A robust Simpson’s 3/8 numerical integration method to solve singularly perturbed delay differential and differential–difference equations having boundary layer at one end

In this study, we develop a numerical integration method to solve singularly perturbed delay differential and differential–difference equations both characterized by a boundary layer at one end. Applying Taylor’s series, we approximate the second-order delay differential and differential–difference equations with an asymptotically equivalent first-order differential equation. To solve this equation, we employ the composite Simpson’s 3/8 rule, which leads to the formulation of a three-term recurrence relation. The resulting tri-diagonal system of equations is efficiently solved using the Thomas algorithm. We perform numerical experiments to investigate the effect of small shifts in the delay parameter on the boundary layer solution, focusing on cases where the boundary layer forms either at the left or right boundary. In addition, we carry out a thorough error analysis to assess the accuracy of the method and provide a discussion of its stability and convergence properties. Our numerical results show that the proposed method yields highly accurate approximations, closely matching the exact solutions.