Stationary problems with various nonlocal boundary conditions

Abstract

In this paper we investigate one-dimensional stationary differential problems with various nonlocal boundary conditions and finite difference schemes for them. The boundary conditions of the first, second and third type are considered and nonlocal part of those boundary conditions can be integral type or Samarskii and Bitsadze type. The base idea is to use fundamental solutions of the stationary problems with classical boundary conditions. We find necessary and sufficient solvability conditions for such problems. Stability estimates are proved in the maximum norm. The convergence of the second order finite difference schemes is proved.