  We consider spectral approximations to the conservative form of the two-sided
Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs)
with nonhomogeneous Dirichlet (fractional and classical, respectively) and
Neumann (fractional) boundary conditions. In particular, we develop a spectral
penalty method (SPM) by using the Jacobi poly-fractonomial approximation for
the conservative R-L FDEs while using the polynomial approximation for the
conservative Caputo FDEs. We establish the well-posedness of the corresponding
weak problems and analyze sufficient conditions for the coercivity of the SPM
for different types of fractional boundary value problems. This analysis allows
us to estimate the proper values of the penalty parameters at boundary points.
We present several numerical examples to verify the theory and demonstrate the
high accuracy of SPM, both for stationary and time dependent FDEs. Moreover, we
compare the results against a Petrov-Galerkin spectral tau method (PGS-$\tau$,
an extension of [Z. Mao, G.E. Karniadakis, SIAM J. Numer. Anal., 2018]) and
demonstrate the superior accuracy of SPM for all cases considered.
