LEGENDRE SERIES SOLUTIONS OF FREDHOLM INTEGRAL EQUATIONS

AbstractA matrix method for approximately solving linear Fredholm integral equations of the second kind is presented. The solution involves a truncated Legendre series approximation. The method is based on first taking the truncated Legendre 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 Legendre coefficients. In addition, some equations considered by other authors are solved in terms of Legendre polynomials and the results are compared.


INTRODUCTION
In this paper we consider the Fredholm integral equations of the second kind where ) (x y is the function to be determined.The constant λ , the kernel function ) , ( t x K and the function ) (x f are given.We assume that the range of the variables is . The solution of equation ( 1) is expressed as the truncated Legendre series where ) (x P r is the Legendre polynomial and of degree r [6], or in the matrix form where and N r a r , , 1 , 0 , K = are coefficients to be determined.

METHOD FOR SOLUTION
To obtain the solution of equation (1) in the form of expression (2) we can first deduce the following matrix approximations corresponding to the Legendre series expansions of the functions Then we can put series (4) in the matrix form where We now consider the kernel function 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) corresponding to the functions and ) (t y , respectively, into equation (1), and then simplifying the result equation, we have the matrix equation { }A and I is the unit matrix; the elements of the fixed matrix Q are given by [1,2]   Thus the unknown coefficients N r a r , , 1 , 0 , K = are uniquely determined by equation ( 11) and thereby the integral equation (1) has a unique solution.This solution is given by the truncated Legendre series (2).

ACCURACY OF SOLUTION
We can easily check the accuracy of the method.Since the truncated Legendre 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 (In all figures different line shows the exact solution and the numerical solution).
and seek the solution ) (x y in Legendre series By using the expansions for the powers r x in terms of the Legendre polynomials ) (x P 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 in equation ( 11) and then simplify to obtain . By using the expansions for the powers r x in terms of the Legendre polynomials ) (x P r [1], we easily find the representations ( ) and hence, from relations ( 5) and ( 7), the matrices , we obtain the fixed matrix Next, we substitute these matrices in equation ( 11) and then simplify to obtain We give numerical analysis for 10 , 9 = N in Table 1 and Fig. 1.We give numerical analysis for various N values in Table 3 and Fig. 3, 4.  ) with exact solution x e 2 is given in Table 4 and Fig. 5.

Example 3 .
is the exact solution[5,8].Let us now take the equation
[2,7]dre series of degree N in both x and t of the form[2,7] [5,8]d we can study the following linear Fredholm integral equation[5,8]

Table 1 .
Comparing the solutions and error analysis which has been found for

Table 2 .
Error analysis of Example 5 for the x values.

Table 3 .
Error analysis of Example 6 for the x values and comparison of present method, exact and the

Table 4 .
Error analysis of Example 7 for the x values.