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

Abstract

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.

More by Aziza Mirzakulova

• Boundary Layers in Two‐Point Boundary Value Problems for Doubly Singularly Perturbed Third‐Order Integro‐Differential EquationsMathematical Methods in the Applied Sciences · 2026
• Asymptotic Expansion of the Solution for a Singularly Perturbed Boundary Value Problem with Unbounded Boundary ConditionLobachevskii Journal of Mathematics · 2026
• ASYMPTOTIC CONVERGENCE OF THE CAUCHY PROBLEM SOLUTION FOR AN INTEGRO–DIFFERENTIAL EQUATION WITH A SMALL PARAMETERHerald of Kazakh-British technical university · 2026 · Citations: 1
• ASYMPTOTIC ESTIMATES OF THE SOLUTION OF THE BVP FOR SINGULARLY PERTURBED INTEGRO-DIFFERENTIAL EQUATIONSJournal of Mathematics Mechanics and Computer Science · 2026