Initial-boundary-value problems for discrete linear evolution equations

We present a transform method for solving initial-boundary-value problems (IBVPs) for linear semidiscrete (differential-difference) and fully discrete (difference-difference) evolution equations. The method is the discrete analogue of the one recently proposed by A. S. Fokas to solve IBVPs for evolution linear partial differential equations. We show that any discrete linear evolution equation can be written as the compatibility condition of a discrete Lax pair, namely, an overdetermined linear system of equations containing a spectral parameter. As in the continuum case, the method employs the simultaneous spectral analysis of both parts of the Lax pair, the symmetries of the evolution equation and a relation, called the global algebraic relation, that couples all known and unknown boundary values. The method applies for differential- difference equations in one lattice variable as well as for multi-dimensional and fully discrete evolution equations. We demonstrate the method by discussing explicitly several examples.