Numerical Solution of Singularly Perturbed Delay Differential Equations with Layer Behavior

and Applied Analysis 3 Table 1: The maximum error for ε = 0.1 and for different δ for Example 1. δ Max error in [11] Max error of presented method 0.01 0.01182463 0.0045 0.03 0.01515596 0.0090 0.06 0.02584799 0.0070 0.08 0.08313177 0.0300


Introduction
In recent years, there has been a growing interest in the singularly perturbed delay differential equation (see [1][2][3][4]).A singularly perturbed delay differential equation is an ordinary differential equation in which the highest derivative is multiplied by a small parameter and involving at least one delay term.Such types of differential equations arise frequently in applications, for example, the first exit time problem in modeling of the activation of neuronal variability [5], in a variety of models for physiological processes or diseases [6], to describe the human pupil-light reflex [7], and variational problems in control theory and depolarization in Stein's model [8].Investigation of boundary value problems for singularly perturbed linear second-order differentialdifference equations was initiated by Lange and Miura [5,9,10]; they proposed an asymptotic approach in study of linear second-order differential-difference equations in which the highest order derivative is multiplied by small parameters.Kadalbajoo and Sharma [11][12][13][14] discussed the numerical methods for solving such type of boundary value problems.Amiraliyev and Erdogan [15] and Amiraliyeva and Amiraliyev [16] developed robust numerical schemes for dealing with singularly perturbed delay differential equation.
In the present work we suggest a technique similar to the one which was used in [17,18] for solving singularly perturbed differential-difference equation with delay in the following form (see [13]): where  is small parameter, 0 <  ≪ 1, and  is also a small shifting parameter, 0 <  ≪ 1, (), (), (), and () are assumed to be smooth, and  is a constant.For  = 0, the problem is a boundary value problem for a singularly perturbed differential equation and then as the singular perturbation parameter tends to zero, the order of the corresponding reduced problem is decreased by one, so there will be one layer.It may be a boundary layer or an

Function Approximation
Consider the problem (1).Divide the interval [ 0 ,   ] into a set of grid points such that where ℎ = (  − 0 )/,   = 1,  0 = 0 and  is a positive integer. Let Then, for  ∈   , the problem (1) can be decomposed to the following suboptimal control problems: where   ().
Our strategy is using Bezier curves to approximate the solutions   () by V  () where V  () is given below.Individual Bezier curves that are defined over the subintervals are joined together to form the Bezier spline curves.For  = 1, 2, . . ., , define the Bezier polynomials V  () of degree  that approximate the action of   () over the interval [ −1 ,   ] as follows: where is the Bernstein polynomial of degree  over the interval [ −1 ,   ] and    is the control points (see [17]).By substituting (5) in (3), one may define  1, () for  ∈ [ −1 ,   ] as Let V() = ∑  =1  1  ()V  () where  1  () is the characteristic function of V  () for  ∈ [ −1 ,   ].Beside the boundary conditions on V(), at each node, we need to impose continuity condition on each successive pair of V  () to guarantee the smoothness.Since the differential equation is of first order, the continuity of  (or V) and its first derivative give where V ()  (  ) is the th derivative V  () with respect to  at  =   .
Thus, the vector of control points    ( = 0, 1,  − 1, ) must satisfy (see [17]) Ghomanjani et al. [17] proved the convergence of this method where ℎ → 0. Now, the residual function can be defined in   as follow: where ‖ ⋅ ‖ is the Euclidean norm and  is a sufficiently large penalty parameter.Our aim is solving the following problem over  = ⋃  =1   : The mathematical programming problem ( 11) can be solved by many subroutine algorithms.Here, we use Maple 12 to solve this optimization problem.

Numerical Results and Discussion
Consider the following examples which can be solved by using the presented method.
Example 1.First we consider the problem (see [11]) under the boundary conditions  A boundary layer exists on left side of the interval.For this problem, the exact solution is where Also, we have plotted the graphs of the exact and computed solution of the problem in Figure 1.The maximum errors are shown in Table 1.
Example 2. Next we consider the problem (see [11]) under the boundary conditions where Also, we have plotted the graphs of the exact and computed solution of the problem in Figure 2. The maximum errors are shown in Table 2.

Conclusions
We have described a numerical algorithm for solving BVPs for singularly perturbed differential-difference equation with small shifts.Here, we have discussed both the cases by using Bezier curves, when boundary layer is on the left side and when boundary layer is on the right side of the underlying

Figure 1 :
Figure 1: Graphs of the exact and computed solution of the BVP with  = 0.1 and  = 0.01 for Example 1.

Table 1 :
The maximum error for  = 0.1 and for different  for Example 1.