Numerical analysis of singularly perturbed delay differential turning point problem

In this paper, we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts. Similar boundary value problems are encountered while simulating several real life processes for instance, first exit time problem in the modelling of neuronal variability. A rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem. In the development of numerical methods for constructing an approximation to the solution of the problem, a special type of mesh is generated to tackle the delay term along with the turning point. Then, to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term, an exponential fitting parameter is used. Several numerical examples are presented to support the theory developed in the paper.