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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 boundary, the Cauchy conditions are specified, meanwhile, the second of them has a discontinuity of the first kind at one point. The smooth boundary condition, which has the first and the second order derivatives, 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 are established under which a piecewise-smooth solution exists. The problem with matcing 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
В работе были сформулированы условия согласования, при выполнении которых существует классическое решение задачи в случае достаточной гладкости ее условий
Summary
Что ψ 1 ∈C1([0, x*]), ψ 2 ∈C1([x*,∞)), ψ 2 (x) = ψ2 (x) для x ∈ (x*,∞), ψ 1(0) = ψ1. Если выполняются условия гладкости для заданных функций f C1(Q), C2 ([0, )), 1 C1([0, x*]), 2 C1([x*, )), C([0, )), то существует единственное классическое решение задачи (1), (3), (4) в смысле определения 1, и оно представляется формулами (11), (13), (18), (19) и (23).
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