Nonlocal problems with local Dirichlet and Neumann boundary conditions

We present novel governing operators in the theory of peridynamics (PD) which will allow the extension of PD to applications that require local boundary conditions (BC). Due to its nonlocal nature, the original PD governing operator uses nonlocal BC. The novel operators agree with the original PD operator in the bulk of the domain and simultaneously enforce local Dirichlet or Neumann BC. Our construction is straightforward and easily accessible. The main ingredients are antiperiodic and periodic extensions of kernel functions together with even and odd parts of functions. We also present governing operators that enforce antiperiodic and periodic BC and the corresponding compatibility conditions for the right-hand side function in a given operator equation. Finally, we present the basic idea in extending the 1D construction to 2D.