GAUSSIAN QUADRATURE METHOD FOR SOLVING DIFFERENTIAL DIFFERENCE EQUATIONS HAVING BOUNDARY LAYERS

Abstract: In this paper, the Gaussian quadrature method is described for the solution of differential difference problems having boundary layers. The given problem is replaced by an asymptotically equivalent first order differential equation with the perturbation parameter as deviating argument. Then, Gaussian two point quadrature is implemented to solve this first order differential equation with perturbation parameter as deviating parameter. Several numerical problems are illustrated to demonstrate the layer behaviour. Comparison of maximum errors in the solution of the problems is made with other methods available in the literature to demonstrate the applicability of the present method.


INTRODUCTION
Differential-difference equation models have stronger mathematical structure when associated with ODEs for the analysis of biosystem dynamics and they produce improved stability with the nature of the underlying processes and analytical results. Delay differential equations model the problems where there is after effect affecting at least one of the variables involved in the problem as compared to ordinary differential equations which model the problems 2815 GAUSSIAN QUADRATURE METHOD in which variables react to current conditions. Due to these reasons, differential-difference equation models are more preferred to ordinary differential equations. The category of differential-difference equations which have delay/advance and singularly perturbed behaviour is recognized as singularly perturbed differential-difference equations. In general, a singularly perturbed differential-difference equation is an ordinary differential equation, where a small positive parameter multiplies the highest order derivative, including at least one delay and/or advance parameters. Solutions of these equations exhibit variety of interesting phenomenon like rapid oscillations, turning point behaviour, boundary and interior layers. These problems possess the boundary layer characteristics. A boundary layer is an interval or region in which the solution changes rapidly. In these layers the physical variables change extremely rapidly over small domains in space or short intervals of time. This type of differential equations occur in the modelling of numerous practical phenomena in bioscience, engineering, control theory, such as in variational problems in control theory, in describing the human pupil-light reflex, in a variety of models for physiological processes or diseases and first exit time problems in the modelling of the determination of expected time for the generation of action potential in nerve cells by random synaptic inputs in dendrites. Stein [12] was first to study of bistable devices. Derstin et al [2], and variational problems in control theory Glizer [5] where they provide the best and in many cases the only realistic simulation of the observed. Lange and Miura [9,10] gave an asymptotic approach for a class of boundary-value problems for linear second-order differential-difference equations. Kadalbajoo and Sharma [6,7,8], 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. Analytical discussion on these problems ia available in the books O'Malley [11] Elsgolts and Norkin [4] and Driver [3].
In this paper, the Gaussian quadrature method is described for the solution of differential difference problems having boundary layers. The given problem is replaced by an asymptotically equivalent first order differential equation with the perturbation parameter as deviating argument.
Then, Gaussian two point quadrature is implemented to solve this first order differential equation where  is small parameter, Equation (4) is a second order singular perturbation problem. Here, We solve the equation (4) subject to the boundary conditions equation (2) by using the Gaussian two point quadrature.

Problem with Left-end Boundary Layer
Using Taylor's expansion about the point s, we have With these Eqn. (6) and Eqn. (4) we get first order equation with ε as deviating argument: The domain [0, 1] is partitioned into N sub domains of mesh size Using Gaussian two-point quadrature formula, we have ( ) Using the linear interpolation for ( ) ( ) ( ) ( ) 11 , , and Rearranging this equation, we have Eqn. (12) can be rewritten in a three term recurrence relation as follows: The tri-diagonal system is solved efficiently by Thomas Algorithm Angel and Bellman [1].

Problem with Right -end Boundary Layer
Again by Taylor series expansion we have accordingly the Eqn. (4) is reduced to the first order equation with ε as the deviating argument:  (17) we get; We arrange this as the three term recurrence relation: The system of Eqn. (20) is solved by using Thomas algorithm given in Angel and Bellman [1].

NUMERICAL EXPERIMENTS
We