The solution of arbitrary smoothness of the one-dimensional wave equation for the problem with mixed conditions

Abstract

In this paper, we represented an analytical form of a classical solution of the wave equation in the class of continuously differentiable functions of arbitrary order with mixed boundary conditions in a quarter of the plane. The boundary of the area consists of two perpendicular half-lines. On one of them, the Cauchy conditions are specified. The second half-line is separated into two parts, namely, the limited segment and the remaining part in the form of a half-line. The Dirichlet condition is specified on the segment, as well as the Neumann condition is fulfilled on the second part in the form of a half-line. In a quarter of the plane, the classical solution of the problem under consideration is determined. To construct this solution, a particular solution of the original wave equation is established. For the given functions of the problem, the concordance conditions are written, which are necessary and sufficient for the solution of the problem to be classical of high order of smoothness and unique.

Classical Solutions of a Mixed Problem with the Zaremba Boundary Condition for a Mildly Quasilinear Wave Equation

“…In the case of the wave equation, i.e., f (t, x, u, u t , u x ) = f (t, x), a solution of arbitrary smoothness to the problem (1.1) -(1.4) was constructed in [42].…”