  We present a method to solve initial-boundary value problems for linear and
integrable nonlinear differential-difference evolution equations. The method is
the discrete version of the one developed by A. S. Fokas to solve
initial-boundary value problems for linear and integrable nonlinear partial
differential equations via an extension of the inverse scattering transform.
The method takes advantage of the Lax pair formulation for both linear and
nonlinear equations, and is based on the simultaneous spectral analysis of both
parts of the Lax pair. A key role is also played by the global algebraic
relation that couples all known and unknown boundary values. Even though
additional technical complications arise in discrete problems compared to
continuum ones, we show that a similar approach can also solve initial-boundary
value problems for linear and integrable nonlinear differential-difference
equations. We demonstrate the method by solving initial-boundary value problems
for the discrete analogue of both the linear and the nonlinear Schrodinger
equations, comparing the solution to those of the corresponding continuum
problems. In the linear case we also explicitly discuss Robin-type boundary
conditions not solvable by Fourier series. In the nonlinear case we also
identify the linearizable boundary conditions, we discuss the elimination of
the unknown boundary datum, we obtain explicitly the linear and continuum limit
of the solution, and we write down the soliton solutions.
