# Numerical solution of singularly perturbed delay differential equations with layer behavior.

1. Introduction

In recent years, there has been a growing interest in the singularly perturbed delay differential equation (see [1-4]). 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 , in a variety of models for physiological processes or diseases , to describe the human pupil-light reflex , and variational problems in control theory and depolarization in Stein's model . Investigation of boundary value problems for singularly perturbed linear second-order differential difference 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 [11-14] discussed the numerical methods for solving such type of boundary value problems. Amiraliyev and Erdogan  and Amiraliyeva and Amiraliyev  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 ):

[member of]y" (t) + a(t) y' (t - [delta]) + b(t) y(t) = f(t), 0 < t < 1,

y (t) = [phi](t), -[delta] [less than or equal to] t [less than or equal to] 0,

y(1) = [gamma], (1)

where [member of] is small parameter, 0 < [member of] [much less than] 1, and [delta] is also a small shifting parameter, 0 < [delta] [much less than] 1, a(t), b(t), f(t), and [phi](t) are assumed to be smooth, and [gamma] is a constant. For [delta] = 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

The current paper is organized as follows. In Section 2, function approximation will be introduced. Numerical examples will be stated in Section 3. Finally, Section 4 will give a conclusion briefly.

2. Function Approximation

Consider the problem (1). Divide the interval [[t.sub.0], [t.sub.f]] into a set of grid points such that

[t.sub.i] = [t.sub.0] + ih, i = 0, 1, ..., k, (2)

where = h ([t.sub.f] [t.sub.0])/k, [t.sub.f] = 1, [t.sub.0] = 0 and k is a positive integer.

Let [S.sub.j] = [[t.sub.j-1], [t.sub.j]] for h = 1, 2, ..., k. Then, for t [member of] [S.sub.j], the problem (1) can be decomposed to the following suboptimal control problems:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [y.sub.j](t).

We mention that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined where (t - [delta]) [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [[delta]/h] denotes the integer part of [delta]/h.

Let y(t) = [[SIGMA].sup.k.sub.j=1] [[chi].sup.1.sub.j](t)[y.sub.j](t) where [[chi].sup.1.sub.j](t) is the characteristic function of [y.sub.j](t) for t [member of] [[t.sub.j-1], [t.sub.j]]. It is trivial that [[t.sub.0],[t.sub.f]] = [[union].sup.k.sub.h=1] [S.sub.j].

Our strategy is using Bezier curves to approximate the solutions [y.sub.j](t) by [v.sub.j](t) where [v.sub.j](t) is given below. Individual Bezier curves that are defined over the subintervals are joined together to form the Bezier spline curves. For j = 1, 2, ..., k, define the Bezier polynomials [v.sub.j](t) of degree n that approximate the action of [y.sub.j](t) over the interval [[t.sub.j-1], [t.sub.j]] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

is the Bernstein polynomial of degree n over the interval [[t.sub.j-1], [t.sub.j]] and [a.sup.j.sub.r] is the control points (see ). By substituting (5) in (3), one may define [R.sub.1,j](t) for t [member of] [[t.sub.j-1], [t.sub.j]] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Let v(t) = [[SIGMA].sup.k.sub.j=1] [[chi].sup.1.sub.j](t)[v.sub.j](t) where [[chi].sup.1.sub.j](t) is the characteristic function of [v.sub.j](t) for t [member of] [[t.sub.j-1], [t.sub.j]]. Beside the boundary conditions on V(t), at each node, we need to impose continuity condition on each successive pair of [v.sub.j](t) to guarantee the smoothness. Since the differential equation is of first order, the continuity of y (or V) and its first derivative give

[v.sup.(s).sub.j]([t.sub.j]) = [v.sup.(s).sub.j+1]([t.sub.j]), s = 0, 1, h = 1, 2, ..., k - 1, (8)

where [v.sup.(s).sub.j]([t.sub.j]) is the sth derivative [v.sub.j](t) with respect to t at t = [t.sub.j].

Thus, the vector of control points [a.sup.j.sub.r] (r = 0, 1, n - 1, n) must satisfy (see )

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Ghomanjani et al.  proved the convergence of this method where h [right arrow] 0.

Now, the residual function can be defined in [S.sub.j] as follow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where [parallel] * [parallel] is the Euclidean norm and M is a sufficiently large penalty parameter. Our aim is solving the following problem over S = [[union].sup.k.sub.j=1] [S.sub.j]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

The mathematical programming problem (11) can be solved by many subroutine algorithms. Here, we use Maple 12 to solve this optimization problem.

