  Fractional differential equation (FDE) provides an accurate description of
transport processes that exhibit anomalous diffusion but introduces new
mathematical difficulties that have not been encountered in the context of
integer-order differential equation. For example, the wellposedness of the
Dirichlet boundary-value problem of one-dimensional variable-coefficient FDE is
not fully resolved yet. In addition, Neumann boundary-value problem of FDE
poses significant challenges, partly due to the fact that different forms of
FDE and different types of Neumann boundary condition have been proposed in the
literature depending on different applications.
  We conduct preliminary mathematical analysis of the wellposedness of
different Neumann boundary-value problems of the FDEs. We prove that five out
of the nine combinations of three different forms of FDEs that are closed by
three types of Neumann boundary conditions are well posed and the remaining
four do not admit a solution. In particular, for each form of the FDE there is
at least one type of Neumann boundary condition such that the corresponding
boundary-value problem is well posed, but there is also at least one type of
Neumann boundary condition such that the corresponding boundary-value problem
is ill posed. This fully demonstrates the subtlety of the study of FDE.
