Numerical Integration of a Class of Singularly Perturbed Delay Differential Equations with Small Shift

We have presented a numerical integration method to solve a class of singularly perturbed delay differential equations with small shift. First, we have replaced the second-order singularly perturbed delay differential equation by an asymptotically equivalent first-order delay differential equation. Then, Simpson’s rule and linear interpolation are employed to get the three-term recurrence relation which is solved easily by discrete invariant imbedding algorithm. The method is demonstrated by implementing it on several linear and nonlinear model examples by taking various values for the delay parameter δ and the perturbation parameter ε.


Introduction
The singularly perturbed delay differential equations with small shift arise very frequently in the modeling of various physical and biological phenomena, for example, micro scale heat transfer 1 , hydrodynamics of liquid helium 2 , second-sound theory 3 , thermoelasticity 4 , diffusion in polymers 5 , reaction-diffusion equations 6 , stability 7 , control of chaotic systems 8 , a variety of models for physiological processes or diseases 9 and so forth.Hence in the recent times, many researchers have been trying to develop numerical methods for solving these problems.Amiraliyev and Cimen 10 presented numerical method comprising a fitted difference scheme on a uniform mesh to solve second-order delay differential equations.Lange and Miura 11, 12 gave an asymptotic approach for a class of boundary-value problems for linear second-order differential-difference equations.Kadalbajoo and Sharma 13-15 presented numerical approaches to solve singularly perturbed differential-difference equations, which contains negative shift in the convention term i.e., in the derivative term .Lange and Miura 16 considered the boundary value International Journal of Differential Equations problem for a singularly perturbed nonlinear differential difference equation with shift and discussed the existence and uniqueness of their solutions.Furthermore, Kadalbajoo and Sharma 17 have discussed the numerical solution of the singularly perturbed nonlinear differential equations with small negative shifts.
In this paper, we have presented a numerical integration method for solving a class of singularly perturbed delay differential equations with small shift.First, the secondorder singularly perturbed delay differential equation is replaced by an asymptotically equivalent first-order delay differential equation.Then we employed Simpson's rule and linear interpolation to get three-term recurrence relation which is solved easily by discrete invariant imbedding algorithm.The method is demonstrated by implementing it on several linear and nonlinear model examples by taking various values for the delay and perturbation parameters.

Description of the Method
Consider a class of singularly perturbed boundary value problems of the following form: with the interval and boundary conditions where ε is small parameter, 0 < ε 1, and δ is also a small shifting parameter, 0 < δ 1; b x , andf x are bounded continuous functions in 0, 1 , and α, β are finite constants.Further, we assume that a x ≥ M > 0 throughout the interval 0, 1 , where M is positive constant.This assumption merely implies that the boundary layer will be in the neighborhood of x 0.
By using Taylor series expansion in the neighborhood of the point x, we have and consequently, 2.1 is replaced by the following first-order differential equation: where The transition from 2.1 to 2.4 is admitted, because of the condition that ε is small, 0 < ε 1.This replacement is significant from the computational point of view.Further details on the validity of this transition can be found in 18 .
International Journal of Differential Equations 3 Now we divide the interval 0, 1 into N equal subintervals of mesh size h 1/N so that x i ih, i 0, 1, 2, . . ., N.
Integrating 2.4 with respect to x from x i to x i 1 for 1, 2, . . ., N − 1, we get where y i y x i , p i p x i , q i q x i , r i r x i , s i s x i .By using Simpson's rule to evaluate the integral in 2.6 , we get

2.7
By the means of Taylor series expansion and then by approximating y x by linear interpolation, we get In similar way, International Journal of Differential Equations Hence, by making use of 2.8a -2.8e in 2.7 we obtain

2.9
To make 2.9 a three-term recurrence relation, we can express y i 1/2 in terms of y i−1 , y i and y i 1 using Hermite's interpolation as follows:

2.10
In view of 2.4 and 2.10 , we get

2.11
By making use of 2.8a -2.8e in 2.11 and finite difference approximations, we get

2.14
This tridiagonal system is solved by using method of discrete invariant imbedding algorithm which is described in the next section.

Discrete Invariant Imbedding Algorithm
We now describe the Thomas algorithm which is also called discrete invariant imbedding 19 to solve the three-term recurrence relation: Let us set a difference relation of the form where W i W x i and T i T x i are to be determined.From 3.2 , we have

3.3
Substituting 3.3 in 3.1 , we have International Journal of Differential Equations By comparing 3.2 and 3.4 , we get the recurrence relations To solve these recurrence relations for i 1, 2, 3, . . ., N − 1, we need the initial conditions for W 0 and T 0 .If we choose W 0 0, then we get T 0 α.With these initial values, we compute W i and T i for i 1, 2, 3, . . ., N − 1 from 3.5 and 3.6 in forward process and then obtain y i in the backward process from 3.2 .
The conditions for the discrete invariant imbedding algorithm to be stable are see 18-21 In our method, one can easily show that if the assumptions a x > 0, b x < 0 and ε − δa x > 0 hold, then the above conditions 3.7 hold, and thus the discrete invariant imbedding algorithm is stable.

Numerical Experiments
To demonstrate the applicability of the method, we have implemented it on two linear and two nonlinear problems with left-end boundary layers.Computational results are compared with exact solutions wherever exact solutions are available.When exact solution is not available, we have tested the effect of small delay parameter on solution of the problem for different values of δ of o ε .

Linear Problems
Example 4.1.Consider an example of singularly perturbed delay differential equation with left layer: The exact solution is given by  εy x e −0.5x y x − δ − y x 0 with y 0 1, y 1 1.

4.3
For which the exact solution is not known.This example is considered to show the effect of the small shift on the boundary layer solution.
The computational results are presented in Tables 5 and 6 for ε 0.001 and 0.0001 for different values of δ.

Nonlinear Problems
Nonlinear problems are linearized by the quasilinearization process.Then we have applied the present method.

4.5
The exact solution is not known.The computational results are presented in Tables 7 and 8  The exact solution is not known.The computational results are presented in Tables 9 and 10 for ε 0.01 and 0.001 for different values of δ.

Discussions and Conclusions
We have presented a numerical integration method to solve singularly perturbed delay differential equations.The scheme is repeated for different choices of the delay parameter,
for ε 0.01 for different values of δ.Example 4.4.Consider an example of singularly perturbed nonlinear delay differential equation: εy x 2y x − δ e y x 0

Table 5 :
Numerical results of Example 4.2 for ε 0.001, N 100, and different values of δ.

Table 6 :
Numerical results of Example 4.2 for ε 0.0001, N 100, and different values of δ.

Table 7 :
Numerical results of Example 4.3 for ε 0.001, N 100, and different values of δ.

Table 8 :
Numerical results of Example 4.3 for ε 0.0001, N 100, and different values of δ.

Table 9 :
Numerical results of Example 4.4 for ε 0.001, N 100, and different values of δ.

Table 10 :
Numerical results of Example 4.4 for ε 0.0001, N 100, and different values of δ.