  We study the local and global wellposedness of the initial-boundary value
problem for the biharmonic Schr\"odinger equation on the half-line with
inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a
representation formula for the solution of the linear nonhomogenenous problem
by using the Fokas method (also known as the \emph{unified transform method}).
We use this representation formula to prove space and time estimates on the
solutions of the linear model in fractional Sobolev spaces by using Fourier
analysis. Secondly, we consider the nonlinear model with a power type
nonlinearity and prove the local wellposedness by means of a classical
contraction argument. We obtain Strichartz estimates to treat the low
regularity case by using the oscillatory integral theory directly on the
representation formula provided by the Fokas method. Global wellposedness of
the defocusing model is established up to cubic nonlinearities by using the
multiplier technique and proving hidden trace regularities.
