Enhancing Numerical Approximations for Scalar Linear Delay Differential Equations with Constant Delay and Impulsive Self Support Conditions

Abstract

Through the course of this study, a numerical approximation approach was developed for a category of scalar delay differential equations that were subject to impulsive conditions. Each and every one of these equations was taken into consideration. An innovative method of approximation is used, which involves the utilization of delay equations that have piecewise constant arguments. The theoretical convergence of the approximation technique was shown at each and every point, with the exception of the instants of time that were impulsive. Additionally, the numerical method does a decent job of estimating the instants of time that are impulsive. In the event that the β parameter of the problem is positive, we possess the capability to establish the convergence conclusion by imposing an additional condition via the use of an additional condition. Without making the assumption of a self-supporting condition, this condition guarantees that the solution
will cross the line x=c at a temporal interval that is impulsive. The conclusions of the theoretical convergence were shown by the presentation of numerical examples. Through the use of numerical study, we were able to identify the existence of periods in the solution. In this particular solution, the minimal period is defined by being bigger than the delay τ that is associated with the model. We take note of the fact that sufficient conditions are presented to guarantee the existence of periodic solutions; nonetheless, numerical examples illustrate that periodic solutions express themselves under criteria that are considerably less strict than those that guarantee their existence. In light of this, there is a pressing need for more study to be conducted in order to determine whether or not periodic solutions are accessible for this particular set of impulsive equations.