Hermite Series Solutions of Linear Fredholm Integral Equations

A matrix method for approximately solving linear Fredholm integral equations of the second kind is presented. The solution involves a truncated Hermite series approximation. The method is based on first taking the truncated Hermite series expansions of the functions in equation and then substituting their matrix forms into the equation. Thereby the equation reduces to a matrix equation, which corresponds to a linear system of algebraic equations with unknown Hermite coefficients. In addition, some equations considered by other authors are solved in terms of Hermite polynomials and the results are compared.


INTRODUCTION
Fredholm integral equations are widely used for modelling and forecasting in almost all areas of science and engineering.
Fredholm integral equations are usually difficult to solve analytically so it is required to obtain an efficient approximate solution.As we know, much work has been done on developing and analyzing numerical methods for solving linear Fredholm integral equations [2-10-11] , so Hermite polynomials have comparison advantages with other orthogonal polynomials.The subject of the presented paper is to apply the Hermite method for solving linear Fredholm integral equations.
In this paper, we consider the Fredholm integral equations of the second kind where ) (x y is the function to be determined.The constant  , the kernel function ) , ( t x K and the function ) (x f are given.We assume that the interested domain of the variables is . The solution of Eq. ( 1) is expressed as the truncated Hermite series where ) (x H r is the Hermite polynomial of degree r [6], or in the matrix form are coefficients to be determined.

METHOD FOR SOLUTION
To obtain the solution of Eq. ( 1) in the form of expression (2) we can first deduce the following matrix approximations corresponding to the Hermite series expansions of the functions Then we can put series (4) in the matrix form can be approximated by double Hermite series of degree N in both x and t of the form [2,7] then we can put series (6) in the matrix form On the other hand, for the unknown function ) (t y in integrand, we write from expressions (2) and ( 3) ) Substituting the matrix forms (3), ( 5), (7) and (8) where and I is the unit matrix; the elements of the fixed matrix Q are similar by [1,2] In Eq. ( 9), if 0 are uniquely determined by equation (11) and thereby the integral Eq. ( 1) has a unique solution.This solution is given by the truncated Hermite series (2).

ACCURACY OF SOLUTION
We can easily check the accuracy of the method.Since the truncated Hermite series in ( 2) is an approximate solution of Eq. ( 1), it must be approximately satisfied this equation.
Then for each at each of the points i x becomes smaller than the prescribed k  10 .On the other hand, the error function can be estimated by

NUMERICAL ILLUSTRATIONS
We show the efficiency of the presented method using the following examples.Example 1.Let us first consider the linear Fredholm integral equation [2,3] and seek the solution By using the expansions for the powers r x in terms of the Hermite polynomials ) (x H r [6], we easily find the representations   and hence, from relations ( 5) and ( 7), the matrices , we obtain the fixed matrix Next, we substitute these matrices into Eq.( 11) and then simplify to obtain We give numerical analysis for various N values in Table 4 and Figure 3.We give numerical analysis for various N values in Table 5 and Figure 4.

CONCLUSIONS AND DISCUSSIONS
In this paper, the usefulness of the method presented for the approximate solution of Fredholm integral equation ( 1) is demonstrated.To show the accuracy of the method, five integral equations are chosen.A considerable advantage of the method is that the solution is expressed as a truncated Hermite series.This means that, after calculation of the Hermite coefficients, the solution , then there exists the solution ) (x y ; otherwise, the method cannot be used in.On the other hand, it would appear that our method shows to best advantage when the known functions ) (x f and ) , ( t x K have Taylor series about the origin which converge rapidly.To get the best approximating solution of the equation, we must take more terms from the Hermite expansions of functions,

Figure 3 .
Figure 3.The absolute errors of the solutions by using the present method for different values of N .

Figure 4 .
Figure 4.The graphics of the Exact Solution, BPF method, TF Method, Legendre Method and Present method calculated for the values of the interval ] 1 , 0 [ .

Table 1 .
Comparing the solutions and error analysis which has been found for Present method : Hermite Method

Table 2 .
Comparing the solutions and error analysis which has been found for Present method : Hermite Method

Table 3 .
Exact, approximate solution and error analysis of Example 5 for the x values

Table 4 .
Comparing the solutions and error analysis which has been found for

Table 5 .
Error analysis of Example 7 for the x values and comparison of present method, exact and the