Asymptotic Convergence of Solutions for Singularly Perturbed Linear Impulsive Systems with Full Singularity
Aviltay N. Dauylbayev M.
September 2025Multidisciplinary Digital Publishing Institute (MDPI)
Symmetry
2025#17Issue 9
This paper considers impulsive systems with singularities. The main novelty of this study is that the impulses (impulsive functions) and the initial value are singular. The asymptotic convergence of the solution to a singularly perturbed initial problem with an infinitely large initial value, as (Formula presented.) to the solution to a corresponding modified degenerate initial problem is proved. It is established that the solution to the initial problem at point (Formula presented.) has an initial jump phenomenon, and the value of this initial jump is determined. The theoretical results are supported by illustrative examples with simulations. Singularly perturbed problems are characterized by the presence of a small parameter multiplying the highest derivatives in the differential equations. This leads to rapid changes in the solution near the boundary or at certain points inside the domain. In our problem, symmetry is violated due to the emergence of a boundary layer at the initial point and at the moments of discontinuity. As a result, the problem as a whole is asymmetric. Such asymmetry in the behavior of the solution is a main feature of singularly perturbed problems, setting them apart from regularly perturbed problems in which the solutions usually exhibit smoother changes.
differential equations with singular impulses , initial jump , singularly perturbation , small parameter
Text of the article Перейти на текст статьи
Department of Mathematics, Al-Farabi Kazakh National University, Almaty, 050040, Kazakhstan
Department of Mathematics
10 лет помогаем публиковать статьи Международный издатель
Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026