  We derive and analyze the alternating direction explicit (ADE) method for
time evolution equations with the time-dependent Dirichlet boundary condition
and with the zero Neumann boundary condition. The original ADE method is an
additive operator splitting (AOS) method, which has been developed for treating
a wide range of linear and nonlinear time evolution equations with the zero
Dirichlet boundary condition. For linear equations, it has been shown to
achieve the second order accuracy in time yet is unconditionally stable for an
arbitrary time step size. For the boundary conditions considered in this work,
we carefully construct the updating formula at grid points near the boundary of
the computational domain and show that these formulas maintain the desired
accuracy and the property of unconditional stability. We also construct
numerical methods based on the ADE scheme for two classes of fractional
differential equations. We will give numerical examples to demonstrate the
simplicity and the computational efficiency of the method.
