SOLVABILITY OF A BOUNDARY VALUE PROBLEM OF THE SAMARSKY-IONKIN TYPE FOR DIFFERENTIAL EQUATIONS WITH INVOLUTION

ИНВОЛЮЦИЯЛЫ ДИФФЕРЕНЦИАЛДЫҚ ТЕҢДЕУЛЕР ҮШІН САМАРСКИЙ-ИОНКИН ТИПТЕС ШЕТТІК ЕСЕПТІҢ БІРМӘНДІ ШЕШІЛІМДІЛІГІ
РАЗРЕШИМОСТЬ КРАЕВОЙ ЗАДАЧИ ТИПА САМАРСКОГО–ИОНКИНА ДЛЯ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С ИНВОЛЮЦИЕЙ
Usmanov K.I. Nazarova K.Zh. Тurganbaeva Z.
2025 Kazakh-British Technical University

Herald of the Kazakh British Technical UNiversity
2025 #22 Issue 1 173 - 183 pp.

In this paper, we investigate a non-local boundary value problem for a second-order differential equation with an involutive transformation. The aim of this work is to apply the parametrization method developed by Professor D. Dzhumabayev to study the solvability of non-local boundary value problems in the context of differential equations with involutive transformations. It is known that the Cauchy problem for such equations may not have a unique solution. Therefore, parameters (formula presenetd) are introduced, and variables are replaced. The parameter values (formula presenetd) are determined at the midpoint of the interval, which guarantees the existence of a unique solution for the Cauchy problem of the original equation. The performed variable substitution formally divides the problem into two components: the Cauchy problem for the initial equation and a system of linear equations with respect to the introduced parameters. By solving the Cauchy problem and substituting its solution into the boundary conditions, one can obtain a system of linear equations with respect to the parameters. If the matrix of this system is invertible, then the problem has a unique solution. In the case when the matrix is non-invertible, two scenarios are possible: either the boundary value problem is unsolvable, or it has multiple solutions. For the second case, the paper defines the eigenvalues and solvability conditions of the boundary value problem.

Cauchy problem , Dzhumabayevs method , eigenvalues , parameter , system of linear equations , unique solvability

Text of the article Перейти на текст статьи

Akhmet Yassawi International Kazakh-Turkish University, Turkistan, Kazakhstan

Akhmet Yassawi International Kazakh-Turkish University

10 лет помогаем публиковать статьи Международный издатель

Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026