<p>An analysis of the formulations and methods for solving diagnostics problems on a fixed boundary of a solid body is carried out. Based on the weak formulations of boundary value problems for a nonlinear hyperbolic equation (nonstationary wave equation with density, elasticity moodulus, and thermal stress depending on temperature) and a parabolic equation (nonstationary heat conduction equation with volumetric heat capacity and thermal conductivity depending on temperature) with mixed boundary conditions, methodologies for finding a regularized solution to the diagnostics problem of unstable boundary conditions are determined. The specified solution is found, in the space <i>L</i>₂, by the regularization method from the first-kind integral equation in the presence of experimental data at all points of the space–time grid in the closed region <i>G</i> + Γ and, in the space <i>W</i>₂⁽¹⁾, by the iterative regularization method from the weak formulation of the boundary value problem of thermoelasticity in the presence of experimental data only on the boundary Γ.</p>