Müntz-legendre Polynomial Solutions of Linear Delay Fredholm Integro-differential Equations and Residual Correction

In this paper, we consider the Müntz-Legendre polynomial solutions of the linear delay Fredholm integro-differential equations and residual correction. Firstly, the linear delay Fredholm integro-differential equations are transformed into a system of linear algebraic equations by using by the matrix operations of the Müntz-Legendre polynomials and the collocation points. When this system is solved, the Müntz-Legendre polynomial solution is obtained. Then, an error estimation is presented by means of the residual function and the Müntz-Legendre polynomial solutions are improved by the residual correction method. The technique is illustrated by studying the problem for an example. The obtained results show that error estimation and the residual correction method is very effective.


INTRODUCTION
In this study, for the linear delay Fredholm integro-differential equations [1,7] under the boundary conditions (1) = , = 0,1,..., 1, the approximate solution based on the Müntz-Legendre polynomials will be obtained in the form 0 ( ) ( ) Here, ( denote the Müntz-Legendre polynomials [6] defined by 1 ( ) ( 1) , Also, an error problem is constructed by the residual error function and the Müntz-Legendre polynomials of this problem are computed and thus the error function is estimated by these solutions.And then, the approximate solutions are improved by summing the Müntz-Legendre polynomial solutions and the estimated error function [4].

FUNDAMENTAL MATRIX RELATIONS
Let us consider the equation ( 1) and find the matrix forms of each term in the equation.For this purpose let us write the matrix form of the differential part on the left hand side of the equation.First we can write the approximate solution (3) in the matrix form [5] as, Here, the matrix () x L can be written as ( 1) By putting Eq.( 6) into Eq.(5), we have the matrix form (7) The k th-order derivative of Eq.( 7) is given by By placing j xx   in Eq. ( 8), we obtain the matrix form and for 0 j   : ( 1) ( 1) Now let us construct the matrix form of the integral part on the right hand side of the equation.The kernel function ( , ) s K x t can be approximated by the truncated Taylor series [2] and the truncated Müntz-Legendre series 00 ( , ) where ..., sm  We write the expressions in (10) 6), ( 11) and ( 12), By writing the matrix forms ( 9) and (13) into the integral part in the equation, we have the matrix relation we have

T rs rs t t dt h h r s N rs
We put the matrix form (6) into the equation ( 14) we have the matrix relation,

METHOD OF SOLUTION
We are now ready to construct the fundamental matrix equation [8,9] for the Eq.( 1).For this purpose we substitute the relations ( 9) and ( 14) into the Eq.( 1) and thus we obtain the matrix equation The collocation points defined by or briefly the fundamental matrix equation becomes 0 0 () where Briefly, Eq.( 16) can be written in the form Here, Eq.( 17) corresponds to a system of ( 1) N  linear algebraic equations with the unknown Müntz-Legendre coefficients 01 , ,..., N a a a .By using the relation ( 8), the matrix form of the conditions (2) becomes To obtain the solution of Eq. ( 1) under the conditions (2), by replacing the last m rows of matrix (17) by the m row matrices (18) we have the new augmented matrix . Thus, the Müntz-Legendre coefficients matrix A is uniquely determined.Finally, by substituting the determined coefficients 01 , , , N a a a  into Eq.(3), we get the Müntz-Legendre polynomial solution 0 ( ) ( )

ERROR ESTIMATION AND IMPROVED APPROXIMATE SOLUTIONS
In this section, we develop an error estimation for the Müntz-Legendre approximate solution for the problem by means of the residual correction method [10,11] and we improve the approximate solution (19) by using this error estimation.The residual error estimation was presented for the Bessel approximate solutions of the system of the linear multi-pantograph equations [12].For the problem (1)-( 2), we modify the error estimation considered in [10][11][12] , we solve the above problem.In Table 1, we give the absolute errors (actual, estimation and correction) of this problem.

CONCLUSIONS
In this paper, we presented the Müntz-Legendre collocation scheme of the linear delay Fredholm integro-differential equations.In addition, we constructed an error problem by means of the residual error function and this problem is solved by Müntz-Legendre collocation scheme.Hence, the Müntz-Legendre polynomial solution of this error problem is an approximation for the actual error function.Finally, by summing the estimated error function and Müntz-Legendre polynomial solution, the corrected approximate solution is obtained.We gave the application of our works for an example.In this application, we observed from example that actual and estimated errors are very close and the improvement is quite effective.Therefore, we note that the error estimation can be used for measurement of the reliability of the considered problem, when an exact solution of any problem is not available.
Finally we can obtain the matrix relations for conditions by means of the relation(8

Table 1 .
Numerical results of the error functions for