Duhamel-type integral for the initial boundary value problem

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 has been introduced and the theorem of unique existence has been proved. A new formula was proposed, being an analog of the well-known Duhamel integral. The most part of the paper is devoted to the analysis of differential properties of the solution. In particular, the possibility of breaking the second partial time derivative on a certain hyperplane was indicated, and its break value was 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 2D subset. In this case, the known data were considered to be the values of a fixed oscillating point’s position at every moment of time. 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.