This paper addresses the open problem of partial differential equation (PDE)-based dynamic modeling for flexible multibody robotic systems by presenting a screw-theoretic synthesis methodology-developed within a unified Lie-algebraic framework-for serial flexible manipulators with an arbitrary number of links in three-dimensional motion. The proposed approach expresses all dynamic states — rigid-body motion, elastic deformation, and inter-link interaction forces — uniformly within the screw theoretic structure as body-fixed twists and their dual wrenches, treating all physical components consistently within the same geometric framework. Hence, the PDE structure of the deformation field is retained exactly, while the se(3) representation admits the linear-algebraic formulation required for multibody assembly and formal well-posedness analysis. Building on a previously developed single-link screw-theoretic PDE model, joint constraints are enforced as screw-compatibility equations connecting the twist and wrench fields of adjacent links at their connection points, and the per-link models are assembled via a closed-form linear-algebraic stacking procedure. The resulting system matrix exhibits near-tridiagonal block structure, interaction wrenches appear explicitly as algebraic variables, and adding a link requires only appending block rows — making the synthesis automatable for arbitrary $n$. The assembled system is formulated as a semi-explicit index-1 differential-algebraic equation, and well-posedness is established by recasting it in abstract Cauchy form through modal projection. Presented solutions are validated experimentally on a two-link flexible manipulator in three-dimensional motion, confirming implementability and physical consistency. Mathematics Subject Classification (2020) 74Kxx · 74H45 · 70E60