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536.12With the use of the Laguerre and Hankel integral transforms, the solution of a two-dimensional initial-boundary-value heat conduction problem for a two-layer slab under mixed boundary conditions is constructed: one of the surfaces is heated by a heat fl ux distributed axisymmetrically in a circle of radius R and is cooled by the Newton law outside this circle. The solution of the problem is reduced to a sequence of infi nite quasi-regular systems of algebraic equations. The results of numerical analysis of the temperature fi eld in the two-layer slab made from an aluminum alloy and ceramics are presented depending on the relative geometric properties of the components and cooling intensity. Keywords: nonstationary heat conduction, mixed boundary conditions, inhomogeneous body, Laguerre polynomials, paired integral equations.Many areas of current thermal-power engineering encounter problems associated with the creation and use of materials exerting a high resistance to oxidation, corrosion, and thermal shock, which requires a combination of various properties in certain inhomogeneity of their structure [1]. The simplest of these are bodies with coatings typical of which is a characteristic sharp jumplike change in the physicomechanical properties with retained continuity of distribution of physical fi elds [1,2].Another important aspect of engineering investigations in thermal-power engineering is the effort to carry out a more accurate mathematical simulation of real technological processes proceeding, in particular, in coated bodies too. Such modeling often necessitates account for so-called mixed heating conditions allowing for simultaneous heating of the surface over some, often limited, region and cooling outside it.The determination and investigation of nonstationary temperature fi elds in inhomogeneous bodies are the concern of a sizable number of scientifi c publications the review of which can be found, for example, in [3]. It should be noted, however, that in the case of mixed boundary-value conditions, the use of the integral Laplace transform for the solution of such problems leads to basic mathematical diffi culties. In particular, usually one does not succeed in obtaining accurate solutions of the problems that lead to paired integral equations [4]. At the same time, the joint numerical transform of the integral Laplace transform and of the integral (Fourier, Hankel, et al.) transforms with respect to the spatial variable can not only infl uence substantially the accuracy of the results obtained but can also distort the qualitative picture of the phenomenon studied. In work [4], a scheme is suggested for obtaining the solution of a nonstationary mixed problem of heat conduction for a homogeneous half-space based on the use of the method of successive approximations. The work also suggests a synthesis of the results obtained by the potential theory methods. All of the indicated solutions were obtained for infi nite or semi-infi nite homogeneous objects. In work [5], the mixed problem of nonstat...