1. Introduction

A class of functional differential equations which have the characteristics of both classes i.e., delay and/or advanced and singularly perturbed behaviour are known as Singularly Perturbed Differential-Difference Equations (SPDDEs). These differential-difference equation models have richer mathematical framework for the analysis of bio system dynamics, compared with ordinary differential equations. Also, they exhibit better consistency with the nature of the underlying process and predictive results. SPDDEs provide mathematical models in optics, physiology, mechanics, epidemiology, neural networks as well as many other applications Ref. ^{[1, 2, 3, 4]}. As a result, numerical treatment of SPDDEs has received a great deal of attention. Stein ^[5] gave stochastic effects due to neuronal excitation with help of a model that represents a differential-difference equation. Lange and Miura ^{[6, 7, 8, 9, 10]} published a series of papers to study the numerical solution of SPDDEs. They extended the matched asymptotic expansions, initially developed for ordinary differential equations, to obtain approximate solutions to differential difference equations. The same authors ^[7] extended the problems involving boundary and interior layer phenomenon, rapid oscillations and resonance behaviour in ^[6] and ^[10] to the problems having solutions which exhibit turning point behaviour i.e., transition regions between rapid oscillations and exponential behaviour. Kadalbajoo and Sharma ^[11] devised a finite difference scheme to obtain the solution for singularly perturbed differential difference equations of mixed type i.e., which contain negative as well as positive shifts. The same authors ^[12] described a numerical approach based on finite difference method to solve a mathematical model arising from neuronal variability. Sharma and Kaushik ^[13] discussed two approaches, namely, fitted operator and fitted mesh method to solve SPDDEs with small delay as well as advanced, with layer behaviour. Amiraliyev and Cimen ^[14] used an exponentially fitted difference scheme on uniform mesh, accomplished by the method of integral identities, for linear second order delay differential equations. Prathima and Sharma ^[15] presented a numerical method to solve SPDDEs with negative shift and isolated turning point at x=0. Chakravarthy and Rao ^[16] presented a modified fourth order Numerov method for solving SPDDEs of mixed type.

With this motivation, in this paper, we discuss the solution of singularly perturbed differential-difference equations exhibiting dual layer using the higher order finite differences. First, the second order singularly perturbed differential-difference equations is replaced by an asymptotically equivalent second order singular perturbed ordinary differential equation. Then, fourth order stable finite difference scheme is applied to get a three term recurrence relation which is easily solved by Thomas algorithm. Some numerical examples have been solved to validate the computational efficiency of the proposed numerical scheme. To analyze the effect of the parameters on the solution, the numerical solution has also been plotted using graphs. The error bound and convergence of the method have also been established.

2. Description of the Method

Consider the singularly perturbed differential difference equation of mixed type, i.e., with delay as well as advance parameters of the form:

(1)

and subject to the interval and boundary conditions

(2)

(3)

where are bounded and continuously differentiable functions on (0,1), is the singular perturbation parameter; and and are the delay and the advance parameters respectively.

Using Taylor series expansion in the neighborhood of the point x, we have

(4)

(5)

Substituting (4) and (5) into Eq. (1), we obtain an asymptotically equivalent singularly perturbed boundary value problem of the form:

(6)

(7)

(8)

where

and .

The transition from Eq.(1) to Eq. (6) is admitted, because of the condition that and are sufficiently small. Further details on the validity of this transition can be found in Elsgolt’s and Norkin ^[17]. If on the interval , then the solution of Eq. (1) exhibits boundary layers at both ends of the interval , whereas it exhibits oscillatory behaviour for . Here, we have considered the first case where the solutions of the problem exhibit dual layers. Dividing the interval [0, 1] into N equal parts with constant mesh length h, with mesh points:, we have .Assuming that has continuous fourth derivatives on and by making use of ’s expansion, we have the fourth order central differences:

(9)

(10)

where and for .

Substituting (9) and (10) into Eq. (6) we can write the central difference approximation of Eq.(6) in the form that includes all the error terms as follows:

(11)

where .

Now, from Eq.(6) we have

(12)

Differentiating both sides of (12), we get:

(13)

(14)

Using (13) and (14) into (11) we obtain

(15)

Now, approximating the converted error term, which has a stabilizing effect, in (15) by using the central difference formulas (9) and (10) for and we obtain

(16)

Where

is the truncation error and .

Rearranging Eq. (16) we obtain the three term recurrence relation of the form

(17)

for i = 1 (1) N-1.

where

This gives us the tri-diagonal system which can easily be solved by Thomas Algorithm.

