Аннотация
The article consider inhomogeneous and non-linear heat problems by applying the method ofpartial discretization of nonlinear differential equations, derived by Professor A. N.Tyurehodzhayev and methods of mathematical physics connected with the use of integralLaplace transforms. The aim of work is to obtain analytical solutions of boundary-valueproblems of inhomogeneous and nonlinear heat conduction by applying the method of partialdiscretization of nonlinear differential equations, establishing of regularity of heat distributionin the layer, which describe the differential equations in partial derivatives of parabolic typewith variable mechanical and thermal characteristics, in some cases dependent on theunknown function itself. This paper addresses the following objectives: 1) Inhomogeneousproblem 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 ofheat capacity, density and heat conduction coefficient, which depends on the unknownfunction itself.In regards to the problems of heat-conduction fundamental works are those of A.V. Lykov [4-5], L. M. Belyaev and A. A. Ryadno [6-7], V. S. Zarubin [8]. Among foreign authors, who havebeen 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 devotedto the nonlinear problems of heat-conduction and methods of their solving. Application of themethod 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 ofheat conduction with almost random variables and nonlinear thermal characteristics in thelayer using the method of partial discretization of nonlinear differential equations of ProfessorA. Tyurehodzhaev by two variables, along with the integral Laplace transform.