Research of mathematical model of inhomogeneous media the heating process

In the paper mathematical model of inhomogeneous medium "TEN - sand - air" heating is investigated. This model is used in engineering problems to calculate the temperature and thermal characteristics during heating. The methodology of these calculations was developed in works of academician A.N. Tikhonov and A.A. Samarskiy. The considered mathematical model is an initial-boundary value problem for heat equation on a finite interval. Our problem, in contrast to the classical problems, includes three unknowns: one unknown function of two variables in the equation and two unknown functions of a single variable in the boundary conditions. The solution of initial-boundary value problem is found in the form of series of functions. These series are constructed by solving of the corresponding boundary value Sturm - Liouville problem in the Kneser's form. It is proved that the series of functions constructed in this way determines a unique classical solution of the initial-boundary value problem. Uniqueness of solution is proved by energy inequalities method.