One-dimensional heat equation with discontinuous conductance

We study a second-order parabolic equation with divergence form elliptic operator, having a piecewise constant diffusion coefficient with two points of discontinuity. Such partial differential equations appear in the modelization of diffusion phenomena in medium consisting of three kinds of materials. Using probabilistic methods, we present an explicit expression of the fundamental solution under certain conditions. We also derive small-time asymptotic expansion of the PDE’s solutions in the general case. The obtained results are directly usable in applications.