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

Abstract

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 (λ_n) and eigenfunctions {φ_n}. 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

Keywords:

• Inverse problem
• Dirichlet series
• representation of solutions of partial differential equations

AMS:

• 35A08
• 35C99
• 35A22
• 40E99