Uniform asymptotic expansion of the solution for the initial value problem with a piecewise constant argument

The article is devoted to the study of a singularly perturbed initial problem for a linear differential equation with a piecewise constant argument second-order for a small parameter. This paper is considered the asymptotic expansion of the solution to the Cauchy problem for singularly perturbed differential equations with piecewise-constant argument. The initial value problem for first order linear differential equations with piecewise-constant argument was obtained that determined the regular members. The Cauchy problems for linear nonhomogeneous differential equations with a constant coefficient were obtained, which determined the boundary layer terms. An asymptotic estimate for the remainder term of the solution of the Cauchy problem was obtained. Using the remainder term, we construct a uniform asymptotic solution with accuracy O(εN+1) on the θi ≤ t ≤ θi+1, i = 0, p segment of the singularly perturbed Cauchy problem with a piecewise constant argument.