Approximate Solution of the Singular-Perturbation Problem on Chebyshev-Gauss Grid

Matrix methods, now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineers for which the existing methodologies do not give reliable results, these methods are solving them competitively. In this work, a matrix methods is presented for approximate solution of the second-order singularly-perturbed delay differential equations. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The error analysis and convergence for the proposed method is introduced. Finally some experiments and their numerical solutions are given.


Introduction
The boundary-value problems for singularly perturbed delay-differential equations arise in various practical problems in biomechanics and physics such as in variational problem in control theory.These problems mainly depend on a small positive parameter and a delay parameter in such a way that the solution varies rapidly in some parts of the domain and varies slowly in some other parts of the domain.Moreover, this class of problems possess boundary layers, i.e. regions of rapid change in the solution near one of the boundary points.There is a wide class of asymptotic expansion methods available for solving the above type problems.But there can be difficulties in applying these asymptotic expansion methods, such as finding the appropriate asymptotic expansions in the inner and outer regions, which are not routine exercises but require skill, insight and experimentation.The numerical treatment of singularly perturbed problems present some major computational difficulties and in recent years a large number of specialpurpose methods have been proposed to provide accurate numerical solutions [1][2][3][4][5] by Kadalbajoo.This type of problem has been intensively studied analytically and it is known that its solution generally has a multiscale character; i.e. it features regions called "boundary layers" where the solution varies rapidly.And these equa-tions as well as numerical methods have been studied by several authors [6][7][8][9][10].The outer solution corresponds to the reduced problem, i.e., that obtained by setting the small perturbation parameter to zero.In recent years, the Chebyshev method has been used to find the approximate solutions of differential, difference, integral and integro-differential-difference equations [11,12].The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem.
Consider the of singularly-perturbed delay differential equations form where 0 < < 1 x , with the boundary conditions ( ) s x and ( ) f x are sufficiently smooth functions.Our goal is to find an approximate solution expressed polynomial of degree in the form where r unknown coefficients, is the shifted Chebyshev polynomials of the first kind and is chosen any positive integer such that .To obtained a solution (2) of the problem (1), we can use the zeroes of the shifted Chebyshev polynomials of the first kind defined by * 1 ( )

Basic Idea
Polynomials are the only functions that a computer can evaluate exactly, so we make approximate functions

 
, a b   by polynomials.The uniform norm (or Chebyshev norm, maximum norm) is defined by It is a good idea to approximate function by polynomials, because the classical Weierstrass Theorem is a fundamental result in the approximation of continuous functions by polynomials [13][14][15].Proof: See Ref. [13][14][15][16].Definition 2.4.Given an integer then a grid set of points are . Then points are called the nodes of the grid.
and a grid of nodes, , there exists a unique polynomial of degree , I f is called the interpolant (or the interpolating polynomial) of f through the grid X .

 
X N I f can be express in the Lagrange form as is the i-th Lagrance cardinal polynomial associated with the grid X : 0, ( ) , 0 .
nodes the error is given by formula: The Lebesgue constant contains all the information on the effects of the choice of X on   .
Theorem 2.6.For any choice of the grid X , there exist a constant such that > 0 ) Proof: See Ref. [13][14][15][16].Definition 2.7.The nodal polynomial associated with the grid is the unique polynomial of degree (  1 N  and leading coefficient 1 whose zeroes are the , then for any grid X 1 N of  nodes, and for any where and nodal polynomial associated with the grid 1 ( ) . Proof: See Ref. [13][14][15][16].

The Shifted Chebyshev Polynomial of the First Kind
Definition 2.9.The Chebyshev polynomial of the first kind is a polynomial in ( ) x of degree , defined by the relation If the range of the variable x is the interval [ 1,1]   , the range the corresponding variable  can be taken .Since the range is quite often more convenient to use than the range , we map the independent variable Definition 2.10.The grid and this lead to a shifted Chebyshev polynomial of the first kind of degree in * ( ) Let us consider the Equation ( 1) and find the matrix forms of each term of the equation.We first consider the solution and its derivative ( ) x defined by a truncated Chebyshev series.Then we can put series in the matrix form ) Note that the shifted Chebyshev polynomial is neither even nor odd and indeed all powers of x from x in for to be .These polynomials have the following properties: By using (5), we obtained the corresponding matrix relation as follows: 2) It is well known that the relation between the powers n x and the shifted Chebyshev polynomials is Moreover it is clearly seen that the relation between the matrix ( ) X x and its derivative ( ) ( ) where The derivative of the matrix defined in (7), by * ( ) N T x using the relation (8), can expressed as where , *(0) * ( ) = ( ) . By substituting ( 9) into (6), we obtain where (0) ( ) = ( ) where Using relation ( 8) and ( 11), we can write In a similarly way as (10) , we obtain .

