  Efficient and accurate spectral solvers for nonlocal models in any spatial
dimension are presented. The approach we pursue is based on the Fourier
multipliers of nonlocal Laplace operators introduced in a previous work. It is
demonstrated that the Fourier multipliers, and the eigenvalues in particular,
can be computed accurately and efficiently. This is achieved through utilizing
the hypergeometric representation of the Fourier multipliers in which their
computation in $n$ dimensions reduces to the computation of a 1D smooth
function given in terms of $_2F_3$. We use this representation to develop
spectral techniques to solve periodic nonlocal time-dependent problems. For
linear problems, such as the nonlocal diffusion and nonlocal wave equations, we
use the diagonalizability of the nonlocal operators to produce a semi-analytic
approach. For nonlinear problems, we present a pseudo-spectral method and apply
it to solve a Brusselator model with nonlocal diffusion. Accuracy and
efficiency of the spectral solvers are compared against a finite-difference
solver.
