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The article consider inhomogeneous and non-linear heat problems by applying the method of partial discretization of nonlinear differential equations, derived by Professor A. N. Tyurehodzhayev and methods of mathematical physics connected with the use of integral Laplace transforms. The aim of work is to obtain analytical solutions of boundary-value problems of inhomogeneous and nonlinear heat conduction by applying the method of partial discretization of nonlinear differential equations, establishing of regularity of heat distribution in the layer, which describe the differential equations in partial derivatives of parabolic type with variable mechanical and thermal characteristics, in some cases dependent on the unknown function itself. This paper addresses the following objectives: 1) Inhomogeneous problem of heat conduction theory with different dependences of heat conduction coefficient, heat capacity and medium density. 2) Non-linear problem of heat conduction with variable of heat capacity, density and heat conduction coefficient, which depends on the unknown function itself. In regards to the problems of heat-conduction fundamental works are those of A.V. Lykov [45], L. M. Belyaev and A. A. Ryadno [6-7], V. S. Zarubin [8]. Among foreign authors, who have been solving the problem of this kind, we note the work of G. Carslaw and D. Jaeger [9], L.A. Kozdoba [10-11], and other heat-conduction investigators. Work of L. I. Kudryashev and N. L. Menshih [12], a series of articles [13-15], etc. are devoted to the nonlinear problems of heat-conduction and methods of their solving. Application of the method of local potential in the heat conduction problems is described in the works of P. Glansdorff and I. Prigogine [16] R. Schechter [17]. In this article for the first time three were obtained analytical solutions of new problems of heat conduction with almost random variables and nonlinear thermal characteristics in the layer using the method of partial discretization of nonlinear differential equations of Professor A. Tyurehodzhaev by two variables, along with the integral Laplace transform.

ANALYTIC SOLUTIONS OF INHOMOGENEOUS AND NONLINEAR PROBLEMS OF HEAT CONDUCTION THEORY FOR A LAYER