Classical Solutions of Mixed Problem on the Plane for Hyperbolic Equation. The Method of Characteristic Parallelogram

Abstract

This paper regards the mixed problem for wave equation on the plane. The authors of the article prove that the usage of necessary and su - cient homogeneous compatibility conditions guarantees coming to the classical solution in the half-strip. The article gives the classical solution to the one- dimensional wave equation in analytical form if there are Dirichlet conditions at the side edges and conditions of Cauchy type at the bottom of the plane. The Classical solution means function which is de ned in all points presented in the closure of the de ned domain. This function must have all classical derivatives included in the equation and conditions of problem. In case of heterogeneous compatibility conditions correct problem is formulated. In this case the conjugation conditions are added to the problem. Then the classical solution is piecewise smooth in the corresponding subdomains of the half-strip. Also this paper obtains the classical solution of the problem using the method of the characteristic parallelogram.