Speciality:
01.02.04 (Mechanics of deformable solids)
Keywords: mathematical simulation; theory of elasticity; theory of plasticity; theory of functions several complex variable; three-dimensional boundary tasks; thermal-elasticity; the theory of elasticity Cosserat; equations mathematical physics; tasks of mechanics of deformable solids in selfadjoint operator; concept of a stochastic continuum; numerical methods; technological tasks of the theory of plasticity; method local functionals; the quantum theory social-historical processes; simulation of social-humanitarian systems and processes.
Subject:
The method of the decision and exploration of three-dimensional tasks of elasticity, founded on consideration of three-dimensional body as four-dimensional area section by a coordinate hyperplane, is offered. It allows to write down "the equations of an equilibrium a Lame' in the shape four the differential equations. Adding to these equations of a condition independence of required functions from fourth coordinate results to decomposition of system on three equations of the three-dimensional theory of elasticity and the equation of the Laplace for the fourth component. Four-dimensional spaces allows to enter in space of coordinates and displacement two-dimensional complex structure. Complex-valued displacement are searched in the shape of holomorphic breaking-up, i.e. as serieses on degrees complex variable with antiholomorphic factors and on to degrees conjugate complex variable with holomorphic in factors. Substitution of these breaking-ups in the complex equations "The theories of elasticity" and complex conditions of independence from fourth coordinates results in system catching on differential equations concerning holomorphic and antiholomorphic functions. Use breaking-ups in serieses or integrated introducings of these functions allows to put and to solve different boundary tasks. Use biholomorphic maps and holomorphic breaking-ups on systems of functions distinct from degrees complex variable, use of features analytical prolongations of holomorphic functions several complex variable, and also spreading of this method on other models of body, switching and the nonlinear models, are flowing and nearest scientific interests. Use for the same tasks others algebraic structures and their link with used two-dimensional complex structure is interest also.
Main publications:
A. I. Aleksandrovich. Primenenie teorii funktsii dvukh kompleksnykh peremennykh k resheniyu prostranstvennykh zadach teorii uprugosti // Izvestiya AN SSSR. MTT, #2, 1977.
A. I. Aleksandrovich, A. Yu. Rodionov. Issledovanie anizotropnykh i termouprugikh zadach metodami kompleksnogo analiza // Sbornik MOIP. Voprosy mekhaniki tverdogo i deformiruemogo tela. M., "Nauka", 1987.
A. I. Aleksandrovich, P. A. Kuvshinov, D. F. Titorenko. Reshenie uravnenii trekhmernoi teorii uprugosti metodom golomorfnogo razlozheniya kompleksnykh peremeschenii po stepennym funktsiyam i funktsiyam Besselya. Izvestiya RAN. MTT. 2001. # 2. S. 31–41.
A. I. Aleksandrovich, A. K. Kornoukhov, F. Popielas. Matematicheskoe modelirovanie protsessa osesimmetrichnoi shtampovki koltsevoi plastiny s tonkim rezinopodobnym pokrytiem // Izdatelstvo RAN. Problemy mashinostroeniya i nadezhnosti mashin. # 4, 1998. S. 61–68.
A. I. Aleksandrovich. Kontseptsiya kvantovogo opisaniya evolyutsii sotsialno-ekonomicheskikh sistem // Sbornik "Matematicheskoe modelirovanie istoricheskikh protsessov". M., 1996.
