Inverse Boundary Conditions Interface Problems for the Heat Equation with Cylindrical Symmetry

In this paper, we study inverse interface problems with unknown boundary conditions, using point observations for parabolic equations with cylindrical symmetry. In the one-dimensional, two-layer interface problem, the left interval $0<r<l_1$, i.e., the zero degeneracy, causes serious solution difficulty. For this, we investigate the well-posedness of the direct (forward) problem. Next, we formulate and solve five inverse boundary condition problems for the interface heat equation with cylindrical symmetry from internal measurements. The finite volume difference method is developed to construct second-order schemes for direct and inverse problems. The correctness of the proposed numerical solution decomposition algorithms for the inverse problems is discussed. Several numerical examples are presented to illustrate the efficiency of the approach.