An inverse problem for the equation of membrane's vibration

Abstract

A mathematical model for membrane's vibration process is used in this paper. The model is based on seeking a solution of the second-order hyperbolic differential equation. A new inverse problem is set and investigated in two versions. In the first version the known data are as follows: the coefficient defining the phase velocity, the starting data of the Cauchy problem, the Cauchy problem solution on two given planes, derivatives of the solution along the vector being normal to these planes. The challenge has been in localizing the support of the right-hand side of the equation for vibrations. The algorithm permitting to find the bounded domain containing the unknown support was designed. In the second version the algorithm refers to the case where the coefficient defining the phase velocity is unknown but an interval of its possible values is known. A series of runs was performed to illustrate the proposed model.