Numerical Approach of High-order Linear Delay Difference Equations with Variable Coefficients in Terms of Laguerre Polynomials

This paper presents a numerical method for the approximate solution of mthorder linear delay difference equations with variable coefficients under the mixed conditions in terms of Laguerre polynomials. The aim of this article is to present an efficient numerical procedure for solving mth-order linear delay difference equations with variable coefficients. Our method depends mainly on a Laguerre series expansion approach. This method transforms linear delay difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system Maple. Key WordsLaguerre Polynomials and Series, Delay Difference Equations, Laguerre Collocation Method 1.INTRODUCTION Orthogonal polynomials occur often as solutions of mathematical and physical problems. They play an important role in the study of wave mechanics, heat conduction, electromagnetic theory, quantum mechanics and mathematical statistics. They provide a natural way to solve, expand, and interpret solutions to many types of important delay difference equations. Representation of a smooth function in terms of a series expansion using orthogonal polynomials is a fundamental concept in approximation theory, and forms the basis of spectral methods of solution of delay difference equations. Laguerre polynomials ) (x Ln constitute complete orthogonal sets of functions on the semi-infinite interval ) , 0 [  . In this paper, we are concerned with the use of Laguerre polynomials to solve delay difference equations. In recent years, the studies of difference equations, i.e. equations containing shifts of the unknown function are developed very rapidly and intensively. It is well known that linear delay difference equations have been considered by many authors[1-11]. The past couple decades have seen a dramatic increase in the application of delay models to problems in biology, physics and engineering[12-15]. In the field of delay difference equation the computation of its solution has been a great challenge and has been of great importance due to the versatility of such equations in the mathematical modeling of processes in various application fields, where they provide the best simulation of observed phenomena and hence the numerical approximation of such equations has been growing more and more. Based on the obtained method, we shall give sufficient approximate solution of the linear delay B. Gürbüz, M. Gülsu and M. Sezer 268 difference Eq.(1). The results can extend and improve the recent works. An example is given to demonstrate the effectiveness of the results.In recent years, Taylor and Chebyshev approximation methods have been given to find polynomial solutions of differential equations by Sezer et al. [16-22]. In this study, the basic ideas of the above studies are developed and applied to the mthorder linear delay difference equation ( which contains only positive shift in the unknown function) with variable coefficients[23,p.228,p.229]     m k k t f k t y t P 0 ) ( ) ( ) ( , N k k   , 0 (1) under the initial conditions 1 , , 1 , 0 , 0 , ) ( ) 0 ( 1 0            m r b t b y b y a m k r rk rk   (2) where ) (t Pk and ) (t f are analytical functions; i rk rk and b a  , are real or complex constants. The aim of this study is to get solution as truncated Laguerre series defined by ) ( ) ( 0 t L a t y n N n n    ,           n r r r n t r n r t L 0 ! ) 1 ( ) ( (3) where ) (t Ln denotes the Laguerre polynomials, ) 0 ( N n an   are unknown Laguerre polynomial coefficients, and N is chosen any positive integer such that m N  . The rest of this paper is organized as follows. We describe the formulation of Laguerre polynomials required for our subsequent development in section 2. Higher-order linear delay difference equation with variable coefficients and fundamental relations are presented in Section 3. The new scheme are based on Laguerre collocation method. The method of finding approximate solution is described in Section 4. To support our findings, we present result of numerical experiments in Section 5. Section 6 concludes this article with a brief summary. Finally some references are introduced at the end. 2. PROPERTIES OF THE LAGUERRE POLYNOMIALS A total orthonormal sequence in ] , ( 2 b L  or ) , [ 2  a L can be obtained from such a sequence in ) , 0 [ 2  L by transformations t=b-s and t=s+a, respectively. We consider ) , 0 [ 2  L . Applying the Gram-Schmidt process to the sequence defined by ... , , , 2 / 2 2 / 2 / t t t e t te e    We can obtain an orthonormal sequence ) ( n e . It can be shown that ) ( n e is total in ) , 0 [ 2  L and is given by ) ( ) ( 2 / t L e t e n t n   , n=0,1,2,... where the Laguerre polynomial of order n is defined by 1 ) ( 0  t L , ) ( ! ) ( t n n n t n e t dt d n e t L   , n=1,2,3,... (4) Numerical Approach of High-Order Linear Delay Difference Equations 269 That is           n