Representations of the solutions of partial differential equations of parabolic and hyperbolic types by means of time observations

In some general situation, the solution of a partial differential equation of parabolic type satisfying some general boundary condition is represented in the form by using eigenvalues (λn) and eigenfunctions {φn}. In this paper, it is shown the principle of telethoscope that the solution u(t,I) is determined and represented by the observatior where r is any fixed large positive constant and E is a small set on the closure of the space domain of the differential equation Furthermore, a general corresponding version for the solutions of differential equations of hyperbolic type is derived