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A transform method for solving initial-boundary-value problems (IBVPs) for evolu tion linear partial differential equations was recently developed by A.S.Fokas.Here we show that an analogue of Fokas method can be formulated to solve IBVPs for evolu tion linear differential-difference equations.We first show that, as in the continuum case, any such equation can be written as the compatibility condition of a discrete Lax pair, namely, an overdetermined linear system of differential-difference equations containing a spectral parameter.