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Русская версия

**Natalya Grigorievna Bilchenko**

Candidate of Science in Physics and Mathematics

Researcher of Research Laboratory #16,

Department of Heat Engineering and Power Engineering Machinery,

Kazan National Research Technical University (KNRTU-KAI)

named after A. N. Tupolev.

The problems of mathematical modeling of optimal heat protection of permeable surfaces of hypersonic aircraft in non-equilibrium dissociating gas are considered. The group theory approach to the optimization of systems with distributed parameters is applied. Conservation laws on all operators of the group admissible by a system of nonlinear differential equations of parabolic type are constructed.

- conservation law
- first integral
- heat and mass transfer
- hypersonic flows
- infinitesimal operator
- laminar boundary layer
- Lie group
- non-equilibrium dissociation
- optimal control

- Bilchenko N. G., Garaev K. G., Derbenev S. A. " On the problem of optimal control of electroconductive liquid laminar boundary layer in magnetic field, " Russian Aeronautics (Izvestiya VUZ Aviatsionnaya Tekhnika), 1994, Vol. 37, no. 1, pp. 23-27
- Bilchenko N. G., Garaev K. G. " On optimum control of laminar boundary layer of electroconductive gas by supersonic flow conditions, " Proceedings of 12th NATIONAL HEAT TRANSFER CONFERENCE (UIT), L'Aquila, Italy, 1994, pp. 213-224
- Bilchenko N. G. " Computation experiments in the problems of heat and mass transfer optimal control on permeable surfaces in hypersonic flows, " (in Russian), Proceedings of Voronezh State University. Series: Physics. Mathematics, 2015, no. 1, pp. 83-94
- Bilchenko N. G. " Computation experiments in the problems of heat and mass transfer optimal control on permeable spherical surfaces in hypersonic flows, " (in Russian), Proceedings of Voronezh State University. Series: System Analysis and Information Technologies, 2015, no. 1, pp. 5-8
- Bilchenko N. G. " Computation experiments in the problems of heat and mass transfer optimal control on permeable surfaces in hypersonic flows: comparative analysis of " simple" blow laws use, " (in Russian), Proceedings of Voronezh State University. Series: Physics. Mathematics, 2015, no. 1, pp. 95-102
- Bilchenko N. G. " A. A. Dorodnitsyn's method in the problems of permeable surfaces in electroconductive gas laminar boundary layer heat and mass transfer optimal control, " (in Russian), Proceedings of Voronezh State University. Series: System Analysis and Information Technologies, 2016, no. 1, pp. 5-14
- Bilchenko N. G. " Computational experiments in the problems of heat and mass transfer optimal control on permeable surfaces in electro-conductive gas laminar boundary layer, " (in Russian), Proceedings of Voronezh State University. Series: System Analysis and Information Technologies, 2016, no. 3, pp. 5-11
- Sirazetdinov T. K. " Optimization of Distributed Parameter Systems, " (in Russian). Nauka Publishers, Moscow, 1977, 478 p
- Bilchenko N. G., Garaev K. G. " On the existence of the first integral in an optimal problem with distributed parameters, " Russian Mathematics (Izvestiya VUZ. Matematika), 1993, Vol. 37, no. 12, pp. 30-33
- Garaev K. G. " On a corollary of the Noether theorem for the two-dimensional problem of the Mayer type, " Journal of Applied Mathematics and Mechanics, 1980, Vol. 44, no. 3, pp. 316-320
- Lie S., Scheffers G. " Vorlesungen ü ber Differentialgleichungen mit bekannten infinitesimalen Transformationsgruppen. " Leipzig, Teubner, 1891, 568 s
- Ovsyannikov L. V. " Group Analysis of Differential Equations. " Academic Press, N. Y., 1982, 416 p
- Noether E. " Invariante Variationsprobleme, " Nachrichten von der Kö niglichen Gesellschaft der Wissenschaften zu Gö ttingen, Mathematisch-Physikalische Klasse, Heft 2, 1918, ss. 235-257
- Garaev K. G. " Optimal control of heat and mass transfer in a laminar compressible-gas boundary layer on permeable surfaces, " Fluid Dynamics, 1988, Vol. 23, no. 3, pp. 399-406
- Bilchenko N. G., Ovchinnikov V. A. " On a Lie group that admits a nonlinear system of partial differential equations of parabolic type, " Russian Mathematics (Izvestiya VUZ. Matematika), 1997, Vol. 41, no. 1, pp. 66-71
- Bilchenko N. G. " On the Conservation Laws in the Problem of Electroconductive Gas Laminar Boundary Layer Optimal Control, " (in Russian), Differential Equations and Control Processes, 2018, no. 1, pp. 77-91. [ http://www.math.spbu.ru/diffjournal/ ]
- Bilchenko N. G., Garaev K. G., Ovchinnikov V. A. " Symmetry group and conservation laws in the problem of the optimal control of a laminar boundary layer of an unbalanced-dissociating gas, " Russian Mathematics (Izvestiya VUZ. Matematika), 1999, Vol. 43, no. 11, pp. 9-17
- Dorrance W. H. " Viscous hypersonic flow. " New York, McGraw-Hill, 1962, X + 334 p
- Belov S. V. " Porous metals in engineering. " (in Russian), [ Poristye metally v mashinostroenii ]. Moscow: Mashinostroenie, 1981, 247 p
- Dorodnitsyn A. A. " Laminar Boundary Layer, " (in Russian), [ Laminarnyj pogranichnyj sloj v szhimaemom gaze ]. Sb. Teoret. rabot po aerodinamike, Moscow: TsAGI, Oborongiz, 1957, pp. 140-173
- "Theory of optimum aerodynamic shapes", ed. by A. Miele. New York, London, Academic press, 1965, 478 p