ASYMPTOTIC EXPANSION OF THE SOLUTION OF THE VALLEE POUSSIN PROBLEM ON A SEGMENT
DOI:
https://doi.org/10.52754/16948610_2026_2_22Keywords:
small parameter, singularly perturbed Vallee Poussin problem, turning point, unstable spectrum, smooth outer solutionAbstract
Relevance. The article studies a system of inhomogeneous singularly perturbed n differential equations. Systems of this type are currently among the most important and are widely used for modeling complex processes in optimal control theory, fluid mechanics, and electrodynamics.
Among such systems, the Valle-Poussin problem is studied as a relevant example. A distinctive feature of the problem under consideration is the presence of a small parameter multiplying the highest derivative, as well as the instability of the spectrum of the coefficient matrix of the linear part of the system at three points of the considered interval. The main objective of this work is to investigate the influence of these three specific points on the behavior of the solution.
A uniform asymptotic expansion of the solution to the boundary value problem is constructed by means of the method of generalized boundary functions developed by K. Alymkulov. An asymptotic expansion of the solution is obtained, and an estimate for the remainder term is derived.
In the article, the Valle-Poussen problem was studied for the case where the number of differential equations in the system is four, and the results obtained for this case were presented in a generalized form for a system with n differential equations.
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