In the case of inhomogeneous materials, the thermophysical characteristics cannot be determined experimentally. To determine such characteristics, mathematical methods of identification based on the apparatus of coefficient inverse heat conduction problems are used. The inverse problems of heat conduction are most often solved in an extreme formulation. In this case, as a rule, gradient methods are used to minimize the residual functional. However, such methods have a number of disadvantages. ewline An alternative method for solving the coefficient inverse heat conduction problem for an inhomogeneous rod is proposed. Based on the weak formulation and the linearization method, operator equations are obtained for solving the inverse heat conduction problem. Thermophysical characteristics are restored on the basis of the iteration process, at each stage of which the Fredholm integral equations of the first kind are solved. The direct problem of thermal conductivity for a rod is solved by the method of reduction to the integral Fredholm equation of the second kind. ewline The initial approximation in the iterative process was determined in two ways in the class of positive bounded linear functions. The first method is based on the use of a priori information on the boundaries of the change in thermophysical characteristics. The second method is based on the use of the Galerkin projection method. A comparative analysis of the methods for finding the initial approximation is carried out. Computational experiments on restoring various laws of change in thermophysical characteristics were carried out. It was found out that the reconstruction procedure is resistant to noisy input information.

Поставлена коэффициентная обратная задача об определении неоднородных теплофизических характеристик стержня. На основе слабой постановки и метода линеаризации получены операторные уравнения для решения обратной задачи теплопроводности. Теплофизические характеристики восстанавливаются на основе итерационного процесса, на каждом этапе которого решаются интегральные уравнения Фредгольма 1-го рода. Рассмотрено два способа определения начального приближения в простых классах функций. Проведен сравнительный анализ двух подходов. Выполнены вычислительные эксперименты по восстановлению неоднородных теплофизических характеристик в классах степенных, экспоненциальных и тригонометрических функций.