Диф. ур-я

2017

DOI: 10.1134/s0374064117010071

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Классическое Решение Первой Смешанной Задачи Для Гиперболического Уравнения Второго Порядка В Криволинейной Полуполосе С Переменными Коэффициентами

В. И. Корзюк¹,

И. И. Столярчук²

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Classical Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential in a Curvilinear Quadrant

Korzyuk,

Rud'ko

2023

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For the telegraph equation with a nonlinear potential in a curvilinear quadrant, we consider a mixed problem with the Cauchy conditions on a spatial half-line and the Dirichlet condition on a noncharacteristic curve. The solution of the problem is constructed by the method of characteristics in an implicit analytical form as a solution of integral equations. We study the solvability of these equations depending on the initial data and their smoothness. For the problem under consideration, the uniqueness of the solution is proved and conditions under which there exists a classical solution are established. A mild solution is constructed in the case of insufficiently smooth data of the problem.

Classical Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential in a Curvilinear Quadrant

Korzyuk,

Rud'ko

2023

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For the telegraph equation with a nonlinear potential in a curvilinear quadrant, we consider a mixed problem with the Cauchy conditions on a spatial half-line and the Dirichlet condition on a noncharacteristic curve. The solution of the problem is constructed by the method of characteristics in an implicit analytical form as a solution of integral equations. We study the solvability of these equations depending on the initial data and their smoothness. For the problem under consideration, the uniqueness of the solution is proved and conditions under which there exists a classical solution are established. A mild solution is constructed in the case of insufficiently smooth data of the problem.

Classical Solution of the Second Mixed Problem for the Telegraph Equation with a Nonlinear Potential

Korzyuk,

Rud'ko

2023

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For the telegraph equation with a nonlinear potential, we consider a mixed problem in the first quadrant in which the Cauchy conditions are specified on the spatial semiaxis and the Neumann condition is set on the temporal semiaxis. The solution is constructed by the method of characteristics in an implicit analytical form as a solution of some integral equations. The solvability of these equations is studied, as well as the dependence of the solutions on the smoothness of the initial data. For the problem under consideration, the uniqueness of the solution is proved and conditions are established under which a classical solution exists. If the matching conditions are not met, then a problem with conjugation conditions is constructed, and if the data is not smooth enough, then a mild solution is constructed.

Solution of some half-strip problems in quadratures for the string vibration equation

Jokhadze,

Kharibegashvili

2024

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