  We study nonlocal equations from the area of peridynamics on bounded domains.
In our companion paper, we discover that, on $\mathbb{R}^n$, the governing
operator in peridynamics, which involves a convolution, is a bounded function
of the classical (local) governing operator. Building on this, we define an
abstract convolution operator on bounded domains. The abstract convolution
operator is a function of the classical operator, defined by a Hilbert basis
available due to the purely discrete spectrum of the latter. As governing
operator of the nonlocal equation we use a function of the classical operator,
this allows us to incorporate local boundary conditions into nonlocal theories.
  For the homogeneous wave equation with the considered boundary conditions, we
prove that continuity is preserved by time evolution. We give explicit solution
expressions for the initial value problems with prominent boundary conditions
such as periodic, antiperiodic, Neumann, and Dirichlet. In order to connect to
the standard convolution, we give an integral representation of the abstract
convolution operator. We present additional "simple" convolutionsbased on
periodic and antiperiodic boundary conditions that lead Neumann and Dirichlet
boundary conditions.
  We present a numerical study of the solutions of the wave equation. For
discretization, we employ a weak formulation based on a Galerkin projection and
use piecewise polynomials on each element which allows discontinuities of the
approximate solution at the element borders. We study convergence order of
solutions with respect to polynomial order and observe optimal convergence. We
depict the solutions for each boundary condition.
