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Algorithms for the gradient method of solution of the inverse problem on determination of the nonlinear thermal-conductivity coefficients are given. Results of numerical experiments are discussed.Introduction. In [1], we consider the problem of functional identification of the nonlinear thermal-conductivity coefficient λ(T). Behind the approach proposed is the gradient method of numerical solution of inverse heat-conduction problems [2][3][4]. We note that, in the traditional approach to finding λ(T), one uses a finite-dimensional approximation of a coefficient by the system of basis functions [4], whereas in [1], we propose a method of solution of inverse heat-conduction problems without preliminary approximation of the functions sought; this method uses new representations of the operator conjugate to the internal-superposition operator, which makes it possible to obtain formulas convenient for numerical calculations of the values of the conjugate operator.In the present work, we propose algorithms of functional identification of the coefficient λ(T), describe computational experiments, and discuss calculation results.Computational Formulas. We give the formulas from [1], which are necessary for numerical realization of the algorithms of solution of inverse heat-conduction problems. The system of equations for finding λ(T) has the form

UDC 536.2Consideration is given to the gradient methods of solution of the inverse heat-conduction problem on determination of the nonlinear coefficient λ(T) without its preliminary finite-dimensional approximation.Introduction. Gradient methods of numerical solution of inverse heat-conduction problems have been developed in many works, mainly in [1][2][3]. In particular, the problem of identification of the nonlinear thermal-conductivity coefficient λ(T) has been considered in [3][4][5][6]. In [1-3, 7, 8], gradient methods have been used for restoration and evaluation of the power of heat sources.One problem frequently arising when gradient methods are used is numerical realization of the values of conjugate (adjoint) operators. For example, in the case of identification of λ(T), the operator conjugate to the internal-superposition operator (other names [9]: the substitution operator, the weighted-shift operator, the operator of replacement of a variable, and the composite operator) is present in the scheme of the method of conjugate gradients. The wellknown approach presented in [3] leads to a complex and difficult-to-control procedure of computation of the values of the operator conjugate to the internal-superposition operator. Therefore, a finite-dimensional approximation of the sought nonlinear coefficients by any system of basis functions has been used in [3] and in subsequent works, thus reducing inverse heat-conduction problems to a problem of restoration of a finite number of parameters. In this connection, such approaches to solution of inverse heat-conduction problems are frequently called parametric ones.In the present work, we consider heat-conduction problems without preliminary approximation of the functions sought. Such an approach is conventionally called functional (or finite-dimensional) identification. Functional identification of the nonlinear thermal-conductivity coefficient by gradient methods is based on new representations of the operator conjugate to the internal-superposition operator; these representations enable one to obtain formulas of the values of a conjugate operator, which are convenient for numerical calculations. We note that similar representations were used earlier in the theory of controlled integro-differential and functional-differential systems [10][11][12].The results of the work are presented in two papers. In the first paper, we describe the algorithm of functional identification of the coefficient λ(T); the emphasis is on finding the gradient of the square of the residual functional for λ(T) in the space L 2 [T (1) , T (2) ] of functions summable with the square and in the Sobolev space W 2 [T

A direct relationship between the theory of inverse problems of mathematical physics and the theory of structural properties of dynamic systems is established based on which the inverse problems of mathematical physics and heat conduction are classified and some of the works on them are reviewed. Introduction. The foundations of the theory of inverse problems of mathematical physics were laid in the 1950s-1960s in the works of A. N. Tikhonov, M. M. Lavrent'ev, I. M. Gel'fand, B. M. Levitan, M. G. Krein, V. A. Marchenko, L. D. Faddeev, and many other mathematicians. The special properties of inverse problems are that, unlike primal problems, they do not possess the property of correctness in the sense of Adamar. In this connection, A. N. Tikhonov and his followers have developed the theory of regularization of ill-posed problems and have proposed stable methods of their solution [1-12].In thermophysics, inverse problems occur as problems of either diagnostics of the thermophysical parameters and internal and (or) boundary sources of the processes of transfer or control and synthesis of the above parameters and sources. We emphasize that the "investigation methodology based on solution of inverse problems is one new line in studying heat-and mass-exchange processes and in processing and optimizing thermal regimes of technical objects and technological processes [13]." The problems and methods of solution of the inverse problems of heat exchange have been presented in detail in [14] (this monograph is now classical).In the present work, we review the basic classes of inverse problems of mathematical physics and, in particular, inverse problems of heat conduction; the inverse problems are organized in accordance with the classification (proposed in [15]) of inverse problems of mathematical physics. The basis for the classification used is the scheme of cause-and-effect relations of dynamic systems. The notion of a dynamic system is fundamental for primal problems of mathematical physics. The theory of structural properties and characteristics of systems, such as controllability, observability, reversibility, realizability, and others, has also been developed within the framework of dynamic systems [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. It turned out that these characteristics are directly related to the formulation of a number of classical inverse problems of mathematical physics and inverse problems of heat conduction [15]. Thus, the classification of inverse problems of mathematical physics that is presented in this review links the theory of inverse problems to the theory of dynamic systems in the space of states, which contributes to the interdisciplinary exchange of results and, in particular, to the use of the methods of the theory of dynamic systems in the theory of inverse heat-conduction problems.It should be noted that investigations of the inverse problems of mathematical physics are the focus of numerous works, including monographs (see, e.g., [1,2,8,13,14,). Therefore, in this review, ...