Non-standard method of solving direct and inverse problems for hyperbolic equations

We consider a non-standard method that has been used for solving parabolic heat equations, but never to solve hyperbolic equations describing oscillatory processes. This technique was developed by Abraham Temkin (1919-2007) in the 1960s and the concept summary is described in the monograph by A. Temkin, “Inverse Methods of Heat Conduction”, Moscow: Energija Press, 1973; 464 p. (in Russian). The method is based on the fact that for non-stationary heat conduction with non-stationary boundary conditions, the influence of initial conditions on the temperature distribution decreases. And after a while, one can assume that the temperature distribution is determined only by a change of boundary conditions over time. Hyperbolic equations have the same property, so it is useful to check whether this method applies to hyperbolic equations. When applying Temkin’s method, we seek a solution in the form of a series where each term is a product of a derivative of the given boundary condition and an unknown function P of a space variable. Plugging the series into the given differential equation yields a system of ordinary differential equations. When solving this, we find the spatial functions P. Further, we compare the classical solution with the solution obtained by this method. The spatial functions are either polynomials or expressions that contain a polynomial as an addend, depending on the geometry of the domain and the type of the boundary conditions. Such a solution allows us to formulate the inverse problem to find the speed of propagation, knowing amplitudes of oscillations at an intermediate point of the domain. The method proposed here allows us to obtain simple formulas for approximate solution of the inverse problem.