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.

Keywords

the mathematical model of heat inhomogeneous medium; solution of initial-boundary value problem; method of energy inequalities.

References

1. Samarskiy А.А. [On a Problem of Heat Propagation]. Vestnik MGU [Bulletin of the MSU], 1947, no. 3, pp. 85-102. (in Russian)
2. Telkov M.G., Naimov A.N. [The Algorithm for Calculating the Regular Temperature Mode in During Heating of an Inhomogeneous Medium]. Materialy 6-oy mezhdunarodnoy nauchno-tekhnicheskoy konferentsii INFOS [Proceedings of the 6th International Scientific and Technical Conference INFOS], Vologda, 2011, pp. 193-197. (in Russian)
3. Telkov M.G., Timoshenko P.O., Sukhanov I.A., Naimov A.N., Sinitsyn A.A. [Investigation of the Temperature Mode in During the Heating of an Inhomogeneous Medium 'TEN-sand-to-air']. Fundamentalnye issledovaniya [Fundamental Researches], 2012, no. 11-12, pp. 458-462. (in Russian)
4. Telkov M.G., Naimov A.N. [The Existence and Completeness of Eigenvectors of the Problem Sturm - Liouville in the Kneser's Form]. Materialy vserossiyskoy nauchno-tekhnicheskoy konferentsii 'Vuzovskaya nauka - regionu' [All-Russian Scientific and Technical Conference 'University Science - the Region'], Vologda, 2012, vol. 1, pp. 177-181. (in Russian)
5. Naimov A.N., Sinitsyn A.A. [On the Eigenvalues of a Boundary Value Problem Sturm - Liouville]. Materialy vserossiyskoy nauchno-tekhnicheskoy konferentsii 'Vuzovskaya nauka - regionu' [All-Russian Scientific and Technical Conference 'University Science - the Region'], Vologda, 2014, vol. 1, pp. 152-158. (in Russian)
6. Oleynik O.A. Lektsii ob uravneniyakh s chastnymi proizvodnymi [Lectures on Partial Differential Equations], Moscow, Binom, 2005. 260 p.