  The main goal of this paper is applying the cut-off projection for solving
one-dimensional backward heat conduction problem in a two-slab system with a
perfect contact. In a constructive manner, we commence by demonstrating the
Fourier-based solution that contains the drastic growth due to the
high-frequency nature of the Fourier series. Such instability leads to the need
of studying the projection method where the cut-off approach is derived
consistently. In the theoretical framework, the first two objectives are to
construct the regularized problem and prove its stability for each noise level.
Our second interest is estimating the error in $L^2$-norm. Another
supplementary objective is computing the eigen-elements. All in all, this paper
can be considered as a preliminary attempt to solve the heating/cooling of a
two-slab composite system backward in time. Several numerical tests are
provided to corroborate the qualitative analysis.
