A new finite integral transform pair for hyperbolic conduction problems in heterogeneous media

The study of heat conduction in situations involving very short times cannot be set about with the help of Fourierʹs law. This model which cannot be justified in case of local non-equilibrium, leads, in this case, to results which do not agree with the observations. When the continuity hypothesis is posed, Fourierʹs law may nevertheless be adjusted. Among the different proposed models the Cattaneo–Vernotte model is one of the most credible. It is used in this work. The resulting hyperbolic problem in a non-homogeneous medium is posed in its primitive form of coupled system of partial differential equations where the unknowns are the temperature and the flux density, i.e. a four-component vector field. Thanks to a dot product which is suited to these fields, the existence of two orthogonal families of complex vector-eigenfunctions is shown. A finite-integral transform technique which is based on this dot product is then applied to the primitive system. It produces an infinite set of uncoupled ordinary differential equations: a separated expansion form of the solution vector field is thus obtained. For multifilmed media (1-D cases), it is shown that the notion of transfer matrix which is familiar in pure diffusion context generalizes naturally.