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Information × Registration Number 0210U000581, 0107U001021 , R & D reports Title The analitical methods of integrations and investigation of nonlinear mathematical models of applied problems popup.stage_title Head Tychynin Valentyn Anatol'evich, Registration Date 25-02-2010 Organization Pridneprovsk State Academy of Civil Engineering fnd Architecture popup.description2 The nonlinear d'Alembert and Liouville equations, three-dimensional nonlinear equation Slid, the equation of a convection-diffusion and a nonlinear thermal conduction and some systems of such equations are investigated. For them nonlocal symmetries are investigated, formulae of generating of solutions are constructed. Nonlinear superposition principles for equations which are reduced to the linear one are discovered. For study of nonlocal symmetries of DEs nonlocal transformations of variables (NTV) which connect two defined equations have been used. Discovered NTVhave been used for construction of corresponding formulas of generating of solutions both in case of nonlocal invariance of the equation, and in that case, when these equations different. If one of the equations - linear (in)homogeneous, nonlocal transformation allows to construct formulas of nonlinear nonlocal superposition of solutions of other (nonlinear) equation. The described approach to study of nonlocal symmetries DEs of hyperbolic type has allowed to receive expected outcomes: to construct B?klund transformation (BT)and autot B?klund ransformation (ABT) (including conditional) for the equation of d'Alembert in two-and three-dimensional cases, to receive formulae of generating of solutions and superposition of solutions of these equations (including on a certain subset). For Liouville and Slid equations have been discovered NTV which linearize them. It has allowed to receive formulae of generating of solutions and superposition of solutions, anzatze Lie and nonlocal anzatze. Examples of generating and superposition of solutions to the studied equations are reduced. For parabolic type DEs it was possible to describe wide classes of the one-dimensional equations of a nonlinear thermal conduction and a convection-diffusion-reaction, invariant with respect to defined NTV and also the equations which are reduced to the linear. Formulas of generating of solutions and nonlinear superposition are constructed. Cases when one of the equations has additional (exrta) symmetries were considered too. Such symmetries were used for construction and generating of solutions to the investigated equation. Further these outcomes have been generalized on a class of systems of the one-dimensional equations of a nonlinear thermal conduction and a convection-diffusion with two unknown functions for which generating of solutions formulas and nonlinear superposition of solutions were received. (These outcomes were not expected to be received within the given work. But they are natural development and generalization of previous.) In the fulfilled work BT and ABT (including conditional) to the many-dimensional linear heat equation, the formulae of generating of solutions and nonlinear superposition to the many-dimensional equations of a nonlinear thermal conduction, in particular, Burgers equations also were constructed. Formulas of generating of solutions, anzatze Lie and nonlocal anzatze were constructed. In most cases solutions of a many-dimensional problem required performance of additional conditions, that is research of conditional nonlocal symmetries of named above equations. The received outcomes as a whole answer the technical project stated in inquiry. 2) The law of temperature change in each stratum in time as a first approximation is defined. It is considered, that the temperature of each stratum is the same for all points its thickness and is different for each stratum. The equation of temperature changing is obtained in assumption that heat comes through the upper and lower edges by a gas convection. The differential equation of cross-section nonlinear oscillations of pressed double plate is derived . The solution of the equation by means of a method of a variable scale is fulfilled. The quasistatic regime of transverse moving of the double plate when the plate heats up in a gas stream is considered. The behavior of restoring force is investigated at heating and cooling of the plate, influence of an eccentricity of application of contracting force, rigidity of elastic fastening, preliminary longitudinal compression, influence of an elastic reinforcement are investigated. The program of an evaluation of restoring force formulas by means of the COMPUTER is created. The quasistatic moving regime of double plate which consists of a stratum of a brass and invar is considered. It is revealed, that at certain parameters of system it jumps over at heating and makes inverse jump over during cooling. This effect can be used for creation of a thermal engine based on a new principle of work. Conditions with which should satisfy parameters of system for a possibility of its work as an element of a thermal engine are defined. The problem of construction of exact solution to equation of heat distribution in a cross-section direction of the double plate which is under the influence of hot gas is fulfilled. The analytical solution in the generalized Fourier series is received. The program of evaluation of expansion terms by means of the COMPUTER is made. The example of distribution of temperature in double plate completed of porcelain and magnesium stratums is considered. Product Description popup.authors Баєв С.В. Петрова О.В. Тичинін В.А. popup.nrat_date 2020-04-02 Close