The paper analyses the transverse eigenvalue problem of nonconventional Sturm-Liouville type associated to the steady-state heat conduction in 3-D two-component slabs with imperfect thermal contact. In particular, it describes how the physical insight deriving from the transverse direction of six suitable 'homogeneous parallelepipeds' inherent to the considered two-layered parallelepiped is capable of providing useful and reasonably accurate information about the best bracketing bounds (lower and upper) for the roots (eigenvalues) of the transverse eigencondition. This information, in fact, enables one to establish starting points (initial guesses) for a root-finding iteration (e.g., Muller's method) so that convergence of the iteration may absolutely be guaranteed. Representative test examples are computed to illustrate the accuracy, reliability, and efficiency of the proposed fully automated solution algorithm. (C) 2003 Elsevier Ltd. All rights reserved.