3. 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 )

[member of]y" (t) + y'(t - [delta]) - y(t) = 0, 0 < t < 1, (12)

under the boundary conditions

y(t) = 1, -[delta] [less than or equal to] t [less than or equal to] 0,

y (1) = 1. (13)

A boundary layer exists on left side of the interval. For this problem, the exact solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

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 )

[member of]y"(t) - y'(t - [delta]) - y(t) = 0, 0 < x < 1, (16)

under the boundary conditions

y(t) = 1, -[delta] [less than or equal to] t [less than or equal to] 0,

y (1) = -1. (17)

A boundary layer exists on right side of the interval. For this problem, the exact solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

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.

4. 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 interval. Numerical examples show that the proposed method is efficient and very easy to use.

http://dx.doi.org/10.1155/2014/731057

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the anonymous reviewers for their careful reading, constructive comments, and nice suggestions which have improved the paper very much.

References

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 M. K. Kadalbajoo and K. K. Sharma, "Parameter-uniform fitted mesh method for singularly perturbed delay differential equations with layer behavior," Electronic Transactions on Numerical Analysis, vol. 23, pp. 180-201, 2006.

 P. Rai and K. K. Sharma, "Numerical analysis of singularly perturbed delay differential turning point problem," Applied Mathematics and Computation, vol. 218, no. 7, pp. 3483-3498, 2011.

 C. G. Lange and R. M. Miura, "Singular perturbation analysis of boundary value problems for differential-difference equations. V: small shifts with layer behavior," SIAM Journal on Applied Mathematics, vol. 54, no. 1, pp. 249-272, 1994.

 M. C. Mackey and L. Glass, "Oscillations and chaos in physiological control system," Science, vol. 197, pp. 287-289, 1997.

 A. Longtin and J. G. Milton, "Complex oscillations in the human pupil light reflex with "mixed" and delayed feedback," Mathematical Biosciences, vol. 90, no. 1-2, pp. 183-199, 1988.

 V. Y. Glizer, "Asymptotic analysis and solution of a finite-horizon [H.sub.[infinity]] control problem for singularly-perturbed linear systems with small state delay," Journal of Optimization Theory and Applications, vol. 117, no. 2, pp. 295-325, 2003.

 C. G. Lange and R.M. Miura, "Singular perturbation analysis of boundary value problems for differential-difference equations," SIAM Journal on Applied Mathematics, vol. 42, no. 3, pp. 502-531, 1982.

 C. G. Lange and R. M. Miura, "Singular perturbation analysis of boundary value problems for differential-difference equations. VI: small shifts with rapid oscillations," SIAM Journal on Applied Mathematics, vol. 54, no. 1, pp. 273-283, 1994.

 M. K. Kadalbajoo and K. K. Sharma, "Numerical analysis of singularly perturbed delay differential equations with layer behavior," Applied Mathematics and Computation, vol. 157, no. 1, pp. 11-28, 2004.

 M. K. Kadalbajoo and K. K. Sharma, "Numerical treatment of a mathematical model arising from a model of neuronal variability," Journal of Mathematical Analysis and Applications, vol. 307, no. 2, pp. 606-627, 2005.

 M. K. Kadalbajoo and K. K. Sharma, "A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations," Applied Mathematics and Computation, vol. 197, no. 2, pp. 692-707, 2008.

 P. Rai and K. K. Sharma, "Numerical study of singularly perturbed differential-difference equation arising in the modeling of neuronal variability," Computers & Mathematics with Applications, vol. 63, no. 1, pp. 118-132, 2012.

 G.M. Amiraliyev and F. Erdogan, "Uniform numerical method for singularly perturbed delay differential equations," Computers & Mathematics with Applications, vol. 53, no. 8, pp. 1251-1259, 2007.

 I. G. Amiraliyeva and G. M. Amiraliyev, "Uniform difference method for parameterized singularly perturbed delay differential equations," Numerical Algorithms, vol. 52, no. 4, pp. 509-521, 2009.

 F. Ghomanjani, M. H. Farahi, and M. Gachpazan, "Bezier control points method to solve constrained quadratic optimal control of time varying linear systems," Computational & Applied Mathematics, vol. 31, no. 3, pp. 433-456, 2012.

 F. Ghomanjani, A. Kilicman, and S. Effati, "Numerical solution for IVP in Volterra type linear integro-differential equations system," Abstract and Applied Analysis, vol. 2013, Article ID 490689, 4 pages, 2013.

F. Ghomanjani, (1) A. Kilicman, (2) and F. Akhavan Ghassabzade (1)

(1) Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

(2) Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to A. Kilicman; akilic@upm.edu.my

Received 4 September 2013; Accepted 24 December 2013; Published 16 January 2014

```
TABLE 1: The maximum error for [member of] = 0.1 and for
different [delta] for Example 1.

5        Max error      Max error of
in      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

TABLE 2: The maximum error for [member of] = 0.1 and for
different [delta] for Example 2.

8         Max error of
presented method

0.01          0.007
0.03          0.022
0.06          0.023
0.08          0.025
```
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