3. Thomas Algorithm

We give a brief description of Thomas algorithm for solving the tri-diagonal system. Consider the system:

(18)

subject to the boundary conditions

(19)

(20)

We set

(21)

for ,

where and which are to be determined.

From (21), we have:

(22)

By substituting (22) in (18) and comparing with (21) we get the recurrence relations:

(23)

(24)

To solve these recurrence relations for , we need the initial conditions for and . For this we take. We choose so that the value of . With these initial values, we compute and for from (23) and (24) in forward process, and then obtain in the backward process from (20) and (21).

4. Error Analysis

In this section we discuss the error analysis of the method. Writing the tri-diagonal system (18) in matrix-vector form, we get

(26)

where, is a tri diagonal matrix of order N-1, with

is a column vector with , where with local truncation error

(27)

where

We also have

(28)

denotes the actual solution and

is the local truncation error.

From (26) and (28), we get

(29)

Thus, we obtain the error equation

(30)

where .

Let be the sum of elements of the i^th row of A, then we have

is a column vector with , where with local truncation error

Since and , for sufficiently small the matrix A is irreducible and monotone Ref. ^[18]. Then it follows that exists and its elements are non negative.

Hence, from Eq. (30), we get

(31)

and

(32)

Let be the element of . Since , from the theory of matrices we have

(33)

Therefore,

(34)

We define and .

From (23), (27), (28) and (30), we get

which implies

(35)

where .

Therefore,

Hence, the method gives a fourth order convergence for uniform mesh as given in Ref. ^[19].

5. Numerical Examples

To validate the computational efficiency of the scheme, we have applied it to two cases of the problem (1)-(2,3). These cases have been chosen because they have been widely discussed in literature and because approximate solutions are available for comparison.

and subject to the conditions

The exact solution of such boundary value problems having constant coefficients (i.e. and are constants) is given by:

(36)

where

Example 1: Consider the model problem given by equations (1)-(3) with

The exact solution of the problem is given by Eq. (36). The numerical results are given in Table 1-Table 4 for =0.1, 0.01 and different values of the delay and advance parameters.

Table 1. Numerical solution of Example 1 for ε=0.1; δ= 0.07 and N=100

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Table 2. Numerical solution of Example 1for ε=0.1; δ= 0.05 and N=100

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Table 3. Numerical solution of Example 1 for ε=0.01; δ= 0.005 and N=100

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Table 4. Numerical solution of Example 1 for ε=0.01; δ= 0.007 and N=100

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Example 2: Consider the model problem given by equations (1)-(3) with

The exact solution of the problem is given by Eq. (36). The numerical results are given in Table 5- Table 8 for =0.1, 0.01 and different values of the delay and advance parameters.

Table 5. Numerical solution of Example 2 for ε=0.1; δ= 0.07 and N=100

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Table 6. Numerical solution of Example 2 for ε=0.1; η=0.05 and N=100

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Table 7. Numerical solution of Example 2 for ε=0.01; η=0.007 and N=100

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Table 8. Numerical solution of Example 2 for ε=0.01; δ=0.005 and N=100

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Figure 1. Numerical solution of Example 1 for ε=0.1 and δ=0.07

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6. Discussion and Conclusion

In this paper, we have discussed the solution of singularly perturbed differential-difference equations exhibiting dual layer using the higher order finite differences. First, the second order singularly perturbed differential-difference equations is replaced by an asymptotically equivalent second order singular perturbed ordinary differential equation. Then, fourth order stable finite difference scheme is applied to get a three term recurrence relation which is easily solved by Thomas algorithm. The error bound and convergence of the method have also been established. Some model examples (i.e., examples in which both the negative and positive shifts are non zero) have been solved to demonstrate the applicability of the proposed approach, by taking various values of the delay, advanced and perturbation parameters. These cases have been chosen because they have been widely discussed in literature Ref. ^{[20, 21, 22]} and also because the exact solutions are available for comparison. The numerical solutions are presented in tables and compared with the exact solution. To analyze the effect of the parameters on the solution, the numerical solution has also been plotted using graphs. From the graphs, it is observed that by varying either or the thickness of the left boundary layer decreases and that of the right boundary layer increases. The effect of negative shift is more dominant than the positive shift.

Figure 2. Numerical solution of Example 1 for ε=0.1 and η=0.05

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Figure 3. Numerical solution of Example 1 for ε=0.01 and δ= 0.007

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Figure 4. Numerical solution of Example 2 for ε=0.1 and δ=0.07

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Figure 5. Numerical solution of Example 2 for ε=0.1 and η=0.05

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Figure 6. Numerical solution of Example 2 for ε=0.01 and η=0.007

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