Intensive wave power and steel quenching 3-D model for cylindrical sample. Time direct and reverse formulations and solutions

In this paper we develop mathematical models for three dimensional hyperbolic heat equations (wave equation or telegraph equation) with inner source power and construct their analytical solutions for the determination of the initial heat flux for cylindrical sample. As additional conditions the temperature and heat flux at the end time are given. In some cases we give expression of wave energy. In some cases we give expression of wave energy. Some solutions of time inverse problems are obtained in the form of first kind Fredholm integral equation, but others has been obtained in closed analytical form as series. We viewed both direct and inverse problems at the time. For the time inverse pr…