Abstract
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.
Highlights
In this study, for the linear delay Fredholm integro-differential equations [1,7] mPk (x) y k (x k ) = g(x) Ks (x,t) y s (x s )dt, 0 x,t 1 (1) k =00 s=0 under the boundary conditions m 1ajk y(k) (0) bjk y(k) (1) = j, j = 0,1,..., m 1, (2)k=0 the approximate solution based on the Müntz-Legendre polynomials will be obtained in the formN yN (x) an Ln (x) . (3)
An error problem is constructed by the residual error function and the Müntz-Legendre polynomials of this problem are computed and the error function is estimated by these solutions
The residual error estimation was presented for the Bessel approximate solutions of the system of the linear multi-pantograph equations [12]
Summary
For the linear delay Fredholm integro-differential equations [1,7]. k=0 the approximate solution based on the Müntz-Legendre polynomials will be obtained in the form. For the linear delay Fredholm integro-differential equations [1,7]. K=0 the approximate solution based on the Müntz-Legendre polynomials will be obtained in the form. An error problem is constructed by the residual error function and the Müntz-. Legendre polynomials of this problem are computed and the error function is estimated by these solutions. The approximate solutions are improved by summing the Müntz-Legendre polynomial solutions and the estimated error function [4]
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