Generalized Calderon-Ryaben'kii's potentials

Abstract

Calderón–Ryaben'kii potentials provide the foundation for the difference potential method, which is an efficient way for solving boundary-value problems (BVPs) in arbitrary domains. This method allows us to reduce a uniquely solvable and well-posed BVP to a pseudo-differential boundary equation. The general theory of Calderón–Ryaben'kii potentials is considered via the theory of distributions. The definition of Calderón–Ryaben'kii potentials is based on the notion of a clear trace. The criterion of the clear trace is formulated. Partial differential equations of the first order and the second order are considered as particular examples. On the basis of the Calderón–Ryaben'kii potential theory, a solution of the active sound control problem is obtained in a general formulation. For the first time, the solution of the problem takes into account the feedback of the active shielding sources on the input (measurement) data. The exact transfer of the boundary conditions from the original boundary to an artificial boundary is also considered.