Analytical solution of initial value problem for ordinary differential equation with singular perturbation and piecewise constant argument

Abstract

The study of differential equations with piecewise constant arguments has been treated widely in the literature. This type of equation, in which techniques of differential and difference equations are combined, models, among others, some biological phenomena, the stabilization of hybrid control systems with feedback discrete controller or damped oscillators. The first studies in this field have been given in 1984, after this, some papers related with stability, oscillation properties and existence of periodic outcomes have been treated by several authors. The manuscript is crafted as follows: Section 2 outlines the primary methodologies adopted throughout the inquiry. Section 3 is dedicated to obtaining the exclusive outcome to the issue. We formulate a series of difference equations overseeing the vector ( y (θ_i ), y^/(θ_i) ) , i = 1,...,p  which portray the constituents of the outcome. This generalized approach allows for a broader understanding of how to tackle such differential equations across various scenarios. These equations now form a recognized branch of the field of differential equations, and they are frequently used in biological and economic models. Undoubtedly, their applications will continue to increase in the future.