Determination of the thermal conductivity from the heat flux on the surface of a three-dimensional body

The inverse problem of determining the thermal conductivity of a substance, which depends on the temperature, by the well-known heat flux at the boundary of a body is considered and investigated. Consideration is performed based on the Dirichlet boundary-value problem for the three-dimensional nonstationary heat equation in a parallelepiped. The coefficient inverse problem is reduced to a variational problem and is solved numerically with the help of gradient methods of functional minimization. The root-mean-square deviation of the calculated heat flux on the surface of the body from the experimentally obtained flux is chosen as the cost functional. The performance and efficiency of the proposed approach are shown by the example of a series of nonlinear problems, the coefficients of which depend on temperature.