Abstract
This paper examines the Cauchy problem for a singularly perturbed differential equation that has a piecewise constant argument. Various types of singular perturbation problems have been discussed in the literature. Notably, this study considers an extension of Tikhonov’s theorem for singularly perturbed differential equations with a piecewise-constant generalized argument. In our current research, we construct a uniform asymptotic approximation for solutions that is valid over the entire closed interval, utilizing the boundary function method. We develop an asymptotic expansion for the solution of a singularly perturbed initial problem, achieving an arbitrary degree of accuracy for a small parameter. Additionally, we present an algorithm that helps determine the asymptotic terms associated with the solution. We formulate a theorem to estimate the residual term of the asymptotic expansion, which quantifies the difference between the exact solution and its approximation. Furthermore, we observe a humping phenomenon within the boundary layer. Lastly, an illustrative example showcasing modeling is provided.