mathnasТруды Института математики НАН БеларусиProceedings of the Institute of Mathematics of the NAS of Belarus1812-5093Институт математики НАН Беларуси10.67268/1812-5093-2026-34-1-85-95DCQCNWmathnas-142Research ArticleДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ, ДИНАМИЧЕСКИЕ СИСТЕМЫ И ОПТИМАЛЬНОЕ УПРАВЛЕНИЕDIFFERENTIAL EQUATIONS, DYNAMIC SYSTEMS AND OPTIMAL CONTROLКлассическое решение первой смешанной задачи для слабоквазилинейного волнового уравнения: метод неподвижной точкиClassical solution to the first mixed problem for a mildly quasilinear wave equation: a fixed-point approachРудькоЯ. В.RudzkoJ. V.

Минск

Minsk

janycz@yahoo.comИнститут математики НАН БеларусиInstitute of Mathematics of the National Academy of Sciences of Belarus2026300620263418595Copyright © Рудько Я.В., 20262026Рудько Я.В.Rudzko J.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathnas.ejournal.by/jour/article/view/142

Для одномерного слабо квазилинейного волнового уравнения, заданного в первом квадранте, рассматривается смешанная задача, в которой на пространственной полуоси задаются условия Коши, а на временной полуоси задается условие Дирихле. Нелинейность содержит независимые переменные, искомую функцию и ее производные. Решение строится в неявном аналитическом виде как решение некоторых интегро-дифференциальных уравнений. Разрешимость интегро-дифференциальных уравнений доказывается с использованием обобщения теоремы Банаха о неподвижной точке. Для рассматриваемой задачи доказывается единственность решения и устанавливаются условия, при выполнении которых существует ее классическое решение.

For a one-dimensional mildly quasilinear wave equation given in the first quadrant, we consider a mixed problem in which Cauchy conditions are specified on the spatial semi-axis and a Dirichlet condition is specified on the temporal semi-axis. The nonlinearity contains independent variables, the unknown function, and its derivatives. We construct the solution in implicit analytical form as the solution of some integro-differential equations. We prove the solvability of these integro-differential equations using a generalization of the Banach fixed-point theorem. For the problem in question, we prove the uniqueness of the solution and establish the conditions under which its classical solution exists

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The authors declare that there are no conflicts of interest present.