Diagnostics of Mathematical Models of Thermoelasticity. Part 1. Weak Solutions of Boundary-Value Problems and Formulation of the Problems of Diagnostics

The formulations and methods of solution of the problems of diagnostics of thermoelasticity on a stationary boundary of a solid body are considered. A weak formulation of the boundary-value problem is presented for a nonlinear hyperbolic equation (a nonstationary wave equation with density, elasticity modulus, and thermal stress depending on temperature) with mixed boundary conditions. An approximate solution of this problem in the space \({W}_{2}^{\left(1\right)}\) $W_2^{\left(1\right)}$ has been obtained, and an example of solution is given. Based on the weak formulation of the boundary-value problem, problems of diagnostics have been formulated in the sense of the theory of function traces. According to the theorem about the uniqueness of the relationship between the function and its trace in the space \({W}_{2}^{\left(1\right)},\) $W_2^{\left(1\right)},$ a conclusion has been drawn about the need to control the convergence of the calculated wave function to the experimental one only on boundaries with unstable conditions in solving the problem of diagnostics.