A general modal-based numerical simulation of transient heat conduction in a one-dimensional homogeneous slab

We present an analytical general solution of the well-known heat conduction problem in a one-dimensional homogeneous slab by using an expansion of the only delayed part of the temperature field function on an infinite eigen functions set. We propose to split the searched solution of the general problem in an instantaneous part and a delayed one which, with whatever boundary conditions combination, verifies Dirichlet conditions and then avoids the Gibbs problem. A space sampling and a reduction of the eigen basis produce a useful and usable numerical model which may be as accurate as required. The results we present may be used directly to numerically calculate solutions to a given problem and may be reused as a basis for more complex problems as multilayer walls or whole thermal systems studies.