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Abstract
In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. On the bottom of the boundary, the Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at a point. The smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proved, and the conditions under which a piecewise-smooth solution exists are established. The problem with conjugate conditions is considered
Highlights
We study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation
The solution is built using the method of characteristics in an explicit analytical form
И. Классическое решение смешанной задачи для уравнения типа Клейна – Гордона – Фока в полуполосе с косыми производными в граничных условиях / В
Summary
V. The classical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition. Что существуют два решения, тогда для их разности получаем однородное уравнение (1) и однородные условия (3)–(4), из которых следует нулевое решение согласно формулам (10), (12), (15), (18) и (19). Поскольку u( j) ∈C2 (Q( j) ) для каждого j ∈{1, 2,3, 4,5,6}, то для того, чтобы решение было из классов С1(Q(1) Q(2) Q(3) ) и С1(Q(4) Q(5) Q(6) ), должны быть выполнены однородные условия сопряжения на характеристиках x + a=t x∗, x − a=t x∗ и x − at =−x∗ для решения и его производных первого порядка.
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