Coupled partial differential systems with coupled boundary conditions are frequent in quantum mechanical scattering problems, chemical physics, thermoelastoplastic modelling, coupled diffusion problems and other fields. In this paper systems of the type
are considered, where the unknown and are -dimensional vectors, are complex matrices, where no simultaneous diagonalizable hypothesis is assumed, and is a positive stable matrix, such that for all eigenvalues of . The construction of exact and analytical-numerical solutions with apriori bounds is shown. Given an admissible error and a bounded subdomain , an approximate solution whose error with respect to an exact series solution is less than uniformly in is constructed.