Об одном классе решений квазилинейных матричных дифференциальных уравнений
Дата
2020
Науковий керівник
Укладач
Редактор
Назва журналу
Номер ISSN
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Назва тому
Видавець
Одеський національний університет імені І. І. Мечникова
Анотація
Для квазилинейного матричного дифференциального уравнения, коэффициенты которого представимы в виде абсолютно и равномерно сходящихся рядов Фурье с медленно
меняющимися коэффициентами и частотой, получены достаточные условия существования решения аналогичной структуры
Для квазiлiнiйного матричного диференцiального рiвняння, коефiцiєнти якого зображуванi у виглядi абсолютно та рiвномiрно збiжних рядiв Фур’є з повiльно змiнними коефiцiєнтами та частотою, отримано достатнi умови iснування розв’язку аналогiчної структури.
In the mathematical description of various phenomena and processes that arise in mathematical physics, electrical engineering, economics, one has to deal with matrix differential equations. Therefore, these equations are relevant both for mathematicians and for specialists in other areas of natural science. Many studies are devoted to them, in which the solvability of matrix equations in various function spaces, boundary value problems for matrix differential equations, and other problems were investigated. In this article, a quasilinear matrix equation is considered, the coefficients of which can be represented in the form of absolutely and uniformly converging Fourier series with coefficients and frequency slowly varying in a certain sense. The problem is posed of obtaining sufficient conditions for the existence of particular solutions of a similar structure for the equation under consideration. For this purpose, the corresponding linear equation is considered first. It is written down in component-wise form, and, based on the assumptions made, the existence of the only particular solution of the specified structure is proved. Then, using the method of successive approximations and the principle of contracting mappings, the existence of a unique particular solution of the indicated structure for the original quasilinear equation are proved.
Для квазiлiнiйного матричного диференцiального рiвняння, коефiцiєнти якого зображуванi у виглядi абсолютно та рiвномiрно збiжних рядiв Фур’є з повiльно змiнними коефiцiєнтами та частотою, отримано достатнi умови iснування розв’язку аналогiчної структури.
In the mathematical description of various phenomena and processes that arise in mathematical physics, electrical engineering, economics, one has to deal with matrix differential equations. Therefore, these equations are relevant both for mathematicians and for specialists in other areas of natural science. Many studies are devoted to them, in which the solvability of matrix equations in various function spaces, boundary value problems for matrix differential equations, and other problems were investigated. In this article, a quasilinear matrix equation is considered, the coefficients of which can be represented in the form of absolutely and uniformly converging Fourier series with coefficients and frequency slowly varying in a certain sense. The problem is posed of obtaining sufficient conditions for the existence of particular solutions of a similar structure for the equation under consideration. For this purpose, the corresponding linear equation is considered first. It is written down in component-wise form, and, based on the assumptions made, the existence of the only particular solution of the specified structure is proved. Then, using the method of successive approximations and the principle of contracting mappings, the existence of a unique particular solution of the indicated structure for the original quasilinear equation are proved.
Опис
Ключові слова
матрица, дифференциальное уравнение, квазилинейный, матриця, диференцiальне рiвняння, квазiлiнiйний, matrix, differential equation, quasilinear
Бібліографічний опис
Дослідження в математиці і механіці = Researches in mathematics and mechanics