homotopy perturbation method [5, 6], projection method and Nystrom method [7], Spline collocation method [8], Runge-Kutta method [9], sinc method [10], Tau method [11], block-pulse functions, hat basis functions method [12], and operational matrices [13, 14].

In the present paper, we use Haar wavelet method to solve the system of linear Fredholm integral equations (SLFIEs) of the second kind:

and the system of linear Volterra integral equations (SLVIEs) of the second kind:

where

In (1) and (2), the functions and are given, and is the vector function of the solution of systems (1) and (2) that will be determined. Also, we assume that (1) and (2) have a unique solution.

2. Haar Wavelet Method

Let us confine to the time interval . The Haar wavelet family is

Here the notations are as follows:

The integer ,   , indicates the level of the wavelet; is the translation parameter. The integer determines the maximal level of resolution. The index is calculated from the formula ; the minimal value for which (4) holds is (then ); the maximal value is , where . The index corresponds to the scaling function of the Haar wavelet .

Simple calculations show the following.

We have that

consequently, the functions are orthogonal.

Next we discretize the functions by dividing the interval into parts of equal length and introduce the collocation points:

Following Chen and Hsiao [15, 16] the coefficients matrix is introduced (this is a matrix). A function which is defined in the interval can be expanded into the Haar wavelet series:

where is the wavelet coefficients. The discrete form of this equation is

or in a matrix presentation , where and are dimensional row vectors.

3. System of Fredholm Integral Equation

Let us consider SLFIEs (1) at :

Let

Replacing (12) into (10) and (11) we find

where

Next we will evaluate the wavelet coefficients and in the following way.

Collocation Method. Satisfying (12) only at the collocation points (7) we get a system of linear equations:

Therefore the matrix form of this system is given by

Now, we can present the following problems.

Example 1. Consider the following SLFIEs [3, 16]:

Carrying out the integration in (13) we obtain

We apply collocation method, and vectors and can be calculated from (15); the functions and are evaluated from (11).

Computations were carried out for different values of . These results were compared with the exact solution and .

The accuracy of the results (see Table 1 and Figure 1) was estimated by the error function for and :

where is defined by (7).

Table 1: Error of and of Example 1.

Error of function
Error of function

2
8

3
16

4
32

5
64

(a)

(b)

Figure 1: (a) Exact and Haar wavelet solution of at . (b) Exact and Haar wavelet solution of at .

Example 2. Consider the following SLFIEs given by [10, 12]:

Carrying out the integration in (13) we obtain

We apply collocation method, and vectors and can be calculated from (15); the functions and are evaluated from (11).

Computations were carried out for different values of . These results were compared with the exact solution (see Table 2 and Figure 2) and .

Table 2: Error of and of Example 2.

Error of function
Error of function

2
8

3
16

4
32

5
64

(a)

(b)

Figure 2: (a) Exact and Haar wavelet solution of at . (b) Exact and Haar wavelet solution of at .

Example 3. Consider the following SLFIEs given by [10]:

Carrying out the integration in (13) we obtain

We apply collocation method, and vectors and can be calculated from (15); the functions and are evaluated from (11).

Computations were carried out for different values of (see Table 3 and Figure 3). These results were compared with the exact solution and .

By comparing the results obtained with the results found in [3, 10, 12], we find that the results we have obtained are accurate and error rate, which is much less but almost nonexistent compared to other methods, is also shown in tables of error attached and this shows the importance of the method used.

Table 3: Error of and of Example 3.

Error of function
Error of function

2
8

3
16

4
32

5
64

(a)

(b)

Figure 3: (a) Exact and Haar wavelet solution of at . (b) Exact and Haar wavelet solution of at .

4. System of Volterra Integral Equations

Let us consider system of linear Volterra integral equations (SLVIEs) (2) at (:

Its discrete form is

where defined in (7) are the collocation points.

Let

Replacing (25) with (24) we find

The matrices ,   , are now defined as

By computing these integrals the following cases should be distinguished:

Next we will evaluate the wavelet coefficients and in the following way.

Collocation Method. Satisfying (27) and (28) only at the collocation points (7) we get a system of linear equations

The matrix form of this system is

Example 4. Consider the following (SLVIEs) [17]:

Carrying out the integration in (28) we obtain

We apply collocation method, and vectors and can be calculated from (30); the functions and are evaluated from (25).

Computations were carried out for different values of (see Table 4 and Figure 4). These results were compared with the exact solution and .

The accuracy of the results was estimated by the error function for and :

where is defined by (7).

Table 4: Error of and of Example 4.

Error of function
Error of function

2
8

3
16

4
32

5
64

(a)

(b)

Figure 4: (a) Exact and Haar wavelet solution of at . (b) Exact and Haar wavelet solution of at .

Example 5. Consider the following (SLVIEs) [4, 18]:

Carrying out the integration in (28) we obtain

We apply collocation method, and vectors and can be calculated from (29); the functions and are evaluated from (25).

Computations were carried out for different values of (see Table 5 and Figure 5). These results were compared with the exact solution and .

Table 5: Error of and of Example 5.

Error of function
Error of function

2
8

3
16

4
32

5
64

(a)

(b)

Figure 5: (a) Exact and Haar wavelet solution of at . (b) Exact and Haar wavelet solution of at .

Example 6. Consider the following (SLVIEs):

Carrying out the integration in (28) we obtain

We apply collocation method, and vectors and can be calculated from (29); the functions and are evaluated from (25).

Computations were carried out for different values of (see Table 6 and Figure 6). These results were compared with the exact solution and .

Table 6: Error of and of Example 6.

Error of function
Error of function

2
8

3
16

4
32

5
64

(a)

(b)

Figure 6: (a) Exact and Haar wavelet solution of at . (b) Exact and Haar wavelet solution of at .

5. Conclusion

In this work the Haar wavelet method for solution of linear systems integral equations is proposed. A method of solution which is applicable for different kind of integral equations, Fredholm and Volterra systems of integral equations, is worked out. The solution is based on the collocation techniques which are proposed. The elaborated method is very simple and, as it follows from the test problems, high precision of results can be obtained with a small number of calculation points. The calculations show that by doubling the number of the calculation points the error function decreases. This result follows also from analytical considerations. It should be noted that in the case of Haar wavelets we have to solve systems of linear equations with a smaller condition number as for other methods based on piecewise constant approximation.

Conflict of Interests

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

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