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Parameter uniform numerical method for singularly perturbed differential–difference equations with interior layers

The present study is devoted to the numerical study of boundary value problems for singularly perturbed linear second-order differential–difference equations with a turning point. The points of the domain where the coefficient of the convection term in the singularly perturbed differential equation vanishes are known as the turning points. The solution of such type of differential equations exhibits boundary layer(s) or interior layer(s) behaviour depending upon the nature of the coefficient of convection term and the reaction term. In particular, this paper focuses on problems whose solution exhibits interior layers. In the development of numerical schemes for singularly perturbed differential–difference equations with a turning point, we use El-Mistikawy–Werle exponential finite difference scheme with some modifications. Some priori estimates have been established and parameter uniform convergence analysis of the proposed scheme is also discussed. Several examples are considered to demonstrate the performance of the proposed scheme and effect of the size of the delay/advance arguments and coefficients of the delay/advance term on the layer behaviour of the solution.