Duhamel-type integral for the initial boundary value problem

Abstract

The paper considers the initial boundary value problem for the wave equation for the case of three spatial variables. The definition of a generalized solution is introduced and the existence and uniqueness theorem is proved. A new formula is proposed, which is an analog of the well-known Duhamel integral. Most of the article is devoted to the analysis of differential properties of the solution. In particular, the possibility of breaking the second partial derivative in time on a certain hyperplane is indicated, and the value of its break is given. This property allowed us to set the inverse problem of determining the coefficient of the equation and propose an algorithm for solving it under the condition of non-zero internal action on a two-dimensional subset. In this case, the known data were considered to be the values of the position of a fixed oscillating point at each time. By analogy with the study of the Cauchy problem, the descent method is used for the wave equation, which allows applying the results obtained for a smaller number of variables. For physical interpretation, the case of two spatial variables is the most obvious as a study of the process of transverse vibrations of semi-bounded surfaces of the membrane type. Here is a list of some publications by other authors that are close to the topic of our work.

Keywords

• wave equation
• duhamel integral
• inverse problem
• descent method
• cauchy problem