In this study, a practical matrix method based on Laguerre polynomials is presented to solve the higher‐order linear delay differential equations with constant coefficients and functional delays under the mixed conditions. Also, an error analysis technique based on residual function is developed and applied to some problems to demonstrate the validity and applicability of the method. In addition, an algorithm written in Matlab is given for the method. Copyright © 2013 John Wiley & Sons, Ltd.

Abstract:Functional di erential equations have importance in many areas of science such as mathematical physics. These systems are di cult to solve analytically.In this paper we consider the systems of linear functional di erential equations [1][2][3][4][5][6][7][8][9] including the term y(αx + β) and advance-delay in derivatives of y . To obtain the approximate solutions of those systems, we present a matrixcollocation method by using Müntz-Legendre polynomials and the collocation points. For this purpose, to obtain the approximate solutions of those systems, we present a matrix-collocation method by using Müntz-Legendre polynomials and the collocation points. This method transform the problem into a system of linear algebraic equations. The solutions of last system determine unknown coe cients of original problem. Also, an error estimation technique is presented and the approximate solutions are improved by using it. The program of method is written in Matlab and the approximate solutions can be obtained easily. Also some examples are given to illustrate the validity of the method.

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.

Singularly perturbed differential equations are encountered in mathematical modelling of processes in physics and engineering. Aim of this study is to give a collocation approach for solutions of singularly perturbed two-point boundary value problems. The method provides obtaining the approximate solutions in the form of Müntz-Legendre polynomials by using collocation points and matrix relations. Singularly perturbed problem is transformed into a system of linear algebraic equations. By solving this system, the approximate solution is computed. Also, an error estimation is done using the residual function and the approximate solutions are improved by means of the estimated error function. Two numerical examples are given to show the applicability of the method.

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