  A direct discontinuous Galerkin (DDG) finite element method is developed for
solving fractional convection-diffusion and Schr\"{o}dinger type equations with
a fractional Laplacian operator of order $\alpha$ $(1<\alpha<2)$. The
fractional operator of order $\alpha$ is expressed as a composite of second
order derivative and a fractional integral of order $2-\alpha$. These problems
have been expressed as a system of parabolic equation and low order integral
equation. This allows us to apply the DDG method which is based on the direct
weak formulation for solutions of fractional convection-diffusion and
Schr\"{o}dinger type equations in each computational cell, letting cells
communicate via the numerical flux $(\partial_{x}u)^{*}$ only. Moreover, we
prove stability and optimal order of convergence $O(h^{N+1})$ for the general
fractional convection-diffusion and Schr\"{o}dinger problems where $h$, $N$ are
the space step size and polynomial degree. The DDG method has the advantage of
easier formulation and implementation as well as the high order accuracy.
Finally, numerical experiments confirm the theoretical results of the method.
