Computer-aided automatic computation of eigenvalues for multi-layer transient heat conduction problems

A new, accurate, computationally efficient and completely automated procedure is developed to estimate the bounds for the eigenvalues of Sturm-Liouville systems associated to unsteady heat conduction problems of 1-D piecewise-homogeneous slabs. The procedure is based on particular ‘single-layer slabs’ capable to keep nearly entirely the physical insights contained in the starting two-layer composite slab. The eigenvalues of the homogeneous slabs (known in a straightforward manner using explicit equations available in literature) are indeed assumed as lower and upper bounds for the corresponding roots of the two-layer eigencondition. Behaving like this, the bounds result very efficient from a computational viewpoint and their calculation pulls itself free definitely of the time-consuming nature of the ‘manual feedback’ typical of the various methods proposed so far.