Tridiagonal linear systems arise ubiquitously in finite-difference discretizations of diffusion-type partial differential equations (PDEs). The Thomas algorithm is a workhorse direct solver for such systems and is widely embedded in time-integration frameworks. Bound-ary conditions, however, can alter the algebraic structure significantly. In particular, periodic boundaries induce cyclic couplings that violate strict tridiagonality. Leveraging rank-one cor-rections via the Sherman–Morrison identity, we extend a two-dimensional boundary-treatment strategy to three dimensions within an operator-splitting framework. We detail implementations for periodic, Dirichlet, and Neumann conditions, and assess accuracy and stability through heat- equation benchmarks and Allen–Cahn dynamics. The results furnish a practical blueprint for robust boundary enforcement in Thomas-based solvers for 3D time-dependent PDEs.