Initial guesses for computing eigenvalues of Sturm-Liouville problems of multi-dimensional multi-layer unsteady heat conduction

The paper analyses the transverse eigenvalue problem of Sturm-Liouville type associated with the time-dependent heat conduction in two-component rectangular domains. In particular, it describes how the physical insight of a ‘homogeneous rectangular region’ thermally and geometrically equivalent to the considered 2-D two-layered region in the transverse direction is capable of providing useful and reasonably accurate information about the initial guesses for the roots (eigenvalues) of the transverse eigencondition. This information, in fact, enables one to establish starting points for a root-finding iteration (e.g., Müller'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 when the two layers have the same thermal diffusivity and are perfectly jointed.