Abstract
The paper investigates an initial--boundary value problem for the heat equation with a piecewise constant coefficient and a Caputo fractional time derivative of order $\alpha$ ($0<\alpha<1$). The domain under consideration is the interval $(0,l)$, which contains a strictly interior discontinuity point $x=x_{0}$. From the physical viewpoint, this problem models the propagation of a temperature field in a thin rod of length $l$. The rod is composite and consists of two segments, $(0,x_{0})$ and $(x_{0},l)$, which possess different thermophysical properties. At the contact point of the two media, $x=x_{0}$, ideal contact conditions are set, implying the continuity of temperature and the continuity of heat flux when passing from one medium to the other. The main objective of the work is to justify the solution of the posed initial–boundary value problem by the method of separation of variables (the Fourier method). The application of this method leads to the necessity of studying the corresponding spectral problem for an ordinary differential operator with a discontinuous coefficient in the highest derivative.