ABSTRACT
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 [4-
5], 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.