Прохоренко, М. В. та Прохоренко, С. В. та Мороз, М. В. та Соляк, Л. В. та Prokhorenko, M. V. та Prokhorenko, S. V. та Moroz, M. V. та Solіak, L. V. (2019) ЗАДАЧА З ІМПУЛЬСНОЮ ДІЄЮ ДЛЯ ОДНОГО РІВНЯННЯ КОЛИВАНЬ СТРУНИ. Вісник Національного університету водного господарства та природокористування (3(87)). с. 119-127.

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Анотація

Досліджено процес коливання струни з імпульсною дією при досягненні енергії коливання струни заданого критичного рівня. Визначено умови існування розв’язків, для яких імпульсна дія відбувається нескінчену кількість разів та записано вигляд розв’язків.

Title in English

PROBLEM WITH IMPULSE ACTION FOR A SINGLE EQUATION OF THE STRINGS FLUCTUATION

English abstract

The development of modern science and technology requires the study of problems for differential equations describing processes with short-term changes, or which are influenced by external forces, the duration of which can be neglected when compiling relevant mathematical models. Such tasks are found in thermophysics, mechanics, chemical technologies, biology, management theory and other branches of science and technology, where they study processes under the influence of short-term forces and called systems with impulse action. Such systems contain differential equations, an equation describing gaps in the first kind at moments of impulse action and impulse conditions. The presence of impulse action substantially changes and complicates the behavior of the trajectories of such problems, even for relatively simple differential equations. The study of differential equations with impulse action began with the development of nonlinear mechanics and attracted the attention of many researchers to the possibility to really describe the processes in nonlinear systems. In spite of the large number of works devoted to various questions of the theory of differential equations with impulse action, a large number of problems of the qualitative theory of such equations remain open for many cases. There is the influence of impulse action in fixed and non-fixed moments of time. In this paper we consider the process of oscillation of a string with instantaneous increase of energy at moments when the full energy of a string reaches this critical level. That is, the moments of impulse action are not predetermined, but regulated by the process itself. For such a task there are conditions for the existence of solutions for which the impulse action is carried out an infinite number of times and constructed such solutions.

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