A FITTED DEVIATING ARGUMENT AND INTERPOLATION SCHEME FOR THE SOLUTION OF SINGULARLY PERTURBED DIFFERENTIAL- DIFFERENCE EQUATION HAVING LAYERS AT BOTH ENDS

In this paper, problem of singularly perturbed differential-difference equation having boundary layers at both ends is solved and analyzed numerically by fitted method. To do this, original problem is transformed into an asymptotically equivalent singularly perturbed differential equation by Taylor’s series expansion. By introducing deviating argument concept, SPDE is replaced by first order differential equation. Resulting equation having deviating argument is solved with proper choice of fitting factor and interpolation. To demonstrate the applicability of this numerical method, three test examples are solved and numerical results are compared with the available/exact results.


INTRODUCTION
A singularly perturbed delay differential equation is an ordinary differential equation in which the highest derivative is multiplied by a small parameter and containing delay term. In recent years, there has been a growing interest in the numerical treatment of such differential equations. This is due to the versatility of such type of differential equations in the mathematical modeling 857 SOLUTION OF SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATION of processes in various application fields, for e.g., the first exit time problem in the modeling of the activation of neuronal variability, in the study of bistable devices, and variational problems in control theory where they provide the best and in many cases the only realistic simulation of the observed. Stein [12] gave a differential-difference equation model incorporating stochastic effects due to neuron excitation. Lange and Miura [4][5] gave an asymptotic approach for a class of boundary-value problems for linear second-order differential-difference equations.
Kadalbajoo and Sharma [9][10], presented a numerical approaches to solve singularly perturbed differential-difference equation, which contains negative shift in the either in the derivative term or the function but not in the derivative term. Asymptotic-numerical method for singularly perturbed differential difference equations of mixed-type is discussed by Salama and Al-Amery [1]. Erdogan [6], has presented an Exponentially fitted method for singularly perturbed delay differential equations. Venkat and Palli [15] presented a numerical approach for solving singularly perturbed convection delay problems via exponentially fitted spline method. Rao and Chakrravarthy [16][17] have described a finite difference method for singularly perturbed differential-difference equations arising from a model of neuronal variability. Reddy and Chakravarthy [20] presented an initial-value approach for solving singularly perturbed two-point boundary value problems. Reddy et al [19] described a numerical integration method for singularly perturbed delay differential equations. Reddy and Awoke [18] presented a method for solving singularly perturbed differential difference equations via fitted method, In this paper, problem of singularly perturbed differential-difference equation having boundary layers at both ends is solved and analyzed numerically by fitted method. To do this, original problem is transformed into an asymptotically equivalent singularly perturbed differential equation by  [2,7,11,13,14].

DESCRIPTION OF THE FITTED METHOD
Consider a class of differential-difference equation with small shifts of mixed type From Equations (4), (5) and (1), we obtain singularly perturbed differential equation with the boundary conditions  El'sgolt's and Norkin [8].
Since problem exhibits two boundary layers across the interval, we divide the interval [0, 1] into two sub intervals [0, 1 2 ] and [ 1 2 , 1]. Clearly in the interval [0, 1 2 ] the boundary layer will be at the left end i.e. at = 0, and in the interval [ 1 2 , 1] the boundary layer will be at right end i.e. at = 1.

Problem with left end boundary layer in [ , ]
From Taylor's series expansion about the deviating argument √ ′ in the neighbourhood of the point , we have From Equations (9) and (6), we get where The transition from equation (6) to (10) is valid, because of the condition that √ ′ is small. For more details on the validity of the transition, one can refer El'sgolt's and Norkin [8].
Now, we divide the interval [0, 1] into equal parts with constant mesh length ℎ.

Problem with right end boundary layer in [ , 1]
From Taylor's series expansion about the deviating argument √ ′ in the neighbourhood of the point , we have From Equations (24) and (6)  With this equation (38), we now have + 1 equations to solve for the unknowns ( 0 , 1 , … , ).
Using invariant imbedding algorithm also knowns as Thomas algorithm, we get the solution.

NUMERICAL EXPERIMENTS
In this section to demonstrate the applicability of the method , we tested it on three standard model examples and solutions obtained from this scheme are compared with the available/exact solutions.
The exact solution of the differential -difference equation  The exact solution is given by Eqn. (39). Results are shown in Table-

DISCUSSION AND CONCLUSIONS
A fitted numerical scheme is presented for solving singularly perturbed differential-difference equations having layers at both ends. In this scheme deviating argument and Hermite interpolation concepts are used. This scheme is implemented on three standard examples for those numerical solutions are found to be in agreement with available or exact solution.
Numerical, exact results and layer behaviour are presented in their respective figures and tables for different values of the parameters. This scheme is very simple and easy to implement on the class of singularly perturbed differential-difference equations having layers at both ends.