Matrix Representation of the Conditions
Using the relation (10), the matrix form of the conditions given by (2) can be written as where

Method of Solution
We are ready to construct the fundamental matrix equation corresponding to Equation (1).For this propose, firstly substituting the matrix relation ( 10) and ( 13) into (1) we obtained For computing the Chebyshev coefficient matrix A numerically, the zeroes of the shifted Chebyshev polynomials of the first kind defined by (3) are putting above relation (15) and organized.We obtained, So, the fundamental matrix equation is gained The fundamental matrix equation (16) for Equation( 1) corresponds to a system of ( Briefly, the matrix form for conditions (2) are where To obtain the solution of Equation ( 1) under the condi tions (2), by replacing the rows matrices (18) by the last m rows of the matrix (17), we have the required augmented matrix ; ; Thus the coefficients are uniquely determined by Equation (2

Convergence and Error Analysis
and this is the smallest possible value.particular, In for any we have [14]   This is much better than uniform grids and close to the optimal value.

4.
We can easily check the accuracy of the obtained solutained the shifted Chebyshev polynomial of the first kind expansion is an ap-

Checking of Solution
tions as follows: Since the ob proximate solution of Equation ( 1), when the function ( ) N y x and its derivatives are substituted in Equation ( 1), the resulting equation must be satisfied approximately, that is for

Illustrative Example
To demonstrate the effect of delay on the layer behavior of the method, we consider the examples given below and solve them using the pre-of the solution and the efficiency sent method and all of them were performed on the computer using a program written in Maple 9 software in the solving process.We have plotted the graphs of the solution of the problem for different  with different values of  to show the effect of delay on the boundary layer solution.
The maximum errors denoted y  where We solved this problem using the method presented here and compared the result with the exact solution of the problems in Table 1.Also for = 10 N , we have plotted the graphs of the computed solutio f the problem for where We calculated numerical results for = 0.5

Conclusions
In recent years, the delay differential equations have attracted the attention of many sciences and engineers.The Chebyshev expansion methods are used to solve the singularly-perturbed delay differential equations numerically.A considerable advantage of the method is that the Chebyshev polynomial coefficients of the solution are found very easily by using computer programs in Maple 9. Shorter computation time and lower operation count results in reduction of cumulative truncation errors and improvement of demonstrate the validity and applicability of the technique.To get the best approximating solution of the equation, we take more forms from the Chebyshev expansion of functions, that is, the truncation limit N must be chosen large enough.Suggested approximations make this method very attractive and contributed to the good agreement between approximate and exact values in the numerical example.As a result, the power of the employed method is confirmed.We assured the correctness of the obtained solu- the aid of Maple, it provides an extra measure of confidence in the results.
Kumar, "Fitted Mesh B-Spline ethod for Singularly Perturbed Differentialations with Small Delay," Applied Mathe- lution for the considered examples is not available but l result Figur rly-perturbed delay differential equation[3] and (b) given byKadalbajoo ([3], Example 2).In Figure1(a), we show the numerical result using present method and for the method using by Kadalbajoo is shown in Figure1(b).The graphs of the solution of the considered examples for different values of delay are plotted in Figures1(a)-(d) to examine the questions on the effect of delay on the boundary layer behavior of the solution.

2 = 2  5 . 3 .
 , for different values n o  in Figure 2(a) and we ca sily check the accuracy of the obtained solutions in Figure 2(b).Example Let us consider the second-order singula n ea rly-perturbed delay differential Equation[6]

Figure 2
Figure 2. (a) Numerical results of Example 5.2 for various (N = 10); (b) For , numerical results of Example 5.2 for 

Figure 3 .
Figure 3. Numerical results of Example 5.3 for various  .

Table 1 .
splay results for some various N .Moreover, we compare the results with Non-standard finite difference methods (NSFDMs) in Table2and we display results