  In this paper we develop a numerical scheme based on quadratures to
approximate solutions of integro-differential equations involving convolution
kernels, $\nu$, of diffusive type. In particular, we assume $\nu$ is symmetric
and exponentially decaying at infinity. We consider problems posed in bounded
domains and in $\R$. In the case of bounded domains with nonlocal Dirichlet
boundary conditions, we show the convergence of the scheme for kernels that
have positive tails, but that can take on negative values. When the equations
are posed on all of $\R$, we show that our scheme converges for nonnegative
kernels. Since nonlocal Neumann boundary conditions lead to an equivalent
formulation as in the unbounded case, we show that these last results also
apply to the Neumann problem.
