A FITTED METHOD FOR SINGULARLY PERTURBED DIFFERENTIAL- DIFFERENCE EQUATIONS HAVING BOUNDARY LAYERS VIA DEVIATING ARGUMENT AND INTERPOLATION

In this paper, singularly perturbed differential-difference equation having boundary layers at one end (left or right) is considered. In order to obtain numerical solution to these problems, the given second order equation having boundary layer is converted into a singularly perturbed ordinary differential equation using Taylor’s transformation afterwards the resultant singularly perturbed ordinary differential equation is replaced by an asymptotically equivalent to first order differential equation with a small deviating argument. Resulting first order differential equation, is solved by choosing the proper integrating factor (fitting factor) and linear interpolation formulas. The numerical results for several test examples demonstrate the applicability of the method.


INTRODUCTION
Singularly perturbed delay differential equation is a differential equation in which the highest order derivative is multiplied by a small parameter and involving a delay term. This type of equation arises frequently in the mathematical modelling of various practical phenomena for example: in the modelling of the human pupil-light reflex; model of HIV infection; the study of bi-stable devices in digital electronics; variational problem in control theory; first exit time problem in modelling of activation of neuronal variability; immune response; evolutionary biology; dynamics of networks of two identical amplifier; mathematical ecology; population dynamics; the modelling of biological oscillator and in a variety of models for physiological process. For a detailed theory and analytical discussion on delay differential equations having boundary layer one may refer the popular books by Bellman and Cooke [1], Driver [4], El'sgol'ts and Norkin [5], Hale [8], Nayfeh [17], O'Malley [18] and VanDyke [24]. Lange and Miura [12][13] are the first to discuss the behavior of the analytical solution of singularly perturbed differential difference equations. Chakravarthy and Reddy [3] presented an initial-value approach for solving singularly perturbed two-point boundary value problems. Fevzi Erdogan [6] described an exponentially fitted method for singularly perturbed delay differential equations.
Gemechis and Reddy [7] discussed the numerical Integration of a class of Singularly Perturbed Delay Differential Equations with small shift. Kadalbajoo and Sharma [9][10] described the numerical treatment of a mathematical model arising from a model of neuronal variability.
Lakshmi Sirisha and Reddy [11] presented a Fitted second order scheme for solving Singularly Perturbed Differential Difference Equations. Nageshwar Rao and Pramod Chakrravarthy [14][15] described a Fitted Numerov method for singularly perturbed parabolic partial differential equation with a small negative shift arising in control theory. Natesan and Bawa [16] presented a second order numerical scheme for singularly perturbed reaction-diffusion robin problems. Ravi Kanth and Murali [19] described a numerical approach for solving singularly perturbed convection delay problems via exponentially fitted spline method. Reddy and Awoke [20] discussed the solving singularly perturbed differential difference equations via fitted method.
Reddy and Soujanya and Phaneendra [21] presented the Numerical integration method for singularly perturbed delay differential equations. Salama, A.A and Al-Amery [22] presented an 2469 SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS Asymptotic-numerical method for singularly perturbed differential difference equations of mixed-type. Soujanya, Reddy and K. Phaneendra [22] discussed the Numerical Solution of Singular Perturbation Problems via Deviating Argument and Exponential Fitting.
In this paper, singularly perturbed differential-difference equation having boundary layers at one end (left or right) is considered. In order to obtain numerical solution to these problems, the given second order equation having boundary layer is converted into a singularly perturbed ordinary differential equation using Taylor's transformation afterwards the resultant singularly perturbed ordinary differential equation is replaced by an asymptotically equivalent to first order differential equation with a small deviating argument. Resulting first order differential equation, is solved by choosing the proper integrating factor (fitting factor) and linear interpolation formulas. The numerical results for several test examples demonstrate the applicability of the method.
Since 0 < ≪ 1, the transition from Equation (1) to Equation (5) is admitted. For more details on the validity of this transition, one can refer El'sgolt's and Norkin [5]. Here we assume that ( ) = and ( ) = are constants.
For more details on the validity of this transition, one can refer El'sgolt's and Norkin [5]. The behaviour of the boundary layer is given by the sign of ( ) and ( ).

Case (i) : For Left-end boundary layer
Consider equation (5) or (13) with their boundary conditions From Taylor's series expansion about the deviating argument √ ′ in the neighbourhood of the point , we have From equation (19) and (22), we have The transition from equation (19) to (23) is valid, because of the condition that √ ′ is small. For more details on the validity of this transition, one can refer El'sgolt's and Norkin [5].
Equation (23) can be written as We take an integrating factor − to equation (27) and producing (as in B. J. McCartin [2]) On integrating equation (28) from to +1 , we get Using the linear Newton's forward interpolation on [ +1 ], which we insert into the above equation, we get After evaluating the integrals involves in equation (31), we get From finite difference approximation, we have From equation (36) and equation (37), equation (32) becomes This is a tridiagonal system of − 1 equations. We solve this tridiagonal system with given two boundary conditions by Thomas algorithm.

Case (ii) : For Right-end boundary layer
Consider equation (5) or (13) with their boundary conditions From Taylor's series expansion about the deviating argument √ ′ in the neighbourhood of the point , we have From equation (39) and (42), we have The transition from equation (39) to (43) is valid, because of the condition that √ ′ is small. For more details on the validity of this transition, one can refer El'sgolt's and Norkin [5]. Now, we divide the interval [0, 1] into equal parts with constant mesh length ℎ = 1⁄ .
Equation (43) can be written as We take an integrating factor − to equation (47) and producing (as in B. J. McCartin [2]) On integrating equation (48) Using the linear Newton's backward interpolation on [ −1 ], which we insert into the above equation, we get After evaluating the integrals involves in equation (31), we get From equation (56) and equation (57), equation (52) becomes This is a tridiagonal system of − 1 equations. We solve this tridiagonal system with given two boundary conditions by Thomas algorithm.