Published Paper Details:

Delay differential equation refers to an ordinary differential equation (ODE) with a delay parameter. The term "advanced differential equation" is used to describe an ODE with a forward parameter. The term "differential-difference equation" is used to refer to an ODE that includes both delay and advance parameters. Similar to the usage of "advance" and "delay" words, "positive shift" and "negative shift" appear in the literature. When the highest order derivative is multiplied by a small parameter, the boundary layer phenomenon appears in the solution of the differential-difference equation. The boundary layer is an area where the answer is constantly evolving. These issues are typically addressed by means of perturbation techniques, such as the matched asymptotic expansions and WKB approach. Skill, insight, and experimentation are needed for these asymptotic expansions. As a result, scientists have begun to rely on numerical approaches. Due to the presence of the boundary layer, oscillatory/unsatisfactory solutions are obtained if we apply the existing numerical methods with the step-size greater than the perturbation parameter. In addition, the solution is graphically represented so that the effects of the delay and complex factors may be comprehended. It is noticed from the tables and figures that our quantitative approaches produce very good approximation to the exact solution. In addition to being theoretically simpler and easier to use, these quantitative approaches are also easily adaptable for computer implementation with only a small amount of computing effort.