We propose a technique for solving nonstationary heat-conduction problems for thermosensitive bodies with simple nonlinearity (the coefficients of thermal conductivity and the heat capacity per unit volume depend on temperature, but the coefficient of thermal diffusivity is constanO heated by convective heat exchange from the surrounding medium.Nonlinear heat-conduction problems for thermosensitive structural elements made of materials whose coefficient of thermal conductivity ~,, and heat capacity per unit volume c, depend on temperature in such a way that their ratio, the coefficient of thermal diffusivity a, is constant, can be completely linearized using the Kirchhoffvariable [lO] in the case when the temperature or heat flux is prescribed on the surface. We consider a method of solving nonstationary problems when convective heat exchange occurs across the entire surface or part of it and the Kirchhoff variable linearizes the heat-conduction problem only partially.Suppose a thermosensitive body is heated by sources arbitrarily distributed over its volume with density W(x, y, z, "0, and by the surrounding medium, from which heat exchange across the boundary S of the body occurs according to Newton's law. Then the temperature of the body is determined from the heat equation Here t,. is the temperature of the medium surrounding the surface S, or, is the coefficient of thermal emission from this surface, t i is the initial temperature of the body, and n is the exterior normal to the surface.We introduce the notation T = t/t o , and we write the coefficient of thermal conductivity and the heat capacity per unit volume of the material of the body in the form L, (t) = ~o L; (T), c,(t) o . = c, c, (T). Here t o isthe base temperature, Z~ and c o are quantities having dimensions, and the superscript 9 denotes the corresponding dimensionless quantities F = ~,;(T)dT.(3) ri We reduce the problem (1), (2) to the forms.where T(0) is the temperature expressed in terms of the Kirchhoff variable (3). Hence, the original problem has been partially linearized: the heat-exchange condition (5) contains the nonlinear expression for the temperature T(0) on the surface S. Let us assume that the value of the temperature

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We propose a method of solving stationary heat-conduction problems of contacting bodies with coeJficient of thermal conductivity that are linear functions of the temperature and the corresponding problems of thermoelasticity based on the method of perturbations. We give a numerical analysis of the thermal stresses in a two-layer tuba.

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We illustrate methods of constructing analytic-numerical solutions of nonsteady heat-conduction problems for thermosensitive bodies under convective heat transfer, and also two-dimensional steady-state heat-conduction problems for piecewise-homogeneous bodies.As a rule, numerical methods are used to study temperature fields in homogeneous or piecewise-homogeneous bodies with heat transfer whose thermophysical characteristics depend on the temperature (thermosensitive bodies). However, analytic solutions of such problems are needed for qualitative analysis of the thermal state of the body, and also to determine its therm0stregsed state, that is, to solve the corresponding problems of thermoelasticity.Nonsteady heat-conducuon problems for homogeneous thermoseusitive bodies in cases when the temperature or heat flux is prescribed on the boundaries and the coefficient of thermal diffusivity has insignificant dependence on the temperature and can be assumed constant, can be completely linearized using the Kirchhoff variable. The same method can be used to linearize completely heat-conduction problems for nonmetallic crystals, whose thermophysical characteristics are proportional to the cube of the absolute temperature under radiative heat transfer. In other cases of radiative, convective, or mixed heat transfer the introduction of the Kirchhoff variable linearizes the heat-conduction problem only partially. Conditions of thermal contact in the solution of heatconduction problems for piecewise-homogeneous bodies likewise can be linearized only partially using the Kirchhoff variable. In all these cases an additional linearization of the heat exchange, contact, or both must be performed in order to find the solution of the problem Yu. M. Kolyano [4] has proposed approximating nonlinear expressions for the temperature by piecewise constant ones in conditions of convective heat transfer or on a contact surface in the partially linearized problem, while the first author and G. Yu. Garmatii [7] have proposed a piecewise linear function with subsequent determination of the approximation parameters by the method of collocations. However, such methods can be applied only when the expression being approximated is a function of a single variable, that is, for solving onedimensional nonsteady and two-dimensional steady-state problems.We here propose an approach that makes it possible to construct an analytic-numerical solution of nonsteady heat-conduction problems for thermosensitive bodies with heat transfer of arbitrary dimension, and also for twodimensional steady-state problems of heat conduction in piecewise-homogeneous bodies, which are convenient for analyzing thermal regimes.We shall illustrate the method of solving nonsteady heat-conduction problems for thermosensitive bodies with heat transfer of arbitrary dimension using the example of the nonsteady heat-conduction problem for a layer 0 _< z _< 8, being heated by a point heat source with power Q(x) that varies over time and moves along an arbitrary curve in the plane z ...