Modal identification of a boundary input in the two-dimensional inverse heat conduction problem

A method for the approximate solution of a two-dimensional inverse boundary heat conduction problem on a compact set of continuous and continuously differentiable functions is proposed. The method allows us to reconstruct a boundary action that depends on time and a spatial coordinate. A modal description of the object is used in the form of an infinite system of linear differential equations with respect to the coefficients of the expansion of the state function in a series in eigenfunctions of the initial-boundary value problem under study. This approach leads to the restoration of the sought value of the heat flux density in the form of a weighted sum of a finite number of its modal components. Their values are determined from the temporal modes of the temperature field, which are found from the experimental data on the basis of the modal representation of the field. To obtain a modal description of the identified input and the temperature field in the form of their expansions into series in eigenfunctions of the same spatial dimension, the mathematical model of the object in the Laplace transform domain and the method of finite integral transformations are used. On this basis, a closed system of equations with respect to the unknown quantities is formed. The proposed approach allows us to construct a sequence of approximations that uniformly converge to the desired solution with increasing number of considered modal components. The problem of the temperature experimental design is solved. This solution ensures the minimization of the approximation error of the experimental temperature field by its model representation in the uniform metric of estimating temperature discrepancies on the control line at the final moment of the identification